Contemporary Geometry and Related Topics: Proceedings of the Workshop Belgrade, Yugoslavia 15 - 21 May 2002 ( World Scientific )

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Proceedings of the Workshop

CONTEMPORARY GEOMETRY AND RELATEDTOPICS

This page intentionally left blank

Proceedings of the Workshop

CONTEMPORARY GEOMETRY AND RELATED TOPICS BeI g rad e , Y ug oslavia

15 - 21 May 2002

Editors

Neda Bokan University of Belgrade, Yugoslavia

Mirjana DjoriC University of Belgrade, Yugoslavia

Anatoly T. Fomenko Moscow State University, Russia

Zoran RakiC University of Belgrade, Yugoslavia

Julius Wess Universitat Munchen, Germany

vp World Scientific N E W JERSEY

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CONTEMPORARY GEOMETRY AND RELATED TOPICS Copyright 0 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Preface The Workshop Contemporary Geometry and Related Topics, which took place in Belgrade from May 15th t o May 21th, 2002, was organized by the Faculty of Mathematics of the Belgrade University and the Mathematical Institute of SANU (Serbian Academy of Science and Arts). The organizers were motivated and inspired in their work by lengthy and excellent tradition of collaboration between colleagues from the Belgrade University and the Moscow State University, especially those from the Chair of Differential Geometry and Applications, as well as the wonderful opportunity to be the host to several guests with the similar field of their research. Although the beginning was almost spontaneous, the final results were such that we felt an urge to publish the Proceedings. There were 60 participants from 12 countries: Belorussia, Bulgaria, Czech Republic, China, France, Germany, Japan, Italy, FYR Macedonia, Russia, Spain and Yugoslavia. Scientific activities were only in one section with lectures of 45 min and 30 min, short communications of 15 min and a poster section in order to give opportunity to the participants to follow all activities without pressure and t o obtain new information. Although at first some topics seemed diverse, the participants have recognized as very important a possible connection of their research to reach the interdisciplinary one. Therefore, in the similar spirit we have organized the content of the Proceedings in the lexicographic order. The success of a meeting depends on various facts. That is why we would like to express our gratitude to all of those who have contributed to the success of our Workshop and its Proceedings. First of all, we feel deep appreciation for all the participants, whose style of communicating with the organizers and quality of lectures have motivated us to implement our plan with pleasure. As the financial support is always very important, we would like to thank all the institutions who helped the organization of this Workshop. The complete list of sponsors is enclosed. For their support and encouragement we are grateful to Prof. Z. Kadelburg, Prof. A. Lipkovski and Prof. D. RadunoviC, dean and vice-deans of the Faculty of Mathematics in Belgrade, respectively. Referees played a significant role by reading the manuscripts carefully and making very precise and useful comments, that have contributed to the improvement of the final version of all the papers. The cover page is based on the logo of the Woikshop. The logo is made by software L2Primitives written by I. KnezeviC, R. SazdanoviC and dr V

vi

S. VukmiroviC (available at www.mathsource.com). We are indebted t o our colleagues M. AntiC, 2. StaniC and dr S. VukmiroviC, members of the Organizing Committee, who helped us very much, not only in the Workshop but also in preparing the Proceedings. Finally, we are especially indebted t o World Scientific for providing us with the opportunity to publish the Proceedings. The Proceedings contains articles which were received by the Editorial board till March 31st, 2003, and passed the usual referring process. Some articles which had been scheduled to be communicated during the Workshop, but were not, due to various reasons, are also included. The Editors

List of sponsors Ministry of Science, Technology and Development, Republic of Serbia Ministry of Education and Sport, Republic of Serbia Faculty of Mathematics, Belgrade Matematical Institute of SANU, Belgrade Rectorate of University in Kosovska Mitrovica Megatrend University, Belgrade Yugoslav Chamber of Commerce and Industry, Belgrade Prva preduzetnitka banka, Beograd National Theater, Belgrade Tourist Organization of Belgrade, Belgrade C E T , Belgrade Minel, Beograd Sinex, Beograd Sekvenca, Beograd Teledan, Beograd Project MM1646, Geometry, Education and Visualization with Applications, of MNTR

Contents V. V . Balashchenko Invariant Structures Generated by Lie Group Automorphisms on Homogeneous Spaces ..................................................

M. B. Banam On Kenmotsu Hypersurfaces in a Six-Dimensional Hermitian Submanifold of Cayley Algebra.. ....................................

1

.33

N. B1a.M and S. VukmiroviC Para-hypercomplex Structures on a Four-Dimensional Lie Group. . . . . . 4 1

A . V. Bolsinov and B. JovanoviC Integrable Geodesic Flows on Riemannian Manifolds: Construction and Obstructions ......................................

57

M. DjoriC and M. Okumura CR Submanifolds of Maximal CR Dimension in Complex Space Forms and Second Fundamental Form.. .............................

105

M. Carmen Domingo-Juan and V. Miquel Pappus-Guldin’s Formulae versus Weyl’s Tube Formula: Old and Recent Results., .................................................... 117 B. Dragovich Non-Archimedean Geometry and Physics on Adelic Spaces . . . . . . . . . .141 B. Dragovich and Z. RakiC Lagrangian Aspects of Quantum Dynamics on a Noncommutative Space.. .............................................................

159

M. Ferncindez-Lbpez, E. Garcia-Rio and D. N . Kupeli The Mobius Equation: A Local Analytical Characterization of Twisted Product Structures .........................................

173

M. Ferna’ndez, V. Mufioz and J. A . Santisteban Symplectically Aspherical Manifolds with Nontrivial 7r2 and with No Kahler Metrics.. ................................................

187

A . T. Fomenko and P. V. Morozov Some New Results in Topological Classification of Integrable Systems in Rigid Body Dynamics ...................................

201

vii

...

Vlll

F. Gavarini The Crystal Duality Principle: From General Symmetries t o Geometrical Symmetries. ...........................................

.223

2.H u and H. Li Willmore Submanifolds in a Riemannian Manifold, ..................251

Dj.Kadajevich Some Aspects of Visualizing Geometric Knowledge: Possibilities, Findings, Further Research. .........................................

277

V. P. Karassiov Dual Algebraic Pairs and Polynomial Lie Algebras in Quantum Physics: Foundations and Geometric Aspects. .......................

285

E. Mdkowsky and V. VelatkoviC Visualisation and Animation in Differential Geometry . . . . . . . . . . . . . . .301 G. V. Nosovskiy Computer Gluing of 2D Projective Images.. ........................

.319

A . A . Oshemkov On the Topological Structure of the Set of Singularities of Integrable Hamiltonian Systems ................................

.335

Th. Yu. Popelensky and Yu. P. Solovjev Lie-Cartan Pairs and Characteristic Classes in Noncommutative Geometry ......................................................

.351

S. Terzic' On Rational Homotopy of Four-Manifolds

......................

. . . . .375

S. Trapani Dual Maps and Kobayashi Distance of Bounded Convex Domains .... .389 in CC" .......................................................... K. TrenEevski Osculating Spaces and Higher Order Curvature Tensors of Submanifolds in R". ...........................................

.407

Lj. S. VelimiroviC, S. M. Mantic' and M. S. Stankovic' On Commutativity of the Lie Derivative and Covariant Derivative a t a Non-Symmetric Affine Connection Space. ......................

.425

L. Vrancken Special Classes of Three Dimensional Affine Hyperspheres Characterized by Properties of Their Cubic Form.. ...................431

INVARIANT STRUCTURES GENERATED BY LIE GROUP AUTOMORPHISMS O N HOMOGENEOUS SPACES *

VITALY V. BALASHCHENKO Faculty of Mathematics a n d Mechanics Belarusian State University F . Scorina av. 4, Minsk 220080, BELARUS E-mail: [email protected]

We collect the recent results on homogeneous regular @-spaces, specifically, homogeneous k-symmetric spaces. The structure of the commutative algebra of all canonical affinor structures on any regular @-space is explicitly described. This partitions all regular +-spaces into a countable set of disjoint classes with respect t o their algebra of canonical affinor structures. We prove the compatibility of classical canonical structures with natural (pseudo)Riemannian metrics. In particular, this leads t o applying canonical f-structures t o the generalized Hermitian geometry. As an example, we introduce a concept of nearly Kahler f-structures naturally generalizing well-known nearly Kahler structures in Hermitian geometry. A remarkable collection of invariant nearly Kahler f-structures is presented. It is based on the canonical f-structures on homogeneous k-symmetric spaces. As a result, the analogy with homogeneous nearly Kahler manifolds is obtained. Finally, some general and particular examples are considered in the above sense.

1. Introduction An important place among homogeneous manifolds of Lie groups is occupied by the homogeneous @-spaces, that is, the homogeneous spaces generated by automorphisms @ of Lie groups (see Refs. [53], [65], [19], [ 401, [ 111). A special role is played by the regular @-spaces, which were first introduced in Ref. [ 531. As is known [ 531, such @-spacesinclude all k-symmetric spaces = id), are included, in their turn, into the wellknown class of homogeneous reductive spaces. At the same time, a distin-

(ak

* MSC2000: primary 53C15, 53C30; secondary 53C10, 53C35. Keywords :homogeneous space, regular @-space, k-symmetric space, invariant structure, canonical affinor structure, nearly Kahler structure, f-structure. t Partially supported by the Belarus State Program of Fundamental Research ”Mathematical structures”.

1

2

guishing feature of regular @-spaces is that each such space has a natural associated object, the commutative algebra d ( 0 ) [ 111 of canonical affinor structures. This algebra contains well-known classical structures, such as almost complex structures, almost product structures, f-structures of K. Yano (f3 f = 0), h-structures (h3 - h = 0) (see Refs. [ 111, [ 41). The first example in this direction was the canonical almost complex structure on homogeneous 3-symmetric spaces (see Refs. [ 541, [ 651, [ 231). It should be mentioned that the structure became an effective tool and a remarkable example in some deep constructions of differential geometry and global analysis such as homogeneous structures [57], [49], [36], [ 2 1 ] , [43], [ l ] , Einstein metrics [ 501, [ 511, holomorphic and minimal submanifolds [ 461, [ 471, real Killing spinors (see Refs. [ 271, [ 131). The main goal of the paper is to show a fundamental role of canonical structures in the theory of homogeneous regular @-spaces as well as its applications. The paper is organized as follows. In Sections 2 and 3, we collect some basic notions and results about homogeneous regular +-spaces and canonical affinor structures. In particular, a precise description of all canonical structures of classical types on homogeneous k-symmetric spaces is included. Besides, the exact formulae for these structures and the relationship between them on 4- and 5-symmetric spaces are presented. In Section 4, an algebraic structure of the algebra of all canonical structures for arbitrary regular @-space is completely described. This result leads to a natural partition of all regular @-spaces into a countable set of disjoint classes with respect to their algebra of canonical &nor structures (see Section 5). In Section 6, we introduce linear subspaces generated by the canonical reductive supplement m and the automorphism 'p of a Lie algebra g. These objects possess the remarkable property and are used for characterizing many geometric situations. In Section 7, we prove the compatibility of all canonical classical structures P, J , f , h with natural (pseud0)Riemannian metrics on regular @-spaces. A principal particular case is that all canonical structures mentioned are compatible with the naturally reductive metric for homogeneous k-symmetric spaces of semisimple Lie groups. Metric f-structures are included into a general concept of the generalized Hermitian geometry. Some basic classes of metric f-structures are introduced in Section 8. In particular, there are two recent generalizations of the well-known nearly Kahler manifolds.

+

3

In Section 9, invariant nearly Kahler f -structures and Killing f -structures on naturally reductive homogeneous spaces are considered. The algebraic criteria for these structures and relationship between them are presented. It turns out that canonical f -structures on homogeneous k-symmetric spaces provide a vast class of invariant NKf-structures. This is proved in Section 10. In particular, the canonical f-structures on naturally reductive 4- and 5-symmetric spaces are NKf-structures. Finally, in Section 11, we present a number of general and particular examples of homogeneous nearly Kahler f-manifolds. They are the spheres S2,S 5 , and S6 in their non-symmetric representations, the 6-dimensional generalized Heisenberg group, the flag manifold SU(3)/Tm,, and others.

2. Homogeneous regular +-spaces Here we briefly formulate some basic definitions and results related to regular @-spacesand canonical affinor structures on them. More detailed and some relevant information can be found in Refs. [ 111, [ 151, [ 191, [ 281, [391, [40l, [531-[551, [591-[651. Let G be a connected Lie group, @ its (analytic) automorphism. Denote by G' the subgroup of all fixed points of @ and G: the identity component of G'. Suppose a closed subgroup H of G satisfies the condition

G:

c H c G'.

Then G / H is called a homogeneous @-space. Homogeneous @-spaces include homogeneous symmetric spaces (a2 = id) and, more general, homogeneous @-spaces of order k (ak= id) or, in the other terminology, homogeneous k-symmetric spaces [ 40 1. For any homogeneous @-space G / H one can define the mapping

So = D : G / H

-

G/H, XH

-

@(x)H.

It is known [ 531 that Sois an analytic diffeomorphism of G / H . Sois usually called a "symmetry" of G / H at the point o = H . It is evident that in view of homogeneity the "symmetry" S, can be defined at any point p E G / H . More exactly, for any p = T ( Z ) O = zH,q = T ( Y ) O = yH we put

s, = T(Z) so 0

It is easy to show that

0 T(Z-').

4

Thus any homogeneous @-space is equipped with the set of symmetries {S, I p E G / H } . Moreover, each S, is an analytic diffeomorphism of the manifold G / H [ 53 1. Note that there exist homogeneous @-spacesthat are not reductive. That is why secalled regular @-spacesfirst introduced by N. A. Stepanov [ 531 are of fundamental importance. Let G / H be a homogeneous @-space, g and ti the corresponding Lie algebras for G and H , 'p = d a e the automorphism of g. Consider the linear operator A = 9 - i d and the Fitting decomposition g = go$g1 with respect to A , where go and g1 denote 0- and 1-component of the decomposition respectively. Further, let 'p = 'ps 'pu be the Jordan decomposition, where 'ps and 'pu is a semisimple and unipotent component of 'p respectively, 'pscpu = 'pu pS. Denote by gY a subspace of all fixed points for a linear endomorphism y in g. It is clear that = gv = K e r A, ti c go, ti c 8 9 s .

Definition 2.1. A homogeneous @-spaceG / H is called a regular @-space if one of the following equivalent conditions is satisfied: (1) ti = 00. (2) B = ti 63 (3) The restriction of the operator A to Ag is non-singular. ( 4 ) A 2 X = 0 ===+ A X = 0 for all X E g. (5) The matrix of the automorphism 'p can be represented in the form

(fi),

(6)

where the matrix B does not admit the eigenvalue 1.

= g'+'a.

Remark 2.1. The original definition by N. A. Stepanov [ 531 is just the condition (1) above. Equivalent conditions (2)-(5) one can find in Refs. [ 531, [55], [ 561, [ 621, [ 111. The defining property in the form (6) was proposed by the referee of the paper. The author thanks him for this and some other useful comments. We recall two important facts:

Theorem 2.1. ([53]) ( i l ) A n y homogeneous @-space of order k (ak = i d ) is a regular @space. (22) A n y regular @-space is reductive. More exactly, the Fitting decomposition g=t)$m, m=Ag

(2.1)

5

is a reductive one. Decomposition (2.1) is the canonical reductive decomposition corresponding to a regular @-spaceG I H , and m is the canonical reductive complement. We also note that for any regular @-spaceG I H each point p = X H E G I H is an isolated fixed point of the symmetry S, [ 531. Decomposition (2.1) is obviously pinvariant. Denote by 8 the restriction of ‘p t o m. As usual, we identify m with the tangent space T , ( G / H ) at the point o = H . It is important to note that the operator 8 commutes with any element of the linear isotropy group A d ( H ) [ 531. It also should be noted [ 531 that (dS,), = 8.

3. Canonical afFinor structures on regular +-spaces An a f i n o r structure on a manifold is known to be a field of endomorphisms acting on its tangent bundle. Suppose F is an invariant affinor structure on a homogeneous manifold G I H . Then F is completely determined by its value F, at,the point 0,where F, is invariant with respect to A d ( H ) . For simplicity, we will denote by the same manner both any invariant structure on G I H and its value at o throughout the rest of the paper.

Definition 3.1. ([ 121, [ 111) An invariant affinor structure F on a regular @-space G I H is called canonical if its value at the point o = H is a polynomial in 8. Denote by A(8) the set of all canonical affinor structures on a regular @space G / H . It is easy to see that d(8) is a commutative subalgebra of the algebra A of all invariant affinor structures on G I H . Moreover, dim A(8) = deg v 5 dim G / H , where v is a minimal polynomial of the operator 8. Note that the algebra A(8)for any symmetric @-space(@’ = i d ) consists of scalar structures only, i.e. it is isomorphic to R. It should be mentioned that all canonical structures are, in addition, invariant with respect to the ”symmetries” {S,} of G I H [53]. Moreover, from (dS,), = 8 it follows that the invariant affinor structure Fp = (dS,),, p E G I H generated by the symmetries {S,} belongs to the algebra d(8). The most remarkable example of canonical structures is the canonical almost complex structure J = (8 - 8’)/& on a homogeneous 3-symmetric

6

space (see Refs. [54], [65], [as]). It turns out that it is not an exception. In other words, the algebra A(0) contains many affinor structures of classical types. In the sequel we will concentrate on the following affinor structures of classical types: -

almost complex structures J ( J 2 = -1); almost product structures P ( P 2 = 1); f -structures (f 3 + f = 0 ) [ 661; f-structures of hyperbolic type or, briefly, h-structures (h3- h = 0 ) 1341.

Clearly, f -structures and h-structures are generalizations of structures J and P respectively. All the canonical structures of classical type on regular +-spaces were completely described in Refs. [ 121, [ 111, [ 41. In particular, for homogeneous k-symmetric spaces, precise computational formulae were indicated. For future reference we select here some results. Denote by S (respectively, s) the number of all irreducible factors (respectively, all irreducible quadratic factors) over JR of a minimal polynomial v.

Theorem 3.1. ([12], [ l l ] ,[ 4 ] ) Let G / H be a regular @-space. (1) The algebra A(0) contains precisely 2' structures P. (2) G / H admits a canonical structure J i f and only i f s = S. I n this case d(0) contains 2" different structures J . (3) G / H admits a canonical f-structure if and only if s # 0. In this case d ( 0 ) contains 3" - 1 different f-structures. Suppose s = S . T h e n 2" f -structures are almost complex and the remaining 3" - 2" - 1 have non-trivial kernels. (4) The algebra A(0) contains 3' different h-structures. All these structures form a (commutative) semigroup in d(0) and include a subgroup of order 2' of canonical structures P .

+

Further, let G / H be a homogeneous k-symmetric space. Then S = s 1 if -1 E Spec0, and S = s in the opposite case. We indicate explicit formulae enabling us to compute all canonical f -structures and h-structures. We shall also use the notation

u={

n n-1

if if

k=2n+1, k=2n.

7

Theorem 3.2. ([12], 1111, [ 4 ] ) Let G I H be a homogeneous @-space of order k. All non-trivial canonical f-structures o n G I H can be given by the operators

(g y)

2 "

f

=

Ic

Cj sin

m=l

( e m - ek--m

> 7

where <j E {-1; 0; l}, j = 1 , 2 , . . . ,u , and not all coeficients Cj are zero. Moreover, the polynomials f define canonical structures J iff all <j E {-1; 1). All canonical h-structures o n G I H can be given by the polynomials k- 1

h=

1 amem, where: ~

m=O

(a) i f k = 2 n

+ 1 , then

a , = ak-,

=

27rmj CE, cos -. k 1

2 "

k j= 1

(b) i f k = 2 n , then 27rmj

a, = ak-,,, = -

k

j=1

Here the numbers 0, j = 1 , 2 , .. . ,u, take their values from the set {-1; 0; 1) and the polynomials h define canonical structures P iff all .$j E {-I; 1). We now particularize the results above mentioned for homogeneous @spaces of orders 3, 4 , and 5 only. Note that there are no fundamental obstructions to considering of higher orders k (see also Ref. [ 651).

Corollary 3.1. ([12], 1111, [ 4 ] ) Let G I H be a homogeneous @-space of order 3. There are (up to sign) only the following canonical structures of classical type o n G I H : 1 J = - (e - e2), P = 1.

&

We note that the existence of the structure J and its properties are well known (see Refs. [ 541, [ 651, [ 231, [ 331).

8

Corollary 3.2. ([12], 1111, [4]) O n a homogeneous @-space of order 4 there are (up to sign) the following canonical classical structures:

P = g 2 , f = - 1( 9 - 0 ) ,3

h 1 = - 1( 1 - g 2 ) , h 2 = -1 (1+g2). 2 2 2 The operators hl and h2 form a pair of complementary projectors, ie.,: hl h2 = 1, hl = h l , h$ = h2. Moreover, the following conditions are equivalent:

+

4

(1) -1 spec 9; (2) the structure P is trivial ( P = -1); (3) the f -structure is an almost complex structure; (4) the structure hl is trivial (hl = 1); (5) the structure h2 is null.

General properties of the canonical structures P and f on homogeneous 4-symmetric spaces were investigated in Ref. [ l o ] .

Corollary 3.3. ([12], 1111, 141) There exist (up to sign) only the following canonical structures of classical type on any homogeneous @-space of order 5: 1 P = - (e - e2 - e3 e4); dfi J~ = (Y (6 - e4) - p ( e 2 - e3); J~ = p(e - e4) ff (e2- e3);

+

+

(e - e4) + 6 (e2- e3);

fl =

1

hi = 5 ( 1 where

=

fi =

s ( e - e4) - (e2- e3);

1 h2=-(1-P); 2

+P);

q; p e; y q; 6 C. =

=

=

10-2& 10

Besides, the following relations are satisfied: J1 P = J2; f1 P = J1 hl = J2 h1 = f i ; hl P = h i ; h2 P = -h2; f2P=J2h2=-Jlh2=-f2; f i f 2 = h l h 2 = 0 ; hl+h2=P.

In addition, the following conditions are equivalent: (1) spec 9 consists of two elements; (2) the structure P is trivial; (3) the structures 51 and J2 coincide (up to sign); (4) one of the structures f i and f 2 is null, while the other is an almost complex structure coinciding with one of the structures 51 and 52; (5) one of the structures hl and h2 is trivial, while the other is null.

9

We note that homogeneous 5-symmetric spaces and invariant structures on them were studied in Refs. [ 651, [ 91, [ 161-[ 181, [ 581, and in Refs. [ 151, [ 641, [ 651 for &/A4 x Ad in more detail. It should be also mentioned that in the particular case of homogeneous @-spacesof any odd order k = 2 n 1 the method of constructing invariant almost complex structures was described in Ref. [ 401. It can be easily seen that all these structures are canonical in the above sense.

+

4. The algebra A(@) for regular @-spaces Consider the commutative algebra A,(P) consisting of the matrices having the form tl z2

.. .

Z,-l

z1 ... -. .. .. . . . . . . . . 0 0 .. .. 0 0 ... 0

0

z,

zn:l .

z2

z1

where a11 elements r j , j = 1 , 2 , . . . ,n belong to the field P. The following theorem gives the description of the algebraic structure for the algebra d(0).

Theorem 4.1. (IS]) Let G / H be a regular @-space, v the unitary minimal polynomial of the operator 0, u = uyl uZnz . . . u$mu;”+:’ . . . urs its decomposition into unitary irreducible factors over the field R, where deg vj = 2 for j = 1,2,. . . ,m, deg uj = 1 f o r j = m 1,.. . ,s, and all polynomials v1,u2,. . . ,us are painoise mutually disjoint. Then, the algebra d(8)of canonical afinor structures of the space G / H i s isomorphic to the direct sum

+

of real commutative algebras.

Outline of the proof The algebra d ( 0 ) is isomorphic to the subalgebra R [ 61 in the algebra of endomorphisms of the linear space m generated by the elements of IR and 8. It means that R [ 0 ] = (h(6)lh E IR[z]}. Further, the algebra IR [ 81 is isomorphic to the quotient algebra R [ z]/(v), where v is the principal ideal determined by the polynomial v. Now, taking

10

into account the decomposition of v into irreducible factors, we obtain the well-known classical representation

=

R[a:]/(.) R [ a : ] / ( v ; '@...@R[a:]/(v,"s) ) (see, for example, Refs. [ 201 and [ 411). Therefore the problem reduces to an explicit description of the quotient algebra R [ a:]/(v">, where Y is a polynomial of degree 1 or 2 irreducible over the field R.

Lemma 4.1. If .(a:)

= (a:-a)",

where a

E

R, then R [ a:]/(u(z)) 3 A,@).

Indeed, any element r+ (v) of the algebra R [ a:]/(v(z)) is completely determined by a polynomial of the form .(a:) = z ~ + ~ ~ ( a : - a ) + ~ ~ ~ + ~ , ( a : - a ) " ~ ' where z1,. . . ,z , E R. It is directly verified that the natural mapping of R [ a:]/(v(a:)) to An(R) is an isomorphism of the algebras.

Lemma 4.2. If .(a:) [a:]/(.(.)) A?L(C).

=

(x2

+ pa: + q),,

where p 2 - 4q < 0 , then

To prove Lemma 4.2, we recall (see Ref. [ 11, 521) that IR [a:]/(v(x)) contains a unique (to a sign) element J ).( such that J = -l(mod .). For future convenience, set T ( z )= x2 pa: q. Then, we can specify a basis in R [ z]/(v(a:)) determined by the polynomials:

+ + +

el = 1, u1 = J, e2 = T , 212 = JT,

. . . , en = Tn-l,u,

= JT"-'.

As usual, denote by i the imaginary unit. Now, representing an arbitrary element of the given algebra by a polynomial of the form n T

= c ( a j ej

+

bj

where

uj)

u j , bj

E

R

for j = 1, . . . ,n,

j=1

-

we can show that the mapping

R[zl/(v(a:)) - A n ( C ) , r+(v) (zl,-*.l~n), = aj b j i for j = 1 , . . . , n, is an isomorphism of the algebras.

+

where zj This proves Lemma 4.2. Lemmas 4.1 and 4.2 obviously imply the assertion of Theorem 4.1.

0

Let us mention the most important special cases of Theorem 4.1.

--

Corollary 4.1. If G / H is a regular @-space, where @ is a semisimple automorphism of the Lie group G, then

d(B)3 C @ .. . @ C @IR m

@

.. . @ R

s-m

As mentioned, the best known class of regular +-spaces is that of homogeneous k-symmetric spaces (ak= id).

Corollary 4.2. Suppose that GIH is a homogeneous k-symmetric space. Let us denote the number of pairs of conjugate kth roots of unity in the spectrum of the operator 8 by m. Then,

m

A(8) E C @ . . - @ C

if - 1 #spec 8.

I ’ m

Remark 4.1. The isomorphism for the algebra d ( 8 ) constructed in the proof of Theorem 4.1 makes it possible to reconstruct any canonical structure on G I H from the element representing this structure in the direct sum of algebras specified above. This procedure is performed successively with the use of the classical Euclidean algorithm under the assumption that the decomposition of the minimal polynomial v into irreducible factors is known. Moreover, knowing the explicit form of the algebra A(8),we can determine all canonical structures on GIH of a prescribed type (classical or not). For example, the existence of canonical almost tangent structures of various orders (i.e., such that T’= 0 for 1 2 2) on many regular aspaces becomes obvious. In addition, we gain a new method for obtaining the known descriptions of the canonical structures of basic classical types (see Refs. [ 11I, (41). 5. The classes of regular +-spaces

In studying homogeneous manifolds, the spaces under consideration are traditionally divided into two classes, namely, homogeneous spaces of semisimple and solvable types. This division manifests itself especially distinctly in classification problems. As to semisimple regular +-spaces, the classification of 3-symmetric spaces (with simple basic groups) is well known (see Refs. [ 651, [ 231). A similar classification of 4symmetric spaces of compact simple groups is obtained in Ref. [ 291. Many authors considered classifications of arbitrary k-symmetric spaces in the semisimple case (see, e.g., Ref. [ 191); the solvability of such a problem in principle is explained in Ref. [28, Chapter l o ] . At the same time, the methods used in the semisimple case do not apply to the solvable case; for this reason, the regular @-spacescannot be classified

12

in this way. Note that there are many examples of k-symmetric spaces of the solvable type (see, e.g., Refs. [40], [ 5 7 ] ) . The mechanism for classifying all regular @-spacessuggested in this paper is based on assigning the algebra d(B) of canonical affinor structures to each regular @-space ( G / H ,a). This algebra reflects the special geometric properties of G / H that are determined by the automorphism a, and the correspondence between spaces and their algebras requires no additional constraints on the Lie group G. Theorem 4.1 proved above implies that the set of disjoint classes obtained is countable. Consider some simplest cases of algebras d ( 0 ) and the corresponding classes of regular @-spaces. (i) d ( 0 ) R. This means that only scalar structures are canonical; i.e., 0 is a homothetic transformation. In particular, all homogeneous symmetric spaces belong to this class (for these spaces, 8 = -1 is a symmetry in the tangent space T , ( G / H ) ) . (ii) d(0) C. This class includes, e.g., all homogeneous 3-symmetric spaces; the role of the imaginary unit in d(0)is played by the wellknown canonical almost complex structure J = (6' - @)/& (see Refs. [ 541, [ 651, [ 231) mentioned above. Clearly, any k-symmetric space with k 2 3 such that spec 8 comprises two elements also belongs to this class. (iii) d(8) G C @ R. All 4-symmetric spaces for which 0 has maximal spectrum belong to this class. Similarly, it contains all k-symmetric spaces ( k = 2n 2 4) with three-element spectra. Let us mention a number of important special features.

Remark 5.1. All classes of algebras d ( 8 ) are realizable. In other words, for a commutative algebra of the form specified in the statement of Theorem 4.1, a regular @-spacewhose algebra d(8)is isomorphic to the given algebra can be constructed. Remark 5.2. Comparing Corollary 4.2 with Theorem 4.1, we see that the set of regular @-spacesis substantially wider and richer geometrically (in the sense of canonical structures) than the best known subset of homogeneous k-symmetric spaces. Remark 5.3. Different regular @-spaces may have the same underlying structure of the homogeneous space G / H . In this case, they may fall into different classes in the partition suggested above. Examples of such

13

homogeneous spaces are given in Section 11. 6. Linear subspaces generated by 19

Let G/H be a reductive homogeneous space, g = 0 @ m the corresponding reductive decomposition, A, p linear endomorphisms on m, i.e., X , p E End(m). Denote by m(’+) a linear span of all the vectors [ X(X),p(X)],where X Ern. Now let G I H be a regular @-spacewith the canonical reductive decomposition (2.1). Consider linear operators X(8) and p(8), where X and p are polynomials with real coefficients. Let m(’9f’) be the corresponding linear subspace in g. Without loss of generality, we can assume that the polynomials X and p are unimodular and deg X

< deg p < deg Y ,

where Y is the unimodular minimal polynomial of 8. Obviously, any m(>>fi) is a pinvariant subspace in g. Since in the symmetric case (i.e. (P2 = i d ) 8 = -id it follows immediately that all spaces rn(’>f’) are trivial. A subspace of type m(’+) was first introduced in Ref. [ 21. It was the case X(8) = 1, p(8) = 8, and the subspace was denoted by m“. It should be noted that the condition mq C lj is in a way a generalization of the local symmetry condition [ m , m ] C t, for a reductive homogeneous space G / H . For example, for any homogeneous @-spaceof order 3 the condition m“ c lj is valid (see also Theorem 6.2 and Corollary 6.1 below). Other spaces of type m(x+) will appear in Section 9. Now we prove the following important fact.

Theorem 6.1. Let g = t, @ m be the canonical reductive decomposition of a regular @-space G I H . For a n y X(8) and p(8) mf’ll) is a n ideal in g.

Proff. for a Proff. for a

14

Proff. for a Proff. for a Proff. for a Proff. for a Remark 6.1. It is known [ 451 that for any reductive homogeneous space G / H with the reductive decomposition g = IJ @ m the subspace [m,m]h generated by all the vectors [ X, Y I,, where X, Y E m, is an ideal in IJ. It is evident that for any regular @-spacean ideal mr”‘)belongs to the ideal [m,m 1 h In conclusion, we consider some particular case. More precisely, we prove the following

Theorem 6.2. Let G I H be a regular @-space such that the minimal polynomial v of the operator 6’ as quadratic: v(x) = x2 + p x + q. Then any subspace m(A+) coincides with my.

15

I n addition, suppose that the polynomial v i s irreducible and the subspace mp i s non-trivial. T h e n the following conditions are equivalent:

( i ) m'+' c b; (ii) q = 1 (iii) spec O = {E,.F}, where

I E ~ = \El = 1.

Proof. The first assertion is obvious because of degX (i)

< degp < degv = 2.

+ p O + q = 0. For any X E m it follows v[x,ex] = [ O X , O ~= X [] e x , ( - p e - q ) x ] = q [ x , ~ x ] .

* (ii)

By assumption O2

Hence [ X IO X ] E

*

(ii) (iii) conjugate roots is evident.

E

b if and only if

q = 1.

Suppose q = 1. Since p is real, we see that complex and of v belong to the unit circle S1 c C. The inverse 0

Note that Riemannian homogeneous @-spaceswith an irreducible quadratic polynomial v (so-called quadratic s-manifolds) were investigated in Ref.

[421.

Corollary 6.1. Let G I H be a homogeneous @-space of order k such that spec O consists of two elements. T h e n mp c b. Indeed, in this case E and E are primitive k-th roots of unity. 7. Canonical structures P, J, f , h and Riemannian metrics Let g = b e m be the canonical reductive decomposition of a regular @-space G / H . Suppose B is a non-degenerate bilinear symmetric form on m x m and B is invariant with respect to both A d G ( H ) and 8. Then B induces on the manifold G I H the pseudo-Riemannian structure g invariant with respect t o both G and "symmetries" {S,}. The basic example of such a metric is the so-called standard metric. This metric is determined by means of the Killing form B of a semisimple Lie algebra g. The standard metric on any regular @-space G I H of a semisimple Lie group G is naturally reductive [ 53 1. Our main purpose is to investigate compatibility of the canonical structures PIJ , f,h with the above metric g. First we recall some definitions.

16

Let ( M ,g ) be a (pseudo-)Riemannian manifold. ( g , P ) is a (pseudo-)Riemannian almost product structure, if g ( P X , P Y ) = g ( X ,Y ) (see, for example, Ref. [ 671); ( 9 ,J ) is an almost Hermitian structure, if g ( J X ,J Y ) = g ( X ,Y ) (see, for example, Ref. [ 221); (9,f ) is a metric f-structure, if g ( f X ,Y ) g ( X ,f Y ) = 0 (see Ref. [ 341); ( 9 ,h) is a (pseudo-)Riemannian h-structure, if g ( h X ,Y ) = g ( X , h Y ) (see Ref. [ 41).

Proff. for a +

Here X , Y are any vector fields on M . It is easy to see that in the special case K e r f = 0 a metric f-structure ( 9 ,f ) coincides with an almost Hermitian structure as well as in the case K e r h = 0 a (pseudo-)Riemannian h-structure (9,h) coincides with a (pseudo-)Riemannian almost product structure. Now we prove the following

Lemma 7.1. Let V be a vector space over R, p bilinear form on V x V . Suppose p is invariant with respect to a non-singular linear operator I: V-V, i.e. p ( l X , l Y ) = p ( X , Y ) f o r a n y X , Y E V . Then

Proff. for a Proff. for a P(X(l)X,Y ) = A X , X ( W Y )

for a n y polynomial X with real coefficients and any X , Y E V .

This proves the required result.

0

Theorem 7.1. Let ( G I H ,g ) be a (pseudo-)Riemannian regular @-space. Suppose g is invariant with respect to G and { S p } . Let P, J , f , h be canonical structures on G I H such that the corresponding polynomials P(O),J(O),f (O),h(O) satisfy respectively the following conditions: (a) P(e) = P(O-l); (c) f (0) =

-f(W;

(b) J ( e ) = -J(e-l); (d) h(0) = h(O-I).

(7.1)

17

Then respectively (a) ( 9 ,P ) is a (pseudo-)Riemannian almost product structure; (b) ( 9 ,J ) i s an almost Hermitian structure; (c) ( g , f ) is a metric f-structure; (d) ( 9 ,h) i s a (pseudo-)Riemannian h-structure. Proof. (a) For any X , Y E m the condition g ( P X ,P Y ) = g ( X ,Y )is equivalent to g ( P X ,Y ) = g ( X ,P Y ) . Now, applying Lemma 7.1 for 1 = 8 and using the assumption P ( 0 ) = P(O-’), we prove the first assertion. The other assertions (b), (c) and (d) can be proved in the same manner.^ Corollary 7.1. Let ( G I H ,g ) be a (pseudo-)Riemannian homogeneous @space of order k with the metric g as above. Then all the canonical structures P, J , f,h o n G I H are compatible with g (in the sense of Theorem 7.1).

Proof. Indeed, by Theorem 3.2 all the canonical structures P, J , f,h can be given by polynomials satisfying conditions (7.1). It remains to apply Theorem 7.1. 8. Metric f-structures in generalized Hermitian geometry The concept of generalized Hermitian geometry created in the 1980’s (see, for example, Refs. [ 341, [ 371) is a natural consequence of the development of Hermitian geometry and the theory of almost contact structures with many applications. One of its central objects is the metric f-structures of the classical type (f3 f = 0), which include the class of almost Hermitian structures. A fundamental role in the geometry of metric f-manifolds is played by the composition tensor T , which was explicitly evaluated in Ref. [ 341:

+

where V is the Levi-Civita connection of a (pseud0)Ftiemannian manifold

( M ,91,

x7 y

EX(W.

Using this tensor T , the algebraic structure of a so-called adjoint Q-algebra in X ( M ) can be defined by the formula:

x * Y =T(X,Y).

18

It gives the opportunity to introduce some classes of metric f -structures in terms of natural properties of the adjoint Q-algebra (see Ref. [ 341). We enumerate below the main classes of metric f-structures together with their defining properties:

Kf Hf

Kahler f - structure: Hennitian f -structure:

Of

=o;

T ( X , Y )= 0, i.e. X(M) is an abelian &-algebra;

Glf

f -structure of class GI, or G1f -structure:

T ( X , X ) = 0, i.e. X(M) is an anticommutative Q-algebra;

QKf Kill f

quasi-Kahler f -structure: Killing f -structure: nearly Kahler f - structure, or N K f-structure:

Vxf +Txf = 0; Vx(f)X = 0;

NKf

VfX (f)f x = 0.

The classes Kf, Hf, Glf, QKf (in more general situation) were introduced in Ref. [ 341 (see also Ref. [ 521). Killing f-manifolds Kill f were defined and studied in Refs. [ 251, [ 261. The class NKf was first determined in Ref. [ 31 (see also Ref. [ 71). The following relationships between the classes mentioned are evident:

Kf = Hf n QKf; Kf

c Hf c Glf; Kf c Kill f c NKf c G l f .

It is important to note that in the special case f = J we obtain the corresponding classes of almost Hermitian structures (see Ref. [ 241). In particular, for f = J the classes Kill f and NKf coincide with the well-known class NK of nearly Kahler structures. We already mentioned that the main classes of almost Hermitian structures are provided with the remarkable set of invariant examples (see Refs. [ 651, [ 231, [ 331). It turns out that also there is a wealth of invariant examples for the basic classes of metric f-structures. These invariant metric f-structures can be realized on homogeneous k-symmetric spaces with canonical f-structures (see Refs. [ 31, [ 51-[ 71, [ 181, [ 441). In what follows, we mainly concentrate on some classes and remarkable examples of invariant N K f -structures.

9. Invariant NKf-structures on naturally reductive homogeneous spaces Let G be a connected Lie group, H its closed subgroup, g an invariant (pseudo-)Riemannian metric on the homogeneous space G / H . Denote by

19

g and fj the Lie algebras corresponding to G and H respectively. Suppose that G / H is a reductive homogeneous space, g = fj @ m the reductive decomposition of the Lie algebra g. As usual, we identify m with the tangent space T , ( G / H ) at the point o = H . Then the invariant metric g is completely defined by its value at the point 0. For convenience we denote by the same manner both any invariant metric g on G I H and its value at 0.

Recall that ( G / H , g ) is naturally reductive [ 381 with respect to a reductive decomposition g = fj @ m if

g([X,Ylm,Z)= g(X, [Y,z]rn) for all X, Y, Z E m. Here the subscript m denotes the projection of g onto m with respect to the reductive decomposition. Now we can characterize invariant Killing and nearly Kahler f -structures in terms of special subspaces m(x+).

Theorem 9.1. Let ( G I H ,g ) be a naturally reductive homogeneous space, f an invariant metric f -structure, g = fj @ m the corresponding reductive decomposition. (i) f is a Killing f-structure iffm('9f) c fj. (ii) f is a nearly Kahler f-structure z f f m(fif2)c g. Proof. The technique is traditional for invariant structures on reductive homogeneous spaces [45]. For instance, we prove (ii). The condition Vfx ( f )f X = 0 means that

for all vector fields X on G I H . Denote by (Y the Nomizu function of an invariant affine connectionv [ 451. Then, using special vector fields [ 45 ] in a neighborhood of the point o it can be shown that the previous condition is equivalent to that of

4 f X , f 2 X )= f ct.(fX,fX),

x E m.

Since the metric g is naturally reductive,

X,Y E m . Hence we obtain [ fX, f 2 X ] E fj for any X E m. This is nothing but the condition m(f,fz)c fj. Statement (i) can be proved in exactly the same way. 0

20

Remark 9.1. Consider the particular case f = J ( J 2 = -1) of Theorem 9.1. Then both the conditions are reduced to that of m(l,J) c b, i.e. [ X, J X ] E b for all X E m. This is just the well-known criterion for nearly Kahler structures in the case (see Ref. [ 65, p. 1381). Any invariant metric f-structure on G I H determines the orthogonal decomposition m = ml @ m2 such that ml = Irn f , m:, = K e r f .

Theorem 9.2. Let ( G / H ,g , f ) be a naturally reductive homogeneous space with a metric f -structure, m = ml @ m2 the decomposition with respect to f . The following conditions are equivalent: (i) f is a Killing f -structure; (ii) f is an N K f -structure and [ ml, m2 ] c b.

+ X2, XI E ml, X2 E m2, we get [ X , f X I= [ x l , f x l l + X 2 , f X I l .

Proof. Taking any X

= X1

(9.1)

I f f is a Killing f-structure, then by Theorem 9.l(i) the vectors [ X, f X ] and [ X1, f X , ] belong t o b. Fkom (9.1) it follows immediately that [X,, f X , ] E b. Since f is non-singular on ml, we obtain [ml,m2] c b. So, (i) implies (ii). Conversely, suppose f is an N K f -structure and [ ml, m2 ] c 9. By Theorem 9.1 (ii) we have

[ X l , f X l I = [ f Y , f 2 Y 1E b, where Y E m. Sirlce [ X2, f X l ] E by assumption it follows from (9.1) that [ X, f X ] E b. Hence by Theorem 9.1 (i) f is a Killing f-structure. This completes the proof.

0

10. Canonical NKf-structures on regular +-spaces The main purpose of this Section is t o show that naturally reductive regular @-spaceswith some canonical f-structures provide a wide class of homogeneous N K f -manifolds.

Theorem 10.1. Let G I H be a regular @-space, g a naturally reductive metric on G I H with respect to the canonical reductive decomposition g = fj @ m, f a metric canonical f-structure o n G I H . Suppose the f structure satisfies the condition f2

= he f .

Then ( G / H , g ,f ) is a nearly Kahler f-manzfold.

(10.1)

21

Proof. By Theorem 9.1 (ii) it suffices to prove that [ fX,f 2 X ] E for any X E m. Using (lO.l), we have 'p[

fx,f2x] = [ efx, ef2x] = [ f2x, f 3 x=] [ f2x, -fx] = [ fx,f2x].

It follows that [ f X ,f 2 X ]E g. Remark 10.1. We recall that the assumptions for a metric g in Theorem 10.1 are satisfied in the case of a semisimple Lie group G and the standard metric g . Moreover, by Corollary 7.1 this metric is compatible with any canonical f-structure on a homogeneous @-space of order k . Corollary 10.1. Let ( G / H ,g ) be a naturally reductive homogeneous @space of order k = 4n, n 2 1. If {i,-i} C spec 8, then there exists a non-trivial canonical N K f -structure o n G / H .

Proof. The procedure of describing canonical structures on homogeneous @-spacesof finite order is constructive (see Ref. [ 11, $4 and $51). We use it to present a canonical f -structure required. Consider the decomposition m = m l @ mg such that ml and mg are determined by the eigenvalues {i, -2) and all the others of the operator 8 respectively. Now we put f to be the complex structure on ml and vanish on mg. In other words, consider the collection (C1,..., C l , . . ' I

CU),

cj E {-I, 0,1}

for constructing canonical f-structures (see Ref. [ 11, Theorem 5.3]),where c l corresponds to the eigenvalue i of 8. Then we put
,...,u } , m # l .

It is not difficult to see that this f-structure satisfies the condition f 2 = 8f . It remains to apply Theorem 10.1. 0 We formulate the following evident but important particular case of Corollary 10.1:

Corollary 10.2. A n y naturally reductive homogeneous @-space ( G / H ,g ) of order 4 is an N K f manifold with respect to the canonical f -structure f = (8 - 83)/2. Indeed, in this case the only canonical f-structure on G / H (see Corollary 3.2) satisfies condition (10.1). Notice that if K e r f = 0, then G / H is a Hermitian symmetric space (see Ref. [ 65, p. 1361).

22

It should be noted that there exist canonical N K f -structures on regular @-spacessuch that condition (10.1) is not satisfied. We consider in this respect canonical f-structures on homogeneous @-spacesof order 5. First we need some special facts on these spaces.

Lemma 10.1. (191, [ 1 6 ] - [ M I , 1111) Let GIH be a homogeneous @-space of order 5, g = t, @ m the canonical reductive decomposition of g, P the canonical almost product structure on G I H . Consider the decomposition m = ml @ m2, where ml and m2 correspond to the eigenvalues +1 and -1 of the operator P respectively. (1) The following relations hold:

[m1,m11 c b @m2,

[m2,m21

(2) The canonical structures J1,J2, f1, J1

= (1, J),

J2

c t, @m1, [m1,m21 c m. can be represented in the f o r m

f2

= (I,-J),

fl = ( I , ( ) ) ,

f2

= (0, J),

where I and J are special complex structures o n ml and tively. (3) For all XI, Y1 E ml, X2, Y2 E m2 we have

m2

respec-

I [X2,Y2]1 = -[X2,J y211; J [Xl,Yl12 = [ X l , I y112; I [Xl,Y211 = - [ I Xl,Y211 = [Xl,J y211; I [Xl,Y2]2 = - [ I Xl,Y2]2 = -[X1,J Y212. Here subscripts 1 and 2 mean projections of vectors Z E g with respect to the decomposition g = t, @ ml @ m2.

Theorem 10.2. Let G I H be a homogeneous @-space of order 5, g a naturally reductive metric o n G I H with respect to the canonical reductive decomposition g = t, @ m. Then both the canonical structures f l and f 2 o n GIH are N K f -structures. Proof. Consider the canonical f-structure X E m we have

[flX,fi2XI = [ Y l , f l Yll

f1 =

(1,O).

= [Yl,IYl],

where Y1 E ml. By Lemma 10.1 (3), we obtain

[Yl,IY1]2 = J[Y1,Y1]2=0. Besides, from Lemma 10.1 (1) it follows that

[ Y l , I Y l ] E [ m l , m l ] c t, @m2.

Then for any

23

[fix,

Now we can conclude that f : X ] E 6. By Theorem 9.1 (ii) this means that fl is an NKf-structure. The analogous statement for the structure f2 = (0,J ) can be proved in the same manner. 0 It is important to note that there are canonical f-structures on naturally reductive homogeneous k-symmetric spaces that are not NKf-structures.

11. Examples In conclusion, we present some general and particular examples of homG geneous NK f-manifolds. As a rule, we also indicate the corresponding algebras d(B)for homogeneous regular @-spacesunder consideration.

11.1. Riemannian 4 - and 5-symmetric spaces Riemannian homogeneous +-spaces of order 4 of classical compact Lie groups were classified and geometrically described in Ref. [ 29 1. The similar problem for homogeneous @-spaces of order 5 was claimed to be solved in Ref. [ 581. By Corollary 10.2 and Theorem 10.2, it presents a collection of homogeneous NKf-manifolds. Note that the canonical f - structures under consideration are generally non-integrable. From Corollary 4.2 it evidently follows that the algebra d(0) for homogeneous 4-symmetric spaces is isomorphic to C @ W or C. In exactly the same way, for homogeneous 5-symmetric spaces we obtain: d(B)E C @ C or d ( 0 ) E C.

11.2. The spheres S2, S5, S6

It is well known that any standard sphere 9" is a Riemannian globally symmetric space. That is why its algebra of canonical affinor structures is isomorphic to R. However among all the spheres only S2, S5, and S6 can be realized as Riemannian homogeneous @-spaces with automorphisms @ that are not involutions (see Refs. [ 481, [ 401). The sphere S2 2 5 0 ( 3 ) / 5 0 ( 2 ) can be represented as a homogeneous aspace of any order k (k 2). If k > 2, then any invariant affinor structure on S2 is canonical, i.e. d(B)= d (see Ref. [ 111). It is evident now that A = d(Q) E C. In particular, the standard complex structure on 9 ' is determined by the canonical almost complex structure. The sphere S5 represented as a homogeneous space SU(3)/SU(2) admits the structure of a Riemannian homogeneous +-space of order 4 (see Refs.

>

24

[ 401, [ 481). The canonical f-structure f = (0 - 03)/2 for this representation was calculated in Ref. [ 111. It has deficiency 1 and defines an invariant almost contact structure on S5. From Section 5 it evidently follows that the algebra d(0)is isomorphic to R @ @. We note that the fact was obtained in Ref. [ 441 by the straightforward computation. Besides, the canonical f-structure is clearly an NKf-structure. It can be easily shown that f is not a Killing f-structure. It should be mentioned that nevertheless the sphere S5 allows a Killing fstructure. This is a so-called weakly cosymplectic structure (see Refs. [ 141, [ 35]), however this f-structure is not invariant. Moreover, from the results in Refs. [ 251, [ 261 it follows that S5 does not admit any invariant Killing f-structure. The.sphere S6 Z G2/SV(3) is the most significant example in the theory of nearly Kahler manifolds (see, for instance, Ref. [ 221). Its nearly Kahler structure is defined by the canonical almost complex structure of the corresponding homogeneous 3-symmetric space. It follows that d(8) 2 C for this representation of the sphere S 6 . 11.3. Homogeneous +-spaces of order 8 The canonical NKf-structure described in Corollary 10.1 can be effectively constructed for homogeneous @-spacesof any order k = 4n. As an example, consider the case k = 8. Applying the general formula (see Theorem 3.2) for the situation, we obtain that all canonical f-structures can be represented in the form

where

<j

E {-1;O;

1). Put

(2

= 1,
1 4

f = - (e - e3 + e5 - e7)

(11.1)

It is easy to verify that this f-structure satisfies condition (10.1). So, we obtain

Theorem 11.1. Let ( G / H , g ) be a naturally reductive homogeneous @space of order 8 . If ( 2 , -2) c spec 8, then the canonical f-structure (11.1) is a non-trivial NKf-structure o n G I H .

25

11.4. The 6-dimensional generalized Heisenberg group We briefly formulate some notions and results related to the 6-dimensional generalized Heisenberg group ( N , g ) . As to details, we refer to Refs. [ 311, [32l, [571Let V and Z be two real vector spaces of dimension n and m ( m 2 1) both equipped with an inner product which we shall denote for both spaces by the same symbol ( , ). Further, let j : 2 End(V) be a linear map such that

-

{j(u)IC(=

( z ( ( a ( , j(@ = -{a(2I,

5

E

v, a E 2.

Next we put n := V @ 2 together with the bracket defined by [a+x,b+yl=

[Z,Yl

E 2,

( [ ~ , Y l , 4= (j(.).,Y),

where a, b E 2 and IC, y E V . It is a 2-step nilpotent Lie algebra with center Z. The simply connected, connected Lie group N whose Lie algebra is n is called a generalized Heisenberg group. Note that N has a left invariant metric g induced by the following inner product on n: (a+IC,b+y) = (a,b)

+ (.,y),

a,b E 2, z,y €

v

The 6-dimensional generalized Heisenberg group ( N ,9) is of especial interest (see Refs. [ 321, [ 571). The brackets for the Lie algebra n = L ( I c ~ , z ~ , I c @ ~ ,L(a1,az) I c ~ ) were explicitly indicated (see Ref. [ 57, p. 1111): [21r%1

= a17

[ 2 i , % ]= a 2 ,

[%x4]

= -az,

[23,24]

=a~,

all other brackets being zero.

It is known (see Ref. [ 57, p. 112 1) that ( N ,g) is a Riemannian homogeneous @-spaceof order 4. More exactly, the automorphism is determined by means of the isometric automorphism 'p of the Lie algebra n such that 'p4 = id. For convenience, we consider 'p written in the form 'p

: (x1,IC2, 5 3 , 2 4 , a1 I a2)

-

(-247

-537 527 5 1 ,

-@).

By our notations we have 0 = 'p. Directly calculating the canonical fstructure f = (0 - @')>/aon ( N , g ) ,we obtain

f

:

(51,227 IC3,54,a1,

a2)

(-547

-53,227

51,0, O ) .

26

It follows f Iv= cp Iv,f Iz= 0, hence ml = I m f = V, m2 = K e r f = 2. It means that f has deficiency 2. By Corollary 4.2, we immediately obtain:

d(B)2 @@EX. It can be easily shown that f is a metric f-structure on ( N , g ) . It is important to note that ( N ,g) is not a naturally reductive homogeneous space (see Refs. [ 321, [ 57, p. 961). Therefore we cannot use Corollary 10.2 above and have to directly calculate. The Levi-Civita connection V of the metric g was indicated in Ref. [ 311:

i

1

VXY = 5 [Z,Y], Vax= Vxu = - -1 3.( a ) z ,

(11.2)

2

Vab=O,

where a , b E 2,z,y E V , 22 = j ( a l ) q , 23 = j ( 4 2 1 , 2 4 = j(a1)j(a2)21. By a denote as usual the Nomizu function of V. From easy calculations it follows that [ IC, cp z ] = 0 for all z E V . Now put y = f X for all X E n. Then if we recall (11.2), we can get 1

a ( f X , f X )= 5 [Y,Y] = 0,

1

4 f X , f 2 X )= 5 [ Y , c p Y ]

= 0.

Moreover, it follows that the canonical f-structure satisfies the condition Vfx(f)fX = 0, i.e. it is an NKf-structure. However this f-structure is not Killing. Besides, using the results of Ref. [ 101 we can show that f is non-integrable. On the other hand, ( N , g ) is simultaneously a homogeneous &-space of order 3 (see Ref. [ 57, p. 111 I). Denote 8 = 'p, where (p is the corresponding isometric automorphism of n such that 'p3 = id. In the case we obviously obtain: A(@ 2 @. It can easily be checked that the canonical almost complex structure J = (8 is defined by the mapping

e2)/fi

J : (51,22,23,24,al,a2)

In particular, we obtain J l v = f To sum up, we have

-----)

(-24, -23,22,21,a2, -al)

Iv.

Theorem 11.2. The 6-dimensional generalized Heisenberg group ( N ,g ) i s a nearly Kahler f -manifold with respect to the canonical f -structure f = ( 0 - B3)/2 of a homogeneous @-space of order 4. This f-structure i s not integrable on ( N , g ) . The algebras d(B)for 3- and 4-symmetric representations of ( N ,g ) are C and C @ JR respectively.

27

Remark 11.1. Recall (see Ref. [ 231) that a (pseudo-)Riemannian homogeneous @-space ( G / H , g ) of order 3 with the canonical almost complex structure J is nearly Kahler iff the metric g is naturally reductive. As to (pseudo-)Riemannian homogeneous @-spaces ( G / H ,g) of order 4 with the canonical f-structure, the situation is not completely similar. More precisely, by Corollary 10.2 any naturally reductive space of such a kind is a nearly Kahler f -manifold. However the above example illustrates that the inverse is not true. This is just the difference between the cases ( G / H ,g , J ) for k = 3 and ( G / H , g ,f ) for k = 4. Remark 11.2. The above example illustrates two distinct @-spaceshaving the same underlying structure of a homogeneous space. It follows that the same homogeneous space may correspond to various classes in the partition with respect to the algebra d(0)(see Remark 5.3).

11.5 The group of hyperbolic motions of the plane R2

Proff. for a (eii!) Proff. for a Proff. for a Proff. for a a,b,cER}

C={

28

but f is n e i t h e r integrable n o r nearly Kahler. 4 - s y m m e t r i c space is i s o m o r p h i c to C @ R.

T h e algebra d ( 8 ) for this

11.6. The flag manifold SlJ(3)/Tm,, The well-known flag manifold SU(3)/Tm,, is a homogeneous k-symmetric space for all k 2 3 (see Ref. [ 301). For the corresponding automorphisms @ of orders k = 3 , 4 , 5 the algebras d(6)are C, C @ R, C @ C respectively. Besides, there are various NKf-structures of rank 4 that are canonical with respect to these automorphisms.

11.7. The regular @-space with non-semisimple 0 An example of a regular @-spaceG I H with a non-semisimple automorphism is given in Ref. [ 11, 931. The minimal polynomial of the operator 13 for this space has the form v(x) = (x2+2px+1)2,wherep2-1 < 0. According to Theorem 4.1, this means that d ( 8 ) 2 A z ( C ) . Note that this algebra d ( 8 ) is maximal in dimension; namely, dim d ( 8 ) = dim G / H = 4. References 1. E. Abbena and S. Garbiero, Almost Hermitian homogeneous manifolds and Lie groups, Nihonkai Math. J. 4, 1-15, (1993). 2. V. V. Balashchenko, Invariant normalizations and induced connections o n regular 9-spaces of linear Lie groups, Dokl. Akad. Nauk BSSR 23, (3), 209-212, (1979) (Russian. English summary). 3. V. V. Balashchenko, Riemannian geometry of canonical structures on regular @-spaces, Preprint No. 174/1994, Fakultat fiir Mathematik der Ruhr-Universitat Bochum, 1-19, 1994. 4. V. V. Balashchenko, Canonical f -structures of hyperbolic type o n regular @-spaces, Uspekhi Mat. Nauk 53, (4), 213-214, (1998); English translation: Russian Math. Surveys 53, (4), 861-863, (1998). 5. V. V. Balashchenko, Naturally reductive Killing f -manifolds, Uspekhi Mat. Nauk 54, (3), 151-152, (1999); English translation: Russian Math. Surveys 54, (3), 623-625, (1999). 6. V. V. Balashchenko, Homogeneous Hermitian f -manijolds, Uspekhi Mat. Nauk 56, (3), 159-160, (2001); English translation: Russian Math. Surveys 56, (3), 575-577, (2001). 7. V. V. Balashchenko, Invariant nearly Kahler f-structures o n homogeneous spaces, Contemporary Mathematics 288, 263-267, (2001). 8. V. V. Balashchenko, T h e algebra of canonical a f i n o r structures and classes of regular @-spaces, Doklady Akademii Nauk 385, (6), 727730, (2002); English translation: Doklady Mathematics 66, (l), 111-114, (2002).

29

9. V. V. Balashchenko and Yu. D. Churbanov, Invariant structures on homogeneous @-spaces of order 5, Uspekhi Mat. Nauk 45,(l),169-170, (1990); English translation: Russian Math. Surveys 45,(l), 195-197, (1990). 10. V. V. Balashchenko and 0. V. Dashevich, Geometry of canonical structures on homogeneous +-spaces of order 4, Uspekhi Mat. Nauk 49,(4), 153-154, (1994); English translation: Russian Math. Surveys 49,(4), 149150, (1994). 11. V. V. Balashchenko and N. A. Stepanov, Canonical afinor structures of classical type on regular +-spaces, Matematicheskii Sbornik 186, ( l l ) , 3-34, (1995); English translation: Sbornik: Mathematics 186,( l l ) , 15511580, (1995). 12. V. V. Balashchenko and N. A. Stepanov, Canonical afinor structures on regular @-spaces, Uspekhi Mat. Nauk 46,(l), 205-206, (1991); English translation: Russian Math. Surveys 46,( l ) , 247-248, (1991). 13. H. Baum, T . Friedrich, R. Grunewald and I. Kath, Twistor and Killing

14. 15. 16.

17.

Spinors on Riemannian Manifolds, Teubner-Texte zur Mathematik, Vol. 124,Teubner-Verlag, Stuttgart/Leipzig, 1991. D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math., Vol. 509,Springer-Verlag, 1976. A. Bore1 and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80, (2), 458-538, (1958). Yu. D. Churbanov, O n some classes of homogeneous @-spaces of order 5, Izv. Vyssh. Uchebn. Zaved. Mat. 2, 88-90, (1992); English translation: Soviet Math. (Iz. VUZ) 36, (2), (1992). Yu. D. Churbanov, Canonical f-structures of homogeneous @-spaces of order 5, Vestnik BGU. Ser.1: Fiz,Mat,Mech. 1, 51-54, (1994), (in Russian).

18. Yu. D. Churbanov, The geometry of homogeneous @-spaces of order 5, Izv. Vyssh. Uchebn. Zaved. Mat. 2, 70-81, (2002); English translation: Russian Math. (Iz. VUZ) 46,(5), (2002). 19. A. S. Fedenko, Spaces with symmetries (Prostranstva s simmetriyami), Minsk: Izdat. Belorussk. Gos. Univ., 168 p., 1977, (in Russian). 20. F. R. Gantmacher, The Theory of Matrices, Moscow: Nauka, 4 t h ed. 1988, (in Russian); English translation of the 1st ed.: The Theory of M a trices, New York: Chelsea, 1959. 21. S. Garbiero and L. Vanhecke, A characterization of locally 3-symmetric spaces, Riv. Mat. Univ. Parma 2, (5), 331-335, (1993). 22. A. Gray, Nearly Kiihler manifolds, J. Diff. Geom. 4,(3), 283-309, (1970). 23. A. Gray, Riemannian manifolds with geodesic symmetries of order 3, J. Diff. Geom. 7, (3-4), 343-369, (1972). 24. A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura ed Appl. 123,(4), 35-58, (1980). 25. A. S. Gritsans, Geometry of Killing f-manifolds, Uspekhi Mat. Nauk 45, (4), 149-150, (1990); English translation: Russian Math. Surveys 45,(4), 168-169, (1990).

30

26. A. S. Gritsans, On construction of Killing f-manifolds, Izv. Vyssh. Uchebn. Zaved. Mat. 6 , 49-57, (1992); English translation: Soviet Math. (Iz. VUZ) 36,(6), (1992). 27. R. Grunewald, Six-dimensional Riemannian manifolds with a real Killing spinor, Ann. Global Anal. Geom. 8, ( l ) , 43-59, (1990). 28. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, New York: Academic Press, 1978. 29. J. A. Jimenez, Riemannian 4-symmetric spaces, Trans. Amer. Math. SOC. 306,(2), 715-734, (1988). 30. J. A. Jimenez, Existence of Hermitian n-symmetric spaces and of noncommutative naturally reductive spaces, Math. Z. 196, (2), 133-139, (1987). 31. A. Kaplan, Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11,127-136, (1981). 32. A. Kaplan, On the geometry of groups of Heisenberg type, Bull. London Math. SOC.15,35-42, (1983). 33. V. F. Kirichenko, On geometry of homogeneous K-spaces, Mat. Zametki 30,(4), 569-582, (1981); English translation: Math. Notes 30,(1981). 34. V. F. Kirichenko, Methods of generalized Hermitian geometry in the theory of almost contact manifolds, Itogi Nauki i Tekhniki: Probl. Geom. VINITI 18, 25-71, (1986); English translation: J. Soviet Math. 42, (5), (1988). 35. V. F. Kirichenko, Sur la geometrie des varietes approximativement cosymplectiques, C. R. Acad. Sci. Paris, Ser.1 295,(12), 673-676, (1982). 36. V. F. Kirichenko, Hermitian-homogeneous generalized almost Hermitian manifolds, Soviet Math. Dokl. 30, 267-271, (1984). 37. V. F. Kirichenko, Generalized quasi-Kaehlerian manifolds and axioms of CR-submanifolds in generalized Hermitian geometry, I, Geom. Dedicata 51,75-104, (1994). 38. S. Kobayashi and K. Nomizu, Foundations of differential geometry, V.2, Intersc. Publ. J.Wiley & Sons, New York-London, 1969. 39. B. P. Komrakov, Homogeneous spaces generated by automorphisms and invariant geometrical structures, Problems of geometry. Itogi Nauki i Tekhniki. VINITI 7,81-104, (1975), (in Russian). 40. 0. Kowalski, Generalized symmetric spaces, Lecture Notes in Math., Vol. 805,Berlin, Heidelberg, New York: Springer-Verlag, 1980. 41. S. Lang, Algebra, Reading, Mass.: Addison-Wesley, 1965; Russian translation: Algebra, Moscow: Mir, 1968. 42. A. J. Ledger and R. B. Pettitt, Compact quadratic s-manifolds, Comment. Math. Helv. 51,105-131, (1976). 43. A. J. Ledger and L. Vanhecke, On a theorem of Kiric'enko relating to 3-symmetric spaces, Riv. Mat. Univ. Parma 13,(4), 367-372, (1987). 44. L. V. Lipagina, On the structure of the algebra of invariant afinor structures on the sphere S5, Izv. Vyssh. Uchebn. Zaved. Mat. 9,17-20, (1997); English translation: Russian Math. (Iz. VUZ), 41,(9), 15-18, (1997). 45. K. Nomizu, Invariant afine connections on homogeneous spaces, Amer.

31

J. Math. 76,( l ) ,33-65, (1954). 46. S . Salamon, Harmonic and holomorphic maps, Geometry Seminar "Luigi Bianchi" I1 - 1984, Lecture Notes in Math., Springer-Verlag, 1164,161224, (1985). 47. S . Salamon, Minimal surfaces and symmetric spaces, Diff. Geometry. Proc. Colloq. Santiago de Compostela, 103-114, (1985). 48. C . U. Sanchez, Regular s-structure on spheres, Indiana Univ. Math. J. 37,( l ) ,165-180, (1988). 49. T. Sato, Riemannian 3-symmetric spaces and homogeneous K-spaces, Memoirs of the Faculty of Technology, Kanazawa Univ. 12,( 2 ) , 137-143, (1979). 50. K. Sekigawa and J. Watanabe, O n some compact Riemannian 3symmetric spaces, Sci. Reports of Niigata Univ. Ser. A. 19,1-17, (1983). 51. K. Sekigawa and H. Yoshida, Riemannian 3-symmetric spaces dejned b y some outer automorphisms of compact Lie groups, Tensor 40,( 3 ) , 261268, (1983). 52. K. D. Singh and Rakeshwar Singh, Some f ( 3 , ~ ) - s t m c t u r emanifolds, Demonstr. Math. 10,(3-4), 637-645, (1977). 53. N. A. Stepanov, Basic facts of the theory of y-spaces, Izv. Vyssh. Uchebn. Zaved. Mat. 3,88-95, (1967); English translation: Soviet Math. (Iz. VUZ) 11,( 3 ) , (1967). 54. N. A. Stepanov, Homogeneous 3-cyclic spaces, Izv. Vyssh. Uchebn. Zaved. Mat. 12,65-74,,(1967); English translation: Soviet Math. (Iz. VUZ) 11, ( 1 2 ) , (1967). 55. N. A. Stepanov, O n reductivity of coset spaces generated by Lie group endomorphisms, Izv. Vyssh. Uchebn. Zaved. Mat. 2,74-79, (1967); English translation: Soviet Math. (12. VUZ) 11, ( 2 ) , (1967). 56. N. A. Stepanov, Regular cp-spaces of a general linear group, Izv. Vyssh. Uchebn. Zaved. Mat. 6,118-121, (1976); English translation: Soviet Math. (Iz. VUZ) 20,( 6 ) , (1976). 57. F. Tricerri and L. Vanhecke, Homogeneous structures on Riemannian manifolds, London Math. SOC.,Lecture Notes Ser. 83, 1983. 58. Gr. Tsagas and Ph. Xenos, Homogeneous spaces which are defined by difieomorphisms of order 5 , Bull. Math. Soc. Sci. Math. RSR 31, ( l ) , 57-77, (1987). 59. V. I. Vedernikov, Symmetric spaces and conjugate connections, Kazan Gos. Univ. Uchen. Zap. 125,( l ) ,7-59, (1965), (in Russian). 60. V. I. Vedernikov, Symmetric spaces. Conjugate connections as a n o n a l ized connection, Trudy Geom. Semin. Inst. Nauch. Inform. AN SSSR 1, 63-88, (1966), (in Russian). 61. V. I. Vedernikov, O n one special class of homogeneous spaces, Izv. Vyssh. Uchebn. Zaved. Mat. 12,17-22, (1972); English translation: Soviet Math. (Iz. VUZ) 16,( 1 2 ) , (1972). 62. V. I. Vedernikov and A. S. Fedenko, Symmetric spaces and their generalizations, Itogi Nauki Tekh., Ser. Algebra Topologiya Geom. 14,249-280, (1976), (in Russian).

32

63. J. A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces, J . Math. Mech. 14,(6), 1033-1047, (1965). 64. J. A. Wolf, Spaces of constant curvature, New York-St.Louis-San Francisko, McGraw-Hill Book Comp. Vol. XV, 408 p., 1967. 65. J. A. Wolf and A. Gray, Homogeneous spaces defined by Lie group automorphisms, J. Diff. Geom. 2, (1-2), 77-159, (1968). 66. K. Yano, O n a structure defined by a tensor field f of type (1,l) satisfying f3 f = 0, Tensor 14,99-109, (1963). 67. K. Yano and M. Kon, Structures on manifolds, Singapore: World Scientific, 1984.

+

ON KENMOTSU HYPERSURFACES IN A SIX-DIMENSIONAL HERMITIAN SUBMANIFOLD OF CAYLEY ALGEBRA *

M. B . BANARU Smolensk University of Humanities, Gertsen s t r . , 2, Smolensk 214 014, RUSSIA E-mail: [email protected]

In the present work, Kenmotsu hypersurfaces in six-dimensional Hermitian submanifolds of Cayley algebra are considered. The Hermitian structure on such manifolds is induced by the means of the three-fold vector cross products in the octave algebra. A simple criterion of the minimality for a Kenmotsu hypersurface in a six-dimensional Hermitian submanifold of Cayley algebra is established. It is also proved that the type number of a minimal Kenmotsu hypersurface in a six-dimensional Hermitian submanifold of Cayley algebra is necessarily even.

1. Introduction

As it is known, an almost contact metric structure on an odd-dimensional manifold N is defined by the system of the tensor fields {a,5,71, g } on this manifold, where 5 is a vector, 7) is a covector, is a tensor of the type (1,l) and g = ( . , - ) is the Riemannian metric. Moreover, the following conditions are fulfilled: 7)(()

= 1,

cp(5)

= 0,

(ax, @ Y )= ( X ,Y )

7) 0 cp = 0, -

a2= -.zd + 5 63 71,

71(X) 7)(Y),

x,y E " N ) ,

where N(N) is the module of smooth vector fields on N . As an example of an almost contact metric structure we can consider the cosymplectic structure, that is characterized by the following condition:

vcp=o,

Vq=O, * MSC2000: 53C55.

Keywords : Hermitian manifold, almost contact metric manifold, Kenmotsu manifold, hypersurface.

33

34

where V is the Levi-Civita connection of the metric. It has been proved that the manifold, admitting the cosymplectic structure, is locally equivalent to a product M x R,where M is a Kahlerian manifold [ 71. The almost contact metric structure are closely connected to the almost Hermitian structures. For instance, if ( N , { a, [, 7 , g ) ) is an almost contact metric manifold, then an almost Hermitian structure is induced on N x R, (see [ 41). If this almost Hermitian structure is integrable, then the input almost contact metric structure is called normal. As it is known, a normal contact metric structure is called Sasakian [ 41. On the other hand, we can characterize the Sasakian structure by the following condition:

VX(@)Y= (X,Y) [ - q(Y)X,X,YE N(N). (1.1) For example, Sasakian structures are induced on totally umbilical hypersurfaces in a Kahlerian manifold [ 41. We note that the Sasakian structures have many remarkable properties and play a fundamental role in contact geometry. In [ 61, 1972 Katsuei Kenmotsu has introduced a new class of almost contact metric structures defined by the condition:

v!(a)Y= (ax, Y)[ - q(Y)ax, x,Y E N(N).

(1-2) The Kenmotsu manifolds are normal and integrable, but they are not contact, consequently, they can not be Sasakian. In spite of the fact that the condition (1.1) and (1.2) are similar, the properties of Kenmotsu manifolds are to some extent antipodal to the Sasakian manifolds properties [ 91. Note that the new investigation in this field (see [ 91) contains a detailed description of Kenmotsu manifolds as well as a collection of examples of such manifolds. In the present paper, Kenmotsu hypersurfaces in six-dimensional Hermitian submanifolds of Cayley algebra are considered. This work is a continuation of researches of the author, who studied almost contact metric structures on hypersurfaces in six-dimensional submanifolds of Cayley algebra before (see, for example, [ 11, [ 21 and [ 31). We note that such outstanding mathematicians as R. Bryant, E. Calabi, A. Gray and V. F. Kirichenko were engaged in the studying of diverse aspects of geometry of six-dimensional almost Hermitian submanifolds of the octave algebra.

2. Preliminaries We consider an almost Hermitian manifold i.e. a 2n-dimensional manifold with a Riemannian metric g = ( . , . ) and an almost complex

35

structure J . Moreover, the following condition must hold:

( J X ,J Y ) = ( X ,Y ) ,

x,Y E N(M2"),

where N(M2") is the module of smooth vector fields on M2". All considered manifolds, tensor fields and similar objects are assumed to be of the class C". We recall that the fundamental (or Kahlerian) form of an almost Hermitian manifold is determined by

F ( X , Y )= ( X , J Y ) , X , Y E N(M2"). Let (M2n,{ J , g = ( . , .)}) be an arbitrary almost Hermitian manifold. We fix a point p E M2". As T p ( M a n )we denote the tangent space at the point p , { Jpr g p = ( . , . ) } is the almost Hermitian structure at the point p induced by the structure { J , g = ( ., *)}. The frames adapted to the structure (or the A-frames) look as follows

.

( P , E I ~ . .,En, EX,.

.

. 1

EZ),

a,

where E, are the eigenvectors corresponded to the eigenvalue i = and E-, are the eigenvectors corresponded to the eigenvalue -2 (as in [ 21). Here the index a ranges from 1 to n, and we state 2 = a n.

+

The matrix of the operator of the almost complex structure written in an A-frame looks as follows:

-i

In

where Inis the identity matrix; k,j = 1 , . . . ,2n. By direct computing it is easy t o obtain that the matrices of the metric g and of the fundamental form F in the frame adapted to the structure look as follows, respectively:

Let 0 = Rs be the Cayley algebra. As it is well-known [ 5 ] , two nonisomorphic three-fold vector cross products are defined on it by

PI(X, Y,2)= - x(m)+ (X, Y ) Z + ( yZ ) X - (2, X)Y, P 2 ( X 1 Y , Z )= - ( x ~ ) z + ( x , Y ) z + ( Y , z ) x - ( z , x ) Y ,

36

-x

where X, Y,2 E 0, (. , . ) is the scalar product in 0 and X is the conjugation operator. Moreover, any other three-fold vector cross product in the octave algebra is isomorphic t o one of the above-mentioned.

If M 6 c 0 is a six-dimensional oriented submanifold, then the induced almost Hermitian structure {Ja,g = ( . , . ) } is determined by the relation

J a ( X ) = p,(X,e l , e2), a

= 1,2,

where { e l , ez} is an arbitrary orthonormal basis of the normal space of M 6 at the point p , X E T p ( M 6 )(see (51). The submanifold M 6 c 0 is called Hermitian, if the almost Hermitian structure is integrable. The point p E M 6 is called general, if eo $ T'(M6), where eo is the unit of Cayley algebra (see [ S ] ) . A submanifold M 6 c 0, consisting only of general points, is called a general type submanifold (see [S]). In what follows, all submanifolds M 6 c 0 to be considered are assumed t o be of general type. 3. The Main Results

Theorem 3.1. Let N be a Kenmotsu hypersurface in a Hermitian submanifold M 6 c 0, and let G be the second fundamental f o r n of the immersion of N into M 6 . Then N is a minimal submanzfold of M 6 i f and only i f a(<,<)= 0. Proof. Let us use the Cartan structural equations of an almost contact metric structure on a hypersurface in a six-dimensional Hermitian submanifold of Cayley algebra (as in [ 2 I): dw" = wp" A wp

+ B,apurA wp + (&BE3 + i o z ) w p A w

dw, = - W! A wp

+ B& wY A w p + (JZB!, - 20:)

1 + (- B& Jz dw =

(JZB?

-

ia,p)

- JZB$

-

W'

A W,

2i$)

W'

A wa

wp A w

+ (B$ -+ i ~ 3 p ) wA w p

+(~ip-i~~a,P)w~w~. Here {B,"b}and { B & } are the components of Kirichenko tensors [ I ] ; a , b , c = 1 , 2 , 3 ; k , j = 1 , 2 , 3 , 4 , 5 , 6 ;l i = a + 3 .

37

Taking into account that the Cartan structural equations of a Kenmotsu structure look as follows [ 121: dw* =up” A u ~ + w A w ~ , d w , = - W: A W P f w A w,, d w = 0,

we get the conditions, whose simultaneous fulfillment is a criterion for the structure on N to be Kenmotsu:

( a ) B y = 0;

(b)

Jz Bp”3+ i up”= - 6;;

(d) f i l l ? - &B,”p - 2iap” = 0;

(e)

1 ( c ) - -B

Jz

Bzp - zuf

y -+ a c a p

=0

=0

;

(3.1)

and the formulae obtained by the complex conjugation (no need to write them explicitly). From (3.1 .(c)) we have:

Since

we get B z p = 0, and that is why d? = 0. Similarly, from (3.1.(e)) we can obtain a! = 0. Therefore we can rewrite the conditions (3.1) as follows:

(a)

= 0; (b) oaP= 0;

(c) 0:

= 0; (d)

~p= ” iJz BE3 + i 6p”,

(3.2)

and the formulae obtained by the-complex conjugation. Now, let us use a criterion of the minimality of an arbitrary hypersurface (see [ 11 I):

gps ups= 0,

p , s = 1,2,3,4,5.

Knowing how the matrix of the contravariant metric tensor on N looks, [ 121: 00010

01000

38

we obtain gps u p s = gapcJc,p p a s j

+ + gs’PcJ,p + ga&laj + g + g a b a f i+

= g%sp

33 cJ33

g33a33.

By force of (3.2) we have gps oPs= i f i B g 3

+ 22

-

i &BE3

-

2 i + c ~ 3 3= 0 3 3 .

*

That is why gps aPs = 0 033 = 0 . The last equality means that c((,6) = 0. SO, a Kenmotsu hypersurface in a six-dimensional Hermitian submanifold of Cayley algebra is minimal, precisely when the condition a ( ( , () = 0 holds.

As it known (see, for example [ l o ] ) , when we give a Riemannian manifold and its submanifold, the rank of the determined second fundamental form is called the type number. Now, we can state the second main result of this work. Theorem 3.2. Kenmotsu hypersurface in a Hermitian submanifold M 6 0 is minimal if and only i f its type number is equal to 4.

c

Proof. 1. Let N be a minimal Kenmotsu hypersurface in a Hermitian submanifold M6 c 0. Then by force of Theorem 3.1 and (3.2), the matrix of the second fundamental form looks as follows:

(UPS)

AS a s p =

=

q, we have: rank(aP,)= 2 r a n k ( c ~ s p ) .

Now, we use the relation (from [ 11) for Kirichenko tensors of six-dimensional Hermitian submanifolds of Cayley algebra:

Bab= 1 E~~~ Dhc ,

“ J z where

39

{Tzc} are the components of the configuration tensor of the submanifold M 6 c 0 ; 'p = 7,8 ; E~~~ = are the components of third-order Kronecker tensor. Let us compute the components of (uQ). By force of (3.2) we have:

ql = 0: = i f i B i 3 + i d :

=i f i

(h

- €137 D,,) + i = -i

012

+i ;

The matrix (asp) can not be degenerate. In fact,

Knowing (see [ 11) that for an arbitrary Hermitian submanifold M 6 the following identity holds:

c0

we have: det(uz,p) = -1 # 0. So, this matrix is not degenerate, therefore rank(u2;p) = 2 and rank(a,,) = 4, i.e. the type number of N is equal to four. 2. If a Kenmotsu hypersurface N in a Hermitian submanifold M 6 c 0 is not minimal, then according to (3.2) we get:

m k ( f f , , ) = 2 rank(ff5p)+ 1. So, the type number of N is an odd number, it can not be equal to f0ur.u

From Theorem 3.2 we can immediately conclude the following statement. Corollary 3.1. A minimal Kenmotsu hypersurface in a Hermitian submanifold M6 c 0 can not be totally geodesic.

40

References 1. M. Banaru, Six theorems on six-dimensional Hermitian submanifolds of Cayley algebra, Bull. Acad. Sci. Rep. Moldova 34,3-10, (2000). 2. M. Banaru, Two theorems on cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of Cayley algebra, J. Harbin I. T. 8,38-40, (2001). 3. M. Banaru, Hermitian geometry of six-dimensional submanifolds of Cayley algebra, Mat. Sbornik 193,3-16, (2002). 4. D. E. Blair, Contact manifolds in Riemannian geometry, Lect. Notes Math. 509,1-145, (1976). 5. A. Gray, Vector cross products on manifolds, Trans Amer. Math. SOC.141, 465-504, (1969). 6. K. Kenmotsu, A class of almost contact Riemannian manifolds, Tbhoku Math. J. 24,93-103, (1972). 7. V. F. Kirichenko, Sur la g6om6trie des variZtks approximativement cosympletiques, C. R. Acad. Sci. Paris. Ser. I 295,673-676, (1982). 8. V. F. Kirichenko, Hermitian geometry of sax-dimensional symmetric submanafolds of Cayley algebra, Vestnik MGU 3,6-13, (1994). 9. V. F. Kirichenko, On geometry of Kenmotsu manifolds, DAN Math. 380, 585-587, (2001). 10. H. Kurihara, The type number on real hypersurfaces in a quaternionic space form, Tsukuba J. Math. 24,127-132, (2000). 11. A. P. Norden, The theory of surfaces, GITTL, Moscow, 1956. 12. L. V. Stepanova, Contact geometry of hypersurfaces of quasi-Kahlerian manifolds, MSPU (PhD thesis), Moscow, 1995.

PARA-HYPERCOMPLEX STRUCTURES ON A FOUR-DIMENSIONAL LIE GROUP *

NOVICA B L A Z I C ~ AND SRDJAN VUKMIROVIC~ Faculty of Mathematics, University of Belgrade, Studenski trg 16, p . p . 550, 11 000 Belgrade, YUGOSLAVIA E-mails: [email protected]. yu, [email protected]. y u

The main goal is to classify 4-dimensional real Lie algebras g which admit parahypercomplex structures and which are not solvable or are solvable with dimg’ 2. This is a step toward the classification of Lie groups admitting the corresponding left-invariant structure and therefore possessing a neutral, left-invariant, anti-selfdual metric. Our study is related to the work of Barberis who classified real, 4dimensional simply-connected Lie groups which admit an invariant hypercomplex structure.

<

1. Introduction

Our work is motivated by the work of Barberis [ 21 where invariant hypercomplex structures on 4-dimensional real Lie groups are classified (see Section 2 for definitions). In that case the corresponding hermitian metric is positive definite and unique up to a positive constant. Our main goal is to classify 4-dimensional real Lie algebras g which admit para-hypercomplex structures in the case of non-solvable g or g solvable and dimg’ 5 2. This is a first step toward the classification of the corresponding left invariant structures on Lie groups. In this case the corresponding hermitian pseudoRiemannian metric determined by the para-hypercomplex structure is also unique up to a constant, but has to be of signature (2,2). This metric is anti-self-dual (see [ 31). In the paper [ 11 Andrada and Salamon have shown that any para-hypercomplex structure on a real Lie algebra g rise to a hypercomplex structure on its complexification g@(considered as a real Lie * MSC2UUU: 53C50,53C56,32M10, 53C26,53C55. Keywords :para-hypercomplex structure, metric of neutral signature. Work partially supported by the SerbianMinistry of Science, Technologies and Development under contract No M M f 854.

41

42

algebra). He referred to para-hypercomplex structure as complex product structure. We show that there are many more 4-dimensional Lie algebras with parahypercomplex structure (Theorems 3.1 and 3.3) than Lie algebras with hypercomplex structure, even though our classification is not yet completed. The case of solvable Lie algebras with dimg’ = 3 is not finished. By the result of Barberis, only R @ so(3) and aff(C) admit hypercomplex structure. The Lie algebra aff(C) also admits a para-hypercomplex structure. The Lie algebra IR @ so(3) does not admit a para-hypercomplex structure, but the corresponding non-solvable Lie algebra IR @ &(R) admits parahypercomplex structure. Let us remark that Snow [ 41 classified the invariant complex structures on 4-dimensional, solvable, simply-connected real Lie groups where the dimension of commutators is less than three. We compare our results with the classification of Snow, in Section 3. In the continuation of this work we plan to complete the classification on the Lie algebra level, in the missing case, and to determine the corresponding simply-connected Lie groups. It is also interesting to study the geometry of these groups with respect to the corresponding hermitian metric. 2. Preliminaries

Let V be a real vector space. A complex structure on V is an endomorphism J1 of V satisfying the condition

J; = -1. Existence of a complex structure implies that V has to be of an even dimension. A product structure on V is an endomorphism J2 of V satisfying the conditions

Ji = 1,

J2

# f 1.

A para-hypercomplex structure on V is a pair ( J l , J 2 ) of anti-commuting complex structure J1 and product structure J2, i.e. satisfying the relations J; = - 1,

Ji = 1,

51 5 2 = - 5 2 51.

(2.1)

If both structures J1 and J2 are complex then the pair (51,J2) is called a hypercomplex structure on V. In the sequel we concentrate on the case of para-hypercomplex structure.

43

It is customary to denote J3 = J1 J 2 . Note that the structure 53 is a product structure. The Lie subalgebra of End(V) spanned by 51, J 2 and 5 3 is isomorphic to sZ2(R). Any z = (zl,x2,z3) E R3 defines a structure by the formula

J,

:= 21 J1

+ 2 2 J 2 + 5 3 53.

Denote by

( x , y ) = x l y 1 - 2 2 Y2

- z3 937

z = (xl,x2,z3), y = (yl,y2,y3) inner product in R3 = R1>2and by x y = (z2 y3

- x3 y 2 , z 3 y1 - z1 Y3721 Y2

- z2 y l )

the usual cross product. The structure J, is a complex structure provided that 2

2

(z,z) = z1 - x2 - x; = 1

and a product structure provided that 2

( 2 , x )= z ] - 2 2

2

-z;

= -1.

Hence, a para-hypercomplex structure (J1, J 2 ) defines a 2-sheeted hyperboloid S- of complex structures and a 1-sheeted hyperboloid S+ of product structures.

Proposition 2.1. If (J1, J 2 ) is a para-hypercomplex structure on a vector space V, then:

(i) Jz Jy = - ( z , Y ) 1 + J z x y . (ii) The pair (J,, Jy)E S- x S+ is a para-hypercomplex structure if and only if x Iy. Proof. From the relations J1J2=

J~Z-J~J J 1~ J 3,= - J 2 = - J 3 J 1 7

cJ~J~=-J~=-J~J

the statement (i) follows by a direct calculation. Since J, is a complex structure and Jy is a product structure, the pair (J,,Jy)is a para-hypercomplex structure if and only if J, and Jy anticommute. Using the relation (i) and the anti-commutativity of the cross product we have

+

0 = J z Jy JyJ, = -2 ( Z,9 ) 1. Hence, the statement (ii) is proved.

0

44

The para-hypercomplex structures (J1, J2) and (J,, J3/)are called compatible.

An almost para-hypercomplex structure on a manifold A4 is a pair ( J1 ,Jz) of sections of E n d ( T M ) satisfying the relations (2.1). It is a parahypercomplex structure if both structures are integrable, that is, if the corresponding Nijenhuis tensors

a = 1,2, vanish on all vector fields X , Y . In this formula sign - occurs in the case of a complex structure and sign + occurs in the case of a product structure. If M = G is a Lie group we additionally assume that the para-hypercomplex structure is left invariant. This allows us to also describe a parahypercomplex structure on its Lie algebra g. Hence, a para-hypercomplex structure (51,J 2 ) on g satisfies both relations (2.1) and (2.2).

Proposition 2.2. Let o n a Lie algebra g.

( J 1 , J2)

( i ) T h e product structure

be a n integrable para-hypercomplex structure 53

= J1 5 2 i s integrable.

(ii) A n y compatible para-hypercomplex structure (J,, J z I )i s integrable.

Proff. for a where N3 is the Nijenhuis tensor of the product structure J3. To prove (ii) denote by J(, the Nijenhuis tensor corresponding to the structure J,, x = (21,x2 ,2 3 ) . One can check that

holds, where we have used the notation, for instance J2

N2 5 3 ( X ,Y ) = J 2 N2 (J 3 X 7J3Y).

Now, statement (ii) follows using statement (i).

45

Let g be an inner product on the vector space V. A para-hypercomplex structure (J1,J2) on V is called hermitian with respect to g if

g ( J a X , Y )= - g ( X , J a y ) ,

x,y E v

(2.3)

holds, i.e. if both structures J1 and J2 are hermitian. It is easy to prove that a hermitian complex structure is an isometry and a hermitian product structure is an anti-isometry, i.e.

g ( J i X ,J i y )= g ( X ,Y ) ,

g(J2X, J2Y) = - g ( X , Y ) .

Existence of an anti-isometry implies that the inner product g must be of neutral, (n,n) signature. Proposition 2.3. Let (J1,J2) be a para-hypercomplex structure hermitian with respect to the scalar product g o n the vector space V. ( i ) The product structure J3 = J1 J2 is hermitian. (ii) Any compatible para-hypercomplex structure (J,, J g ) is hermitian. Proof. (i) If J1 and J2 are hermitian then J3 is hermitian since we have

( J 3 X , Y ) = ( J 1 J 2 X , Y ) = -(J2X, J 1 Y ) = ( X , J 2 J 1 Y ) = - ( X , J3Y). (ii) Since the condition of any J, to be hermitian is linear with respect to x , the statement (ii) follows from the statement (i). 0 Now, we prove some lemmas which will be useful in the sequel. Lemma 2.1. If (J1,Jz) is a para-hypercomplex structure o n a real 4dimensional vector space V then: (a) There is a n inner product g o n V , unique up t o a non-zero constant, such that the structure (J1,J2) is hermitian with respect t o g . (ii) A n y compatible para-hypercomplex structure (J,,J y ) determines the same inner product g o n V. Proof. First, we prove the existence of such an inner product. If (-, -) is an arbitrary inner product on V , then the inner product

g ( X ,Y ) := ( X ,Y )

+ ( J l X ,J 1 Y ) - ( J z X ,J2Y) - ( J 3 X ,J 3 Y )

(2.4)

satisfies the properties (2.3). To see the uniqueness let g'(., .) be another inner product on V satisfying (2.3). As remarked before both products are of signature (2,2). There

46

exists a vector X which is not null with respect to the both inner products, for instance

g ( X , X ) = 1,

g ' ( X , X ) = X # 0.

The relations (2.1) and (2.3) imply that the vectors X, J l X , J z X , J3X are mutually orthogonal with respect to both inner products. Moreover,

g ( X ,X ) = g ( J l X , J l X ) = 1 = -g( J2X, J 2 X ) = -g( J3X, J 3 X ) g'(X, X ) = g ' ( J l X , J l X ) = X = -gf(J2X, J 2 X ) = -g'(J3X, J 3 X ) . Hence, g ( . , . ) = X g'( . , .), X # 0. (ii) According to Proposition 2.3 the structure ( J z ,JzI)is hermitian with respect to g . The statement follows from the uniqueness of g (up to a non-zero scalar).

Remark 2.1. In the light of Lemma 2.1 we see that the notion of null vector N (such that g ( N ,N ) = 0 ) depends only on the hermitian structure (J1,J2) and not on a particular inner product. From the proof of Lemma 2.1 we also obtain the following.

Lemma 2.2. If (J1,J2) is a is a para-hypercomplex structure o n a real 4-dimensional vector space V then

( X , J l X ,J2X, J 3 X ) is a basis of V

u X is not null.

Lemma 2.3. If J, is a n endomorphism of a 4-dimensional Lie algebra g such that J i = fl and ( X ,J,X,Y, J a y ) is a basis of g then the corresponding Nijenhuis tensor N , vanishes if and only i f N o ( X , Y ) = 0. Proof. One can easily show that N,(J,X,Y) = -J,N,(X,Y). lemma follows from the fact that N, is antisymmetric and bilinear

The

.

0

Lemma 2.4. Let (J1,J2) be a para-hypercomplex structure on a real 4dimensional vector space V and let W c V be a 2-dimensional subspace. Then there exists a compatible para-hypercomplex structure ( J i ,J i ) such that: a) If W is definite (contains no null directions) then Ji W = W.

(ii) If W is Lorentz (contains exactly two null directions) then J i W = W.

47

(iii) If W is totally null (every vector in W is a null vector) then either (a) J ; ( w = l , V = W @ J : W , o r (b) there exists a non-null vector X such that W = R ( J I X + J ; X , X - J l X ) , J ( W ) = W forall J E S * . (iv) If the induced metric on W is of rank 1 ( W contains exactly one null direction N ) then there exists a non-null vector X E W such that N = J1 ' X - J; X . Proof. (i) and (ii): Let ( X , Y ) be a pseudo-orthonormal basis of W (IXI2 = -IYI2 = 1 and ( X , Y ) = 0 with respect to the induced inner product on W ) . Then, according to Lemma 2.2 vectors X , J l X , J2X and J3X form a pseudo-orthonormal basis of V and we have

Y = 21J I X

+

~2

+

J2X

23

J3X

where - occurs if W is Lorentz and The structure

JX = 2 1 J1

with

2;

- x i - 2% = f

l,

+ if W is positive or negative definite.

+

22

J2

+

23

53

preserves W . It is a product structure if W is Lorentz (and we set J i = J z ) and a complex structure if W is definite (and we set Ji = J x ) . The second structure can be chosen such that ( J l , J i ) forms a compatible parahypercomplex structure. Note that there cannot exists a product structure preserving a definite W since a product structure is an anti-isometry. Similarly, a complex structure preserving a Lorentz W cannot exist. (iii): Let N I E W be a null vector. There exists a non-null vector X E V perpendicular to N1. Hence

N 1 = cx J l X

+ p J2X + y J 3 X

and ff

2

- 8 2 - y2 = 0

+

so a # 0 and we may assume that cx = 1. Then J i = PJ2 yJ3 is a product structure, the structure ( J ; , Ji), Ji = J1 is a compatible parahypercomplex structure and we have

+ JiX. Any null vector a X + b J ; X + c J i X + d JAX N1 = J:X

which is orthogonal to the

vector N1 is of the form

N* = a X

+ b J:X + b J;X

f a JiX.

48

Notice that the vector N1 is also of the form N* and that there exist exactly two null planes W* containing the vector N1. They can be written in the form W* = R (N1,N,” = x f J; X ). The plane W - is the +1-eigenspace of the product structure Ji and the vectors N1, N;, J j N l , JjNT are independent, so V = W - @ J i W - and (a) holds. In the case of the plane W+ one easily checks that JiW+ = W + = JiW+ and hence statement (b) follows. (iv): The proof is similar t o the first part of the previous proof (with N1 = N ) . 0 3. Lie algebras admitting a para-hypercomplex structure 3.1. Case when g has a non-trivial center

In the following theorem the additive basis of the Lie algebra g is either ( X ,Y,2,W ) or ( X ,Y,N1, N2). The vectors N , are null vectors. Theorem 3.1. A 4-dimensional Lie algebra g admitting a para-hypercomplex structure and with a non-trivial center Z ( g ) is one of the following:

(i) g is abelian, (22) g 2 R @ 512 (R), (Zii) Z ( g ) = R ( Z ) , [ X , Y ]= Y, [ X , W ]= W, (iv) Z(g) = R ( N l , N 2 ) , P , Y I = N l , (v) Z(8) = R ( N l , N 2 ) , [ X , Y I = x , (vi) Z(8) = R ( N 1 ), [ X , N 2 ]= aN1, [ Y , N 2 ]= aN2, [ X , Y ]= crNI+PN2+UX, a # 0. As a consequence of Levi decomposition theorem and the classification of real semisimple Lie algebras the only non-solvable Lie algebras which are 4-dimensional are Iw @ so(3) and R @ s12(R). Since they both have a non-trivial center, as a consequence of Theorem 3.1 we have the following corollary. Corollary 3.1. The only non-solvable, real 4-dimensional Lie algebra admitting a para-hypercomplex structure is R @ sl2 (R). Remark 3.1. The algebra R @ sl2(R) corresponds to the unique nonsolvable 4-dimensional Lie algebra R @ so(3) which admits hypercomplex

49

structure (Barberis [ 2 ] ) . According to the Snow notation [4], the cases (iii), (iw) and (w) are Lie algebras S3, S1, and S2 respectively. In the case (wi) the derived algebra is the Heisenberg algebra (of dimension 3).

Proof of Theorem 3.1. First, we prove the existence of a para-hypercomplex structure on these algebras. On the abelian Lie algebra and on the algebra EX @ &(EX) a possible structure is

J1Z = X , J1Y = W,

J2Z = Y , J2X = -W.

One can easily check that the structure

J1Z = X , J1Y = W, J2Z = W - Z , J2X = X + Y , J2Y = -Y, J2W = W is a para-hypercomplex structure on the Lie algebra (iii). Finally, the structure

J1Ni = N2, J1X = Y , J2Nl = N2, J2X = Y - N1, J2Y = X

+ N2

is a para-hypercomplex structure on the Lie algebras (iv)-(vi). In order to prove that these are the only Lie algebras with non-trivial center which admit a para-hypercomplex structure we consider two cases. Case 1: there exists a non-null central element Z.

Let (J1,J2) be a para-hypercomplex structure on g and denote

X = JiZ,

Y = JzZ,

W = J3Z.

Then

[ X , Y ]= a Z + b X + c Y + d W .

(3.1)

According to Lemma 2.3 integrability of J1 is equivalent t o

O = N 1 ( Z , Y ) = [ X , W ]- J l [ X , Y ] .

(3.2)

Similarly, the integrability of J2 is equivalent to

O = N z ( X , Z ) 1[ Y , W ]- J 2 [ X , Y ] .

(3.3)

From the relations (3.1), (3.2) and (3.3) we get

[ X , W ]= - b Z + a X - d Y + c w ,

[ Y ,W ]= c z - d X + a Y - bW.

50

The Jacobi identity is equivalent to

0 = ~ [ x , y l+, “~Yl, W l > X l + “ W , X I , Y l = = 2 ( - a 2 - b2 c’) 2 - 2 c d X - 2 d b Y - 2 a d W.

+

If a = b = c = d = 0 then the algebra g is abelian (i). If a = b = c = 0 and d # 0 then after scaling g Z R CB &(R) (ii). If d = 0 and 0 # c2 = a2 b2 then the derived algebra g’ = [g,g] of g is 2-dimensional since

+

c [ y , W ] = a [ x , y ]+ b [ W , X ] It is generated by the vectors W1 = [X, Y], Y1 = [W,XI. The vectors 2, X1 = ( l / c ) X , Y1 and Wl are linearly independent and their commutator relations are given by the relations (iii). Case 2: all central vectors are null vectors. Denote one of them by N . According to Lemma 2.4 (iv), we can assume that N = JIX - J2X for a non-null vector X E 8’. Then the vectors N , JlN, X and J1X form a basis of g and the structure J2 expressed in the terms of that basis reads J2X

= J1X - N ,

J2J1N

=N,

J2J1X

= J1N

+ X,

J2N = J1N. (3.4)

The integrability of the structure J1 gives the following conditions

0 = N1 (X, N ) = [ J l X , J1N ] - J1[X, JiN].

(3.5)

Since the vectors N , JzN, X and J2X form a basis of g, the integrability of the product structure J 2 is equivalent to

0 = N2(X,N ) = [ J l X , J l N ] - J2[ X, J i N ] . The vector [ X, J l N ] is of the form [ X, J l N ] = a N Using the relations (3.5) and (3.6) we get that

(3.6)

+ b J1N + c X + d J1X.

[ x,JlN] = a N + bJlN + 2 b X , [ J i X , J i N ]= - b N + a J l N + 2 b J i X . I f w e w r i t e [ X 7 J 1 X ]= a N + p J 1 N + y X + S J 1 X a n d i m p o s e t h e Jacobi identity on the vectors J1N,X and J1X we get the following system of equations: -4 a b - b2 - 6 b + y a - a2 = 0,

-4 b p + u S + b y = 0, The system has three classes of solutions.

+

b ( u y) = 0, b ( b - 6) = 0.

51

1) a = 0 = b. In this case the only non-zero commutator is

[ x,J l X ] = Q N

+ p J1N + y x + 6 J l X .

+

If y = 0 = 6, the change of the basis Y = J l X , N1 = Q N P J l N , Nz E R ( N , J1 N ) gives the relations (iv). If 6 # 0 then the change Y = ( 1 / 6 ) [ X , J 1 X ]N, I = N , N z = J I N gives the relations (v). The case 6 = 0,y # 0 similarly reduces t o the relations (v). 2 ) b = 6 # 0 , a = -7. This case reduces t o the relations (iii). 0 3) a = y # 0. This immediately gives the commutator relations (vi). 3.2. Case of solvable Lie algebra g and dimg'

5 2.

Theorem 3.2. Let g be a 4-dimensional real Lie algebra admitting a parahypercomplex structure and dimg' = 1. T h e n g is one of the algebras (i), (ii) f r o m Theorem 3.1.

Proof. If g has a non-trivial center then from Theorem 3.1 we get the algebras (i) and (ii). Now, as in [ 21, Proposition 3.2, let 6 = (0) and let X be a non-zero element of g'. There exists Y such that [ Y,X ] = X . Then g decomposes as g = ker(adx) n ker(ady)

Iwx @ W.

<

From the Jacobi identity we get that = ker(adx) nker(ady), a contradiction. Hence solvable g without center and with dimg' = 1 does not exist (this does not depend on the existence of para-hypercomplex structure). In the following theorem the additive basis of Lie algebra g is ( X ,Y,2,W ) . Theorem 3.3. Let g be a 4-dimensional solvable Lie algebra admitting a para-hypercomplex structure and with dimg' = 2. If g has a non-trivial center then it is algebra (ii) f r o m Theorem 3.1. If g does not have a center then g is one of the following:

(a) [ X , Z ] = X , [ X , W ] = Y , [ Y , Z ] = Y , [ Y , W ] = a X + b Y , a , b E l R ,

0) [ X , Z I = x, [Y,Wl = y, (c) [ X , Z ] = X , [ X , W ] = X , [ Y , Z ] = Y , [ Y , W ] = Y . Remark 3.2. Using the notation introduced by Snow [ 41, these Lie algebras are S11, S8 and ,510 respectively. The class S11 contains as a special case the Lie algebra aff(C) which is the unique solvable Lie algebra with 2-dimensional derived algebra which admits hypercomplex structure [ 2 1.

52

Proof. First, we prove the existence of para-hypercomplex structures on these algebras. One can easily check that the structure:

J,X = 2, J1Y = w, JzX = X , J2Y = Y, J 2 2 = -2,J2W = -W is a para-hypercomplex structure on the algebras (a) and (c). The structure

J,X = -Y, J l Z =

-w,JZX = Y, J 2 2 = -w

is a para-hypercomplex structure on the algebra (b). Now, suppose that the center of g is trivial and that ( J l , J 2 ) is a parahypercomplex structure on g. According to Lemma 2.1 and Remark 2.1 the structure (J1,J2) determines the inner product on g = V and the notion of a null vector. As in Lemma 2.4 we have t o consider the cases concerning the rank and the signature of the induced inner product on g’ = W .

Case i) : Induced metric on g‘ is definite. Because of Lemma 2.4 (i) we may assume that g’ is invariant with respect to the complex structure J1, Jlg’ = g’, and g = g’ @ J2g’. Let { X ,J l X = Y } be a basis of g’ and { X ,Y,J z X , J 2 Y ) be a basis of g. The Lie algebra g‘ is abelian since g is solvable and by the integrability of the product structure JZ we have NZ( X ,Jl X ) = 0 and [ J 2 X , J z Y ]= O , [J2X,Y]=[J2Y,X]. Integrability of the complex structure J1, N1 ( X ,J 2 X ) = 0 and [ X ,J 2 X ] = - [ Y , J 2 Y ] .

(3.7)

(3-8)

For arbitrary vectors V and W in g,

[ v,w I = Q(V,W )x

+ P(V,w y,

where Q and 6 , are skew-symmetric bilinear forms on g. From the Jacobi identity we have

4x7 J 2 X ) = P ( X ,JZY),

Q(JZY, X ) = P ( X ,J 2 X )

and the bracket in g is determined by c = a ( X ,J z X ) and d = P ( X , J z X ) as follows:

[ X , J z X ]= - [ Y , J z Y ] = c X + d Y , [ X , J Z Y ]= [ Y , J z X ] = - d X + c Y . Since dimg’ = 2, c2 + d2 # 0 and we may choose

x = (c2 + d2)-1(cX + d Y ) , 2 = (2 + d2)-1(cJ2X

-

dJzY),

Y

= (2

+ d2)-l(-dX + C Y ) ,

w = (2 + dZ)-l(dJzX + CJZY),

53

and hence

[X,Z]=X, [X,W]=Y,

[Y,Z]=Y,

[Y,W]=-X,

so we get the algebra (a) for a = -1, b = 0. Note that g E aff(C). Case ii) : Induced metric on g’ is indefinite, of Lorentz type (-+). Because of Lemma 2.4 (ii) we may assume that g’ is invariant with respect to the product structure 5 2 , J2g’ = g‘, and g = g‘c~J1g’.Let { X , J 2 X = Y } be a basis of g’ and { X ,Y ,J l X , J l Y } be a basis of g. By the integrability of the complex structure J1, N1 ( X ,Y ) = 0 and

[JlX,JlY]=O,

[ J l X , Y ]= [ J l Y , X ] .

(3.9)

Integrability of the product structure J Z ,& ( X , J l X ) = 0 and

[ X , Jix]= [ Y ,J i Y

1.

(3.10)

From the Jacobi identity we have

a ( X ,J l X ) = P ( X , J l Y ) ,

a(J1Y,X

) = -P(X, A X ) ,

and the bracket in g is determined by c = a ( X , J I X )and d = P ( X ,J I X ) as follows:

[ X , J 1 X ] =[ Y , J l Y ]= c X + d Y , Since dimg’ = 2, c2 - d2

[ X , J l Y ]= [ Y , J 1 X ]= d X + c Y .

# 0 and we may choose

x = (c2 - d2)-1(cX + d Y ) ,

y ( d X +C Y ) , = ( 2- d2)-’(-dJ1x + C J l Y ) ,

Y = ( 2- d

2 = (c2 - d y ( C J 1 x - d J l Y ) ,

w

and hence

[X,Z]=X,

[X,T?r]=F,

[Y,Z]=Y,

[Y,W]=X,

and we get algebra (a) for a = 1, b = 0. Case iii): g’ is a totally null plane.

According t o Lemma 2.4 iii) we have to consider two geometrically different cases. In the first case we can assume that J21gt = 1 and g = g’ Jlg’ holds. If ( X ,Y ) is a basis of g‘ we have

+

J2X = X ,

J2Y = Y,

J2JlX = - J i X ,

JZJiY = -J1Y.

One easily checks that the integrability of the complex structure Jl is equivalent to the relations

54

It is interesting that the product structure J2 is automatically integrable. Hence, the possible non-null commutators are

T’ = [ X ,J i X ] = U X + b y ,

+d Y, X‘ = [ X , J l Y ] = e X + f Y. Y‘ = [ Y,J i Y ] = C X

The Jacobi identity is equivalent to the equations

( e - d ) X’

+ f Y‘ - cT’ = 0,

( a - f ) X’

+ b y ’ - eT‘ = 0 ,

(3.11)

or equivalently

e ( e - d ) + c ( f -a)=O,

e f =bc,

af - f2+bd-be=0.

If X’ is a zero vector then we get the algebra (b). Suppose that X‘ is a non-zero vector. If Y’ or T’ is a zero vector then we get an algebra (a) for a = 0 = b. Suppose that none of the vectors X ’ , Y’,2’ is the zero vector. We can suppose that one of the pairs X ’ , Y’ and X ’ , T’ is independent, say X ’ , T’. If the vectors X’ and Y’ are collinear then we get the algebra (a) for a = 0, b = 1. Finally, if the both pairs X‘, T‘ and X ‘ , Y‘ are independent then introduce a new basis ( X ’ , Y‘,Z‘, W‘) satisfying 1

Z ‘ = -1( f J i X - b J l Y ) ,

W‘ = - (- e J1X + a J l Y ) , D D where D = a f - be # 0. In the new basis the commutator relations take the very simple form

[ x’,2‘1 = x‘, [ x’,W ‘ ]= Y’,

[ Y’,2’1 = Y ’ , fc-de ad-bc [Y’,W‘]= x‘+ D Y‘ D Since X’ and Y’ are independent then c f - d e # 0 , that is, a # 0 in the algebra (a). In the second case we can assume that (N1,N2) is a basis of g’ and g‘ is invariant with respect to J1, J2,J3. Then a possible basis of g is ~

N1 = J1X

+ J2X,

N2 = X - J3X, N3 = J l X - J2X, N2 = X

We calculate the structures in terms of that basis:

+ J3X.

55

By the integrability of J3,

J 3 [ N i , N 4 ]= [ N i , N 4 ] ,

J3[N2,N31 = - [ N 2 , N 3 ] .

Thus,

[ N i , N 4 ]= p N i ,

[N2,N3]=

The integrability of J1 and J2 is equivalent to 0 = - [ ~ ~ , ~ 4 ] - ~ ~ 1 + ~ ~ 2 + [ ~ 1 , ~ 3 ]

After imposing the Jacobi identity and simple changes we get the algebra (c). Case iv) : the induced metric on 8’ is of rank 1. Denote by N the null vector belonging to 8’ (which is up to a scaling unique). According to Lemma 2.4 iv) we can choose a product structure J2 such that for the basis ( X ,N ) of g’ one has N = J I X - J2X,

N is null.

(3.12)

Then ( X , N ,J l X , J1N) is a basis of g. One easily calculates the following relations

J2N = J1N.

J2X = J1X - N ,

The integrability of J1 is equivalent to J V ~X[ , N ] = 0 , i.e. to the relations

[ J i X , J i N ] = 0,

[X , J l N ] = [ N , J l X ] .

Since ( X ,N , J2X, J2N) is a basis of g the integrability of the product structure J2 is equivalent to N 2 ( X ,N ) = 0 which gives the condition [ N ,J i N ] = 0.

The commutator relations now read

[X,J1X]=aX+bN,

[X,JlN]=cX+dN,

where a , b, c, d are unknown coefficients. The Jacobi identity is now equivalent to the following relations c=O,

d(a-d)=0.

The case d = 0 gives the algebra with dimg‘ = 1 which we have already discussed. The remaining case a = d # 0 , after the change

Y =N,

Z= J I N ,

1

X =-X, a

1 a

r/t. = - JlX

b a2

- -J i N ,

56

takes the form

[ Y , S ] = O , [ Y , W ] = Y , [ X , Z ] = Y ,[ X , W ] = X of the algebra (a) for a = 0 = b.

0

Acknowledgments

We are grateful to the referee for careful reading of the paper which helped us to greatly improve the contents of the paper and the way of exposing it. References 1. A. Andrada, S. Salamon, Complex Product Structures on Lie Algebras, preprint (2003), arXiv: math .DC/0305102. 2. M. L. Barberis, Hypercomplex Structures on Four-dimensional Lie Groups, Proc. Amer. Math. SOC.128 (4), 1043-1054, (1997). 3. V. DeSmedt, S. Salamon, Anti-selj-dual rnetrics o n Lie groups, Proc. Conf. Integrable Systems and Differential Geometry, Contemp. Math. 308,63-75, (2002). 4. J. E. Snow, Invariant Complex Structures on Four-dimensional Solvable Real Lie Groups, Manuscripta Math. 66,397-412, (1990).

INTEGRABLE GEODESIC FLOWS ON RIEMANNIAN MANIFOLDS: CONSTRUCTION AND OBSTRUCTIONS *

ALEXEY V. BOLSINOV Moscow State University, Faculty of Mechanics and Mathematics, Department of Differential Geometry and Applications, Vorob’evy gory d. 1, 119992 Moscow, RUSSIA E-mail: bolsino&mech.math.msu.su

BOZIDAR JOVANOVIC Mathematical Institute S A N U , Kneza Mihaila 35, 11000 Belgrade, Serbia, YUGOSLAVIA E-mail: [email protected]

This paper is a review of recent and classical results on integrable geodesic flows on Riemannian manifolds and topological obstructions t o integrability. We also discuss some open problems.

Contents 1. Introduction

................................................

.58

1.1. Geodesics on Riemannian manifolds 1.2. Integrable geodesic flows 1.3. Statement of the problem

2. 3.

Classical examples of integrable geodesic flows ............ 62 Topological obstructions to integrability ................... . 6 5 3.1. Case of two-dimensional surfaces 3.2. Topological obstructions in the case of non simply connected manifolds 3.3. Topological entropy and integrability of geodesic flows

4.

Counterexamples

............................................

69

* MSC2000r 70H06, 37535, 53D17, 53D25.

Keywords :geodesic flows, topological entropy, homogeneous spaces, Hamiltonian action of a Lie group, (non-commutative) integrability. 57

58 5.

Geodesic flows on homogeneous spaces and bi-quotients of Lie groups ....................................................... 72 5.1. Non-commutative integrability 5.2. Integrable geodesic flows on G / H and K\G/H

6.

Complete commutative algebras on T * ( G / H )

...............

77

6.1. Mishchenko-Fomenko conjecture 6.2. Integrable pairs

7. 8.

Integrable deformations of normal metrics ................. 81 Methods and examples ..................................... .83 8.1. Argument shift method

8.2. Chains of subalgebras 8.3. Generalized chain method

9. Integrability and reduction ................................. 10. Geodesic flows on the spheres ..............................

.90

.94

1. Introduction 1.1. Geodesics on Riemannian manifolds

We start with basic definitions and the statement of the problem to which our paper is devoted. The first notion is a geodesic line on a manifold. To introduce it, consider the following problem. Imagine a point moving on a tw-dimensional surface in three-dimensional space. What is its trajectory, if there is no external force acting on the point? The classical mechanics gives the following answer: the point will move with velocity of constant absolute value and in such a way that the acceleration (as a vector in R3)is.always orthogonal to the surface. The trajectory of the point is called a geodesic line. It is not hard to verify that such a motion is described by a system of two second-order differential equations. A less obvious fact is that the geodesics are not changed under transformations which do not touch the interior geometry of the surface, or, in the language of differential geometry, the induced Riemannian metric. The other approach to define geodesics uses the interior geometry from the very beginning. Consider the following problem. Given two points on a surface, find the shortest curve on the surface connecting them. Such a curve (if it exists) is just a geodesic. In general, there may exist several such curves or no one. However, if the points are sufficiently close, then the desired curve always exists and is unique. Thus the geodesics can be

59

characterized as locally shortest lines on the surface. This interpretation allows one to define geodesics on an arbitrary manifold endowed with a Riemannian metric. However, the first approach can be applied for the general case as well. Geodesics are the trajectories of points moving by inertia. In other words, the acceleration of a point must be identically zero. One should only explain that in this case by the acceleration we mean the covariant derivative of the velocity vector. Let us recall that such a derivative is naturally defined on every Riemannian manifold and it is called Levi-Civita connection.

It can be easily shown that the geodesics are described by a system of second-order differential equations which can be written in the Hamiltonian form. Let M be a smooth manifold with a Riemannian metric g = ( g i j ) . Consider an arbitrary local coordinate system x l , . . . ,xn and pass from velocities ki to momenta p j by using the standard transformation p j = gijki. Then in the new coordinates x i , p i ( i = 1,.. . ,n) the equations of geodesics read: dxz dH dt dpi '

dpi dt

--

dH

--

dxi '

(1.1)

where H (the Hamiltonian) is interpreted as the kinetic energy:

Here gaj are the coefficients of the tensor inverse to the metric. This system of equations is Hamiltonian on the cotangent bundle T * M (with the standard symplectic form w = C d p i A d z i ) and is called the geodesic flow of the Riemannian manifold ( M ,9). Speaking more precisely, the geodesic flow is the one-parameter group of diffeomorphisms defined by this system of differential equations.

1.2. Integrable geodesic flows We will be interested in the global behavior of geodesics on closed Riemannian manifolds. Namely we want to distinguish the case of the so-called integrable geodesic flows. Before giving the formal definition, we consider some examples. First take the two-dimensional sphere. The geodesics on it are well known: these are equators. All of them are closed and of the same length. Thus,

60

the dynamics of the geodesic flow is very simple. The second example are geodesics on the surface of revolution. In a typical case the behavior of a geodesic is as follows: it moves on the surface around the axis of revolution and at the same time oscillates along this axis. If the geodesic is not closed, then its closure is an annulus-like region on the surface. As we see, the dynamics is quite regular and is a superposition of rotation and oscillation. An analogous picture can be seen on the three-axial ellipsoid. This case is more complicated. Here there are two kinds of annulus-like regions inside which geodesics move. But in the whole, the behavior of geodesics is still regular. Now consider an arbitrary closed surface without any special symmetries (for example, deform a little bit the standard sphere We shall see that the behavior of geodesics lose regularity and becomes chaotic. Regular behaviour is the characteristic property of integrable geodesic flows which are formally defined as follows.

Definition 1.1. The geodesic flow (1.1)is called completely Liouville integrable, if it admits n smooth functions f l ( x , p ) ,. . . ,f n ( x , p ) satisfying three conditions: (1) f i ( x , p ) is an integral of the geodesic flow, i. e., is constant along each geodesic line ( ~ ( tp)( t, ) ) ; (2) f l , . . . ,f n pairwise commute with respect to the standard Poisson bracket on T * M , i.e.,

a

(3)

f l ,...

,f n

ax" a p ,

ax" a p ,

= 0;

are functionally independent on T * M

Remark 1.1. The third condition needs to be commented. The functional independence of the integrals can be meant in three different senses. The differentials of f l , . . . ,f n must be linearly independent: (a) on an open everywhere dense subset, (b) on an open everywhere dense subset of full measure, (c) everywhere except for a piece-wise smooth polyhedron. Instead of these conditions one can assume all the functions to be real analytic. Then it suffices to require tdeir functional independence at least at one point (we will call such a situation analytic integrability).

61

The regularity of dynamics in the case of integrability follows from the following classical Liouville theorem (see [ 41):

Theorem 1.1. Let X" = { fl = c1,. . . ,f n = cn} be a common level surface of the first integrals of a Hamiltonian system. If this surface is regular (i.e., the differentials of fi, . . . ,f n are independent o n it), compact and connected, then (1) X" is difleomorphic to the n-dimensional torus; (2) the dynamics o n this torus is quasi-periodic, i.e., can be linearized in appropriate angle coordinates , . . . ,cpn:

cpl(t)= W l t , . . . ,cpn(t)= writ. Thus, except a certain singular set, the phase space of the system turns out to be foliated into invariant tori with quasi-periodic dynamics. However the dynamics on the singular set may be rather complicated (see below).

1.3. S t a t e m e n t of the problem The general question discussed below can be formulated as follows: Which closed smooth manifolds admit Riemennian metrics with integrable geodesic flows? In other words, we want to divide all manifolds into two classes depending on the fact if there exist or not integrable geodesic flows on them. This question is rather complicated and so far we cannot expect any complete answer. At present there are, in essence, only two general approaches to the problem. The positive answer t o the existence question is strictly individual: we can take a certain manifold (or a certain class of manifolds) and construct on it an explicit example of a metric with integrable geodesic flow. So far there is no other method except the explicit construction. Thus the problem is reduced to studying new constructions of integrable Hamiltonian systems in the particular case of geodesic flows. The second approach deals with topological obstructions to integrability. More precisely, the problem is to find a topological property of a manifold which is not compatible with integrability. Then all the manifold having such a property belong to the second class, i.e., admit no integrable geodesic flows. Notice that the character of first integrals is very important. Usually one consider the integrals of three different types: Cm-smooth, analytic or polynomials in momenta. In the last case on can fix or bound their degrees. Besides, sometimes one has to require some topological restrictions for the

62

structure of the singular set. Each time we deal with a specific problem and obtain a new result, the most important of which we shall try t o mention below. Let us emphasize that in our paper we deal with topological obstructions to integrability only. This means that we are mostly interested in the principal possibility of constructing integrable systems on a given manifold. There is another problem in Hamiltonian dynamics very natural and important, namely, the problem of analytical obstructions to integrability, which can be formulated in a very general way as follows. Given a Hamiltonian function HI is the corresponding Hamiltonian system integrable or not? There are many quite powerful methods to answer this question (see Kozlov [46, 471). One of the most famous and classical is, for example, the PainlevBKovalevskaya test. Among modern approaches to this problem one should mention the papers by Ziglin [ 861, Yoshida [ 84, 851, and Ruiz and Ramis (e.g., see [ 721). Speaking of examples of integrable geodesic flows, in our paper we will confine ourselves to the existence problem. However the complete description of integrable geodesic flows of a certain type is extremely interesting and deep problem. The algebregeometric approach to the classification of integrable geodesic flows has been developed by Adler and van Moerbeke [ 1, 21 (case of SO(4)) and Haine [ 361 (case of S O ( n ) ) .

2. Classical examples of integrable geodesic flows Geodesic flows on two-dimensional surfaces. Classical examples of surfaces with integrable geodesic flows are the twedimensional sphere S2 = +zg + x $ = 1) of constant curvature and the flat torus. Geodesics on the sphere are equators (i.e., sections by the planes passing through its center). Since all geodesics are closed, in this case there are three independent first integrals instead of two. As a ”basis” one can take linear integrals corresponding to infinitesimal rotations of the sphere. For example, the integral corresponding to the rotation about the axis Ox3 has the form

{XI

where p E T,*S2,and 53 = 6’/6’’p is the vector field related to the standard spherical coordinate ‘p on S2: & ( 2 1 , x 2 , 2 3 ) = ( - x 2 , 2 1 , 0 ) . In terms of the tangent bundle these integrals admit the following natural representation

63

as one vector integral:

F ( x , i ) = ( f 1 7 f 2 , f 3 ) = [.,kI where x E S2 c R3 and x E TzS2 are considered as vectors of threedimensional Euclidean space R3. The geometrical meaning of F is clear: this is the vector orthogonal to the plane to which the geodesic belongs. These three linear integrals do not commute, but as two commuting integrals one can take any of them and the Hamiltonian of the geodesic flow

H = (f,”+ fi” + f W . In the case of a flat metric ds2

+

= dxf d x z on the torus, the geodesics are quasiperiodic windings x i ( t ) = ci t ( i = 1 , 2 ) , where x1 mod 27r, x2 mod 27r denote standard angle coordinates. The first commuting integrals are the corresponding momenta p1 and p2. The Hamiltonian of the flow is expressed by pl and p2 in the obvious way: H(xl,x2,p1,p2)= (pf + p i ) / 2 , or more generally, H ( ~ , x 2 , p l , p 2=) ( u p 4 +2bplp2 + c p : ) / 2 . The next classical examples are metrics of revolution and Liouville metrics.

Theorem 2.1. (Clairaut) T h e geodesic flow o n a surface of revolution admits a non-trivial linear integral and, consequently, is integrable. The Clairaut integral has the same form as f 3 in the case of the sphere (if we consider a surface of revolution about the axis 0x3). It admits also the following geometrical description. Let ?c, be the angle between the geodesic and the parallel on the surface of revolution, T be the distance to the revolution axis. Then the Clairaut integral is TCOS?~,. The existence of such an integral is the reflection of the following classical result (Noether theorem): let a Riemannian metric gij admit a one-parameter isometry group @ : M M . Then the corresponding geodesic flow has the linear integral of the form f c ( x , p ) = p ( < ( x ) )where ,

-

is the vector field associated with the one-parameter group 4’ (that is, the corresponding infinitesimal isometry). More generally: if a metric admits an isometry group G, then the corresponding geodesic flow has an algebra of first integrals which is isomorphic to the Lie algebra of the group G.

Theorem 2.2. (Liouville) T h e geodesic flow of the metric

64

admits the non-trivial quadratic integral of the f o r m

and, consequently, i s integrable. This formula, in general, is local. It is not hard to construct an example of a metric on a two-dimensional surface, which admits representation (2.1) in some local coordinates at each point, but on the whole surface, the integrals given by (2.2) cannot be arranged in one globally defined integral. One of such examples is the constant negative curvature metric on a surface of genus g > 1. Locally its geodesic flow admits a quadratic integral (speaking more precisely, there are three independent linear integrals from which one can combine a quadratic one). However, there are no such global integrals (see the section below). Projectively equivalent metrics. There is another interesting class of manifolds closely related to our problem. These are manifolds admitting two projectively (or geodesically) equivalent Riemannian metrics g and i j , that is, metrics having the same geodesics (considered as unparametrized curves) (see Levi-Civita [ 491). If g and i j are in general position, i.e., there exists at least one point at which the operator gij-' has simple spectrum, then the geodesic flows of the both metrics are integrable. Moreover, the first integrals are all quadratic or linear functions. A rather elegant proof of this fact has been obtained by V. Matveev and P. Topalov [52, 791 (it is interesting that the corresponding Hamilton-Jacobi equations admit separation of variables and that the problem can be considered from a biHamiltonian point of view, see [ 181). The open problem is to describe the class of all such manifolds. Geodesics on the ellipsoid. Finally, as one of the most beautiful examples we have the geodesic flow of the standard metric ds2 = ( d s , d s ) on the ( n - 1)-dimensional ellipsoid [ 62, 63, 43, 301

Q = {(x,A-'a:)= 1) c R", A = diag(a1,. . . , a n ) , a1 > a2 > - . . > a,. The problem was solved by Jacobi by separation of variables in elliptic coordinates [ 371. Moser, in his famous papers [ 62, 631, found an L-A pair representation and commuting integrals in Euclidean coordinates by using geometry of quadrics.

65

Theorem 2.3. (Moser, 1980) The functions Fk:

restricted to the tangent bundle T Q { x ,x} Poisson commute and give complete integrability of the geodesic flow on the ellipsoid Q . The geometric interpretation of the integrability is described by the Chasles theorem: the tangent line of a geodesic z ( t ) on Q is also tangent to a fixed set of confocal quadrics Q ( L Y = ~ ){ ( ( A - ai Id)-lx, x ) = l}, cq E R, i = 1,. . . ,n - 2 (for example, see [ 43, 301). 3. Topological obstructions to integrability 3.1. Case of two-dimensional surfaces The first results on topological obstructions to integrability relate to the case of twedimensional surfaces.

Theorem 3.1. (Kozlov, 1979 1451) Two-dimensional surfaces of genus > 1 admit no analytically integrable geodesic flows.

g

The original proof essentially used some delicate properties of analytic functions. But later it was understood that the analyticity condition could be essentially weakened (see Taimanov [ 76, 771). However, it is still not clear if it is possible to omit any additional conditions to the first integrals. Question 3.1. Given a twedimensional surface of genus g > 1, do there exist integrable geodesic flows on it with Cm-smooth integrals? Here is another result related to the same class of surfaces.

Theorem 3.2. (Kolokol‘tsov, 1982 1451) Two-dimensional surfaces of genus g > 1 admit no geodesic flows integrable by means of a n integral polynomial in momenta (the coeficients of this polynomial are assumed to be smooth functions without any analyticity conditions). The idea of the proof of Theorem 3.2 is rather different from that of Kozlov’s theorem. By using the polynomial integral one constructs a certain holomorphic form on the given surface. Then analyzing its zeros and poles one can estimate the genus of the surface.

66

Question 3.2. Is there a multidimensional analog of Kolokol’tsov theorem? In other words, are there topological obstructions to polynomial integrability?

If we confine ourselves to linear and quadratic integrals, then the problem is getting simpler and it is, probably, possible to obtain the complete list of manifolds with linearly and quadratically integrable geodesic flows. The point is that under some additional conditions such Aows admit separation of variables on the configuration space. These variables will have, however, certain singularities and the problem is reduced to studying the topology of manifolds admitting global coordinate systems with special types of singularities. Such an approach has been used by Kiyohara [ 411, but the final answer is not yet obtained. 3.2. Topological obstructions in the case of n o n simply connected manifolds The next fundamental result is due to I. Taimanov [ 76, 771.

Theorem 3.3. (Taimanov, 1987) If a geodesic flow o n a closed manifold M is analytically integrable, then (1) the fundamental group of M is almost commutative (i.e., contains a commutative subgroup of finite index); (2) i f dim H1(M, Q) = d , then H * ( M , Q ) contains a subring isomorphic to the rational cohomology ring of the d-dimensional toms;

(3) zf dim H1(M, Q ) = n = dim M , then the rational cohomology rings of M and of the n-dimensional t o m s are isomorphic.

The idea of the proof is purely topological and the analyticity condition is not essential. In fact, I. Taimanov proved this result under the much weaker assumption that a geodesic flow is geometrically simple. This means that the structure of the singular set where the first integrals are dependent is not too complicated from the topological point of view. Speaking more precisely, to this singular set one should add some new ”cuts” in such a way that the rest becomes a trivial fibration into Liouville tori over a disjoint union of discs. Besides the geometric simplicity condition takes into account some properties of the projection of this ”completed” singular set from the cotangent bundle onto the configuration space (see [ 76, 771).

67

3.3. Topological entropy and integrability of geodesic flows In 1991 Paternain suggested an approach to finding topological obstructions t o integrability of geodesic flows based on the notion of topological entropy [ 67, 681. The topological entropy is a characteristic of a dynamical system on a compact manifold, which measures, in a certain sense, its chaoticity. Since, as a rule, integrable Hamiltonian systems have zero topological entropy, one can proceed as follows. First one may try to estimate the topological entropy of a geodesic flow on a given manifold by purely topological means. Very often one can do that even without any information about the Riemannian metric: if the topology of a manifold is sufficiently complicated (see examples below), then the topological entropy of any geodesic flow has automatically to be positive. The second part of the problem is to prove that the integrability of a geodesic flow implies indeed vanishing of the topological entropy (perhaps under some additional conditions to the first integrals). Recall the definition of topological entropy. Let Ft be a dynamical system on a compact manifold X considered as a one-parameter group of diffeomorphisms. Suppose that we want to approximate this systems up to E > 0 on a segment [ 0, T ] by using only finite number of solutions. In other words, we want to choose a finite number of points xl,. . . ,Z N ( ~ , T )in such a way that for any other point y E X there exists xi satisfying dist (Ft ( y ), Ft (xi))< E for any t E [O,T]. Here dist denotes any metric compatible with the topology of X . Suppose now that N ( E T , ) is the minimal number of such points xi and consider its asymptotics as E -+ 0 and T --+ 00.

Definition 3.1. The topological entropy of the flow Ft is define t o be

ht,(Ft) = lim limsup E+OT-~

In N(E,2')

T

'

The next theorem is the first result showing how the topology of a manifold affect the topological entropy of geodesic flows.

Theorem 3.4. (Dinaburg, 1971 [ 2 6 ] ) I f the fundamental group T ~ ( Mof) M has exponential growth, then the topological entropy of the geodesic flow is positive for any smooth Riemannian metric o n M . In particular, the topological entropy of any geodesic flow on a two-dimensional surface of genus g > 1 is positive.

68

Recall that the topological entropy of a geodesic flow is that of its restriction onto a compact isoenergy surface Q2n-1 = { H ( z , p ) = l} c T * M n , H(z1P ) = IPI2/ 2 .

It is important that the entropy approach works successfully in the case of simply-connected manifolds when Kozlov's and Taimanov's theorems cannot be applied.

Theorem 3.5. (Paternain, 1991 [67, 681) If a simply-connected manifold i s not rationally elliptic, then the topological entropy of any geodesic flow o n it is positive. The rational ellipticity means that the rational homotopy groups of M are trivial starting from a certain dimension N , i.e., n n ( M ) @I Q = 0 for any n > N . The next result not only guarantees the positiveness of the topological entropy for a large class of manifolds, but also allows one to estimate it from below.

Theorem 3.6. (Babenko, 1997 [ 6 ] ) The topological entropy of a geodesic flow o n a simply connected Riemannian manifold ( M ,g ) admits the following estimate

where D h ( M , g ) i s the homology diameter of ( M , g ) . Note that the limit lim SUD n-m

ln(rank r n ( M ) ) n

is equal to zero only for rationally elliptic manifolds, otherwise this is always a positive number. The homology diameter of a manifold depend on the choice of g and is, therefore, a geometric characteristic of a manifold (exact definition can be found in [ 61).

To use the topological entropy as an obstruction to integrability, it was necessary to show vanishing topological entropy for integrable geodesic flows. Under some rather strong additional assumptions this statement holds indeed. Theorem 3.7. (Paternain, 1991 [67, 681) Let ( M ,g ) be a smooth compact Riemannian manifold. Suppose that the geodesic flow o n it is integrable in the class of non-degenerate first integrals. T h e n the topological entropy of this flow vanishes.

69

The non-degeneracy of integrals means the following. First consider the case when x is a singular point for all the first integrals of the system. Then the non-degeneracy is equivalent to the existence of such an integral f that x is non-degenerate in the usual sense of the Morse theory, that is, det

(-)a2f

#O.

axi a x j

If some integrals have non-zero differentials at z, then we make local symplectic reduction by the action of these integrals after which we get to the previous situation. The point x is called non-degenerate for the initial system if it is such for the reduced one. The integrals of a Hamiltonian system are called non-degenerate if every point 5 of a symplectic manifold is non-degenerate. However, the non-degeneracy assumption is rather strong and, in the multidimensional case, holds very rarely. For example, the degeneracy appears in such a natural case as the geodesic flow on the n-dimensional ellipsoid (n 2 3). One of the reasons is the existence of stable degenerate singularities which cannot be avoided by small perturbation and, consequently, are generic.

4. Counterexamples

It has been, however, understood recently that additional assumptions on the first integrals cannot be completely omitted. In other words, in general case neither topological entropy, nor ”complexity” of the fundamental group is an obstruction to integrability of geodesic flows. The first interesting example of an 17exotic71 integrable flow was constructed by L. Butler [ 241.

Theorem 4.1. (Butler, 1998) There is a three-dimensional Riemannian (real-analytic) NIL-manifold ( M 3 ,g ) such that (1) the geodesic flow of the metric g o n M 3 is completely integrable by means of Cm-smooth first integrals (moreover, two of these integrals are real-analytic functions); (2) the fundamental group .rrl(M) is not almost commutative and has

polynomial growth; (3) the topological entropy of the geodesic flow vanishes.

This example shows that in the smooth case the statement analogous to Taimanov’s theorem (Theorem 3.3) fails. Thus, the geometric simplicity

70

assumption introduced by Taimanov is really very important (Butler’s example is not geometrically simple). Besides this is the first example of the situation when a real-analytic flow is integrable, but its integrals cannot be real-analytic functions. The topological structure of M 3 in Butler’s example is quite simple. This is a fibration M 3

T2

S1 over the circle with the torus as a fiber and with the monodromy matrix of the form

A=

(it).

In other words, to reconstruct the manifold one needs to take the direct product T2 x [ 0 , 2 n ]and then to glue its feet T2 x (0) and T2 x (27r) by the linear map given by the matrix A in angle coordinates. Using the idea of Butler, a year later the first author and Taimanov in [ 131 constructed an example of a three-dimensional manifold with integrable geodesic flow having positive topological entropy.

Theorem 4.2. (Bolsinov, Tairnanov, 1999) There is a three-dimensional Riemannian (real-analytic) SOL-manifold ( M 3 ,9) such that ( I ) the geodesic flow of the metric g o n M 3 i s completely integrable by means of Cw-smooth first integrals (moreover, two of these integrals are real-analytic functions); (2) the fundamental group n l ( M ) is not almost commutative and has exponential growth; (3) the topological entropy of the geodesic flow is positive. In this example, M 3 has ”almost the same” structure as in Butler’s example: it suffices to replace the matrix A by an integer hyperbolic matrix, for instance,

The Riemannian metric g on M 3 is described as follows. Let z, y , z be local coordinates on M , where z mod 2n is an angle coordinate on the base S1, and x mod 2 r , y mod 27r are angle coordinates on the fiber T2 . Then

ds2 = dZE where

d-2 s, - a l l ( z )d x 2

+ dZ2,

+ 2 a12(2)d z dy + a22(z)dy2

71

is a flat metric on the fiber T,” over z E S1. The coefficients all,a12, a22 are chosen so that the metric turns out to be smooth on the whole manifold

M. Since the coefficients of the metric do not depend on x and y, the corresponding momenta p , , p , are commuting first integrals of the geodesic flow. The only problem is that p , and p , are not globally defined on M 3 because of non-triviality of the monodromy of the T2-fibration. However, it is possible to construct other functions PI = Fl(p,,p,) and FZ = F2(pz,py) which will be preserved under the monodromy action. As such functions one should take invariants of the action of the cyclic group Z generated by T the linear transformation B-l on the two-dimensional space IR2(pz,py). It is interesting to remark that the orbit structure of this action is such that only one of the invariants is an analytic function. The second can be chosen Cm-smooth, but not real-analytic. The functions F1 and F2 (together with the Hamiltonian) guarantee the complete integrability of the geodesic flow on (M,g). The positiveness of the topological entropy easily follows from Dinaburg’s theorem, but also can be explained directly. The point is that the geodesic flow admits a natural invariant manifold N formed by the geodesics of the form z ( t ) = const, y ( t ) = const, z ( t ) = t. It is easily seen that the union of such geodesics is a submanifold in T * M diffeomorphic to the base M . The geodesic flow q5t restricted onto N preserves the T2-fibers. Moreover the 2r-shift along the flow transforms each fiber into itself q5t=2K : T2 T2 and coincides with the hyperbolic automorphism of the torus given by the matrix B. It is well-known (as one of basic examples) that the topological entropy of such an automorphism is positive and equals to lnX, where X is the maximal eigenvalue of B (see, for example, [ 401). Thus the geodesic flow admits a subsystem with positive topological entropy and, consequently, possesses this property itself.

-

The constructions of Theorems 4.1, 4.2 are naturally generalized to the case of arbitrary dimension [ 14, 251. Note that the topological structure of the singular set in these examples is quite simple. This is a finite polyhedron whose strata are not just only smooth but also real-analytic submanifolds. ”Non-analytic” is the way of how Liouville tori approach this singular set.

It is worth to explain why the above geodesic flows are not geometrically simple in the sense of Taimanov [ 76, 771. The point is that the base of the foliation into Liouville tori is not simply connected. To make it such

72

(as required in the definition of geometric simplicity) we need additional cuts of the base. But this is impossible to do by a "geometrically simple" way: in each tangent space such a cut will consist of infinitely many t w e dimensional planes. The above examples lead to the following questions:

Question 4.1. Which additional properties of the first integrals guarantee vanishing the topological entropy? Question 4.2. Is analyticity such a condition? In other words, is the topological entropy an obstruction to analytic integrability? Question 4.3. Is it possible t o include an arbitrary (as chaotic as one wishes) dynamical system as a subsystem into an integrable Hamiltonian system of higher dimension?

5. Geodesic flows on homogeneous spaces and bi-quotients of Lie groups Since the integrability is closely related to the existence of some symmetries (possibly hidden), for the construction of multi-dimensional integrable examples, as a rule we should use metrics with large symmetry groups. In the final construction the symmetry can be removed by algebraic modification of metrics. The classical example is the geodesic flow of a left-invariant metric on the Lie group SO(3). The geodesic flow of such a metric describes the motion of a rigid body about a fixed point under its own inertia. This problem was solved by Euler. In general, the geodesic flow of a left-invariant metrics on a Lie group G after G-reduction reduces to the Euler equations on g* (the dual space of the Lie algebra g), which are Hamiltonian equations with respect to the LiePoisson bracket on g* [ 41.

A multidimensional generalization of the Euler case has been suggested by Manakov [ 511. Using his idea, Mishchenko and Fomenko proposed the argument shift method (see below) and constructed integrable examples of Euler equations for all compact groups [ 541 and proved the integrability of the original geodesic flows [ 55, 561.

Theorem 5.1. (Mishchenko, Fornenko, 1976) Every compact Lie group admits a family of left-invariant metrics with completely integrable geodesic flows.

73

There are many other important constructions on various Lie algebras (we mention just some of the review papers and books [ 5, 10, 31, 32, 701). In particular, the problem of algebraic integrability of geodesic flows on SO(4) and S O ( n ) is studied in [ 1, 2, 361. Throughout the paper, by a normal G-invariant Riemannian metric on the homogeneous space G I H of a compact group G, we mean the metric induced from a bi-invariant metric on G.

Theorem 5.2. (Thimrn 1981,Mishchenko 1982) Geodesic flows of normal metrics o n compact symmetric spaces G / H are completely integrable. These results are generalized by Brailov, Guillemin and Sternberg and Mikityuk. Brailov applied the Mishchenko-Fomenko construction to the symmetric spaces and obtained families of non G-invariant metrics with integrable flows on symmetric spaces [ 21, 221. Guillemin and Sternberg [ 33, 341 and Mikityuk [ 591 described the class of homogeneous spaces G I H on which all G-invariant Hamiltonian systems are integrable by means of Noether integrals [ 33, 34, 591. It appears that in those cases (G, H ) is a spherical (or Gelfand) pair. If G I H is a symmetric space then (G, H ) is a spherical pair, but there exist spherical pairs which are not symmetric. Note that, for G compact, (G, H ) is a spherical pair if and only if G / H is a weekly symmetric space (see [ 81 ] and references therein). Examples of homogeneous, non (weekly) symmetric spaces GIH with integrable geodesic flows are given by Thimm [ 781, Paternain and Spatzier [ 691 and Mikityuk and Stepin [ 611. Also, to this list now we can add the above mentioned examples constructed by Butler [ 241 and Taimanov and the first author [ 13, 141. Particular examples of non-homogeneous manifolds (bi-quotients of compact Lie groups) with integrable geodesic flows are obtained by Paternain and Spatzier [69] and Bazaikin [ 71. It appears that many of those results can be considered together within the framework of non-commutative integrability. This approach allowed us to obtain general theorems on integrability of geodesic flows on homogeneous spaces and bi-quotients of compact Lie groups [ 15, 161.

5.1. N o n - commutative integrability There are a lot of examples of integrable Hamiltonian systems with n degrees of freedom that admit more than n (noncommuting) integrals. Then

74

n dimensional Lagrangian tori are foliated by lower dimensional isotropic tori (sometimes such systems are called superintegrable). The concept of non-commutative integrability has been introduced by Mishchenko and Fomenko [ 55, 561 (see also [ 64, 20, 311). Let M be a 2n-dimensional symplectic manifold. Let (3, {., .}) be a Poisson subalgebra of ( C " ( M ) , {., .}). Suppose that in the neighborhood of a generic point x we can find exactly 1 independent functions f l , . . . ,fi E 3 and the corank of the matrix {fi, fj} is equal to some constant r . Then numbers 1 and r are called differential dimension and differential index of 3 and they are denoted by ddim 3 and dind 3, respectively. The algebra 3 is called complete if: ddim 3

+ dind 3 = dim M .

In fact, this definition is equivalent to any of the following three conditions. Take a generic point x and 1 independent functions small neighborhood U of x . Then

f1,.

. . ,fi

E

3 in some

. . . ,fi in U are isotropic. (ii) The subspace generated by d f l ( x ) ,. . . ,d f i ( x ) is coisotropic in TZM. (iii) If there is a function f which commute with 3 on U then (i) Common level sets of

f=f(f17"'7fi)

On

fl,

"

In particular, we shall say that the algebra 3 i s complete at x if one of the conditions (i-iii) is satisfied. The Hamiltonian system 2 = X H ( X )is completely integrable in the noncommutative sense if it possesses a complete algebra of first integrals F. Then each connected compact component of a regular level set of the functions f l , . . . ,fi E F is an r-dimensional invariant isotropic torus T ' (see [ 55, 64, 56, 20, 311). Similarly as in the Liouville theorem, in a neighborhood of T ' there are generalized action-angle variables p , q, I , 'p, defined in a toroidal domain 0 = T'{cp} x B,{ I , p , q } , .L

such that the symplectic form becomes w = C;='=, d I i A d q i + C f = , dpi Adqi, and the Hamiltonian function depends only on I1, . . . ,I,. The Hamiltonian equations take the following form in action-angle coordinates: $51 = Wl(I)=

aH

-,

a11

. . . , CpT

= WT(I)=

aH 81,

-,

.

I =p

= q = 0.

75

Example 5.1. Consider a Riemannian manifold (Q, d s 2 ) with closed geodesics: for every 5 E Q, all geodesics starting from 5 return back to the same point. Then one can find a complete algebra of integrals 3 with dind 3 = 1. In other words, the geodesic flow is completely integrable in non-commutative sense [ 161. Note that the concept of noncommutative integrability can be naturally extended to Poisson manifolds ( N ,{., .}). The algebra 3 is complete if ddim 3 + dind 3 = dim N

+ corank {., .},

i.e., the restriction of F to a generic symplectic leaf M of N is complete on M . Also, if a Hamiltonian system x = X H ( Z )on N possesses a complete algebra of first integrals 3,then (under compactness condition) N is almost everywhere foliated by (dind 3 - corank {., .})-dimensional invariant tori with quasi-periodic dynamics. 5.2. Integrable geodesic flows o n G / H and K\G/H Let a connected compact Lie group G act on a 2n-dimensional connected symplectic manifold ( M , w ) . Suppose the action is Hamiltonian, i.e., G acts on M by symplectomorphisms and there is a well-defined momentum mapping @ : M g* (g* is the dual space of the Lie algebra g) such that oneparameter subgroups of symplectomorphisms are generated by the Hamiltonian vector fields of functions fc(z) = @(x)(c), 6 E g, z E M and ~ I ~ =~ {fcl, , ~fc,}. ~ IThen @ is equivariant with respect to the given action of G on M and the co-adjoint action of 6 on g*. In particular, if p E @ ( M )then the co-adjoint orbit O G ( ~belongs ) to @ ( M ) .

-

Consider the following two natural classes of functions on M . Let 3 1 be the set of functions in C m ( M ) obtained by pulling-back the algebra Cm(g*) by the moment map .Fl = @*C"(g*). Let 3 2 be the set of G-invariant functions in C"(M). The mapping h H f h = h o @ is a morphism of Poisson structures: {fhl(4,fh*(4}

=

{hl (P)42(P)Ig*, P = a(.),

where {., . } O * is the Lie-Poisson bracket on g*: { . f ( P M P ) } 0 * = P ( [ d f ( P ) , d g ( P ) l ) , f , g : B*

-

R.

Thus, 3 1 is closed under the Poisson bracket. Since G acts in a Hamiltonian way, 3 2 is closed under the Poisson bracket as well. In other words, 3 1 and 3 2 are Lie subalgebras in C" ( M ) .

76

The second essential fact is that h o @ commute with any G-invariant function (the Noether theorem). In other words: { . F l , 3 2 } = 0. The following theorem, although it is a reformulation of some well known facts about the momentum mapping (see [ 35, 50]), is fundamental in the considerations below. This formulation was suggested by A. S. Mishchenko.

Theorem 5.3. The algebra of functions 3 ddim

(31

+

3 2 )

+ dind

1

+ 3 2 i s complete:

(31 f 3 2 )

= dim M .

Let A c Cm(g*) be a Lie subalgebra. By A we shall denote the pull-back of A by the moment map: A = @ * ( A )= { j h = h o @, h E A } . Let I3 be a subalgebra of 3 2 .

Corollary 5.1. ([ 161) ( i ) A 3 2 is a complete algebra o n M i f and only i f A is a complete algebra o n the generic orbit O c ( p ) c @ ( M )of the co-adjoint action. (ii) If I3 i s complete (commutative) subalgebra and A i s complete (commutative) algebra o n the orbit 0 ~ ( p )for , generic p E 4 ( M ) then A I3 is complete (commutative) algebra o n M .

+

+

Now, let M be the cotangent bundle T*(G/H) with the natural G action. By dsi we shall denote the normal G-invariant Riemannian metric on the homogeneous space GIH, induced by a bi-invariant metric on G. The Hamiltonian of the metric d s i Poisson commute with 3 from Theorem 5.3 we get

1

and F2. Thus,

Theorem 5.4. (1151) Let G be a compact Lie group. The geodesic flows of normal metrics dsg o n the homogeneous spaces G I H are completely integrable ( i n non-commutative sense). Remark 5.1. The dimension of invariant isotropic tori T*G/H is dimpr,(ann,(v)) (see Remark 6.1 in the next section). Note that for 3 1 and 3 2 we can take analytic functions, polynomial in momenta. For example, in the case of Lie groups, these functions are polynomials on g* shifted to T*G by right and left translations, respectively.

A similar construction can be applied to bi-quotients of compact groups. Consider a subgroup U of G x G and define the action of U on G by: (917 92) '

= 91 g T 1 i

(917 92) E

u,

E G.

77

If the action is free then the orbit space G//U is a smooth manifold called a bi-quotient of the Lie group G. In particular, if U = K x H, where K and H are subgroups of G, then the bi-quotient G//U is denoted by K\G/H. A bi-invariant metric on G, by submersion (see section 9), induces a normal Riemannian metric dsg on G / / U . Note that every bi-quotient G//U is canonically isometric to AG\G x G/U, for a certain normal metric on AG\G x GIU. Here AG denotes the diagonal subgroup (see [ 821). On T*(K\G/H) there exist algebras B1 and .F2 analogous to the above algebras on homogeneous spaces. For 3 1 we take all polynomials on g* invariant with respect to the K-action and extend them to right invariant functions on T*G. These functions give well defined functions on T*(K\G/H) since they are invariant with respect to the K x H-action on T'G. Similarly, for .F2 we take all polynomials on g* invariant with respect to the H-action, extend them to left invariant functions on T*G and consider as functions on T*(K\G/H). (When K is trivial these algebras are precisely the algebras described above). It is clear that in such a way we obtain integrals of geodesic flow of dsg.

Theorem 5.5. ([16]) The algebra of functions 3; ddim

(31

+ 3 2 is complete:

+ 3 2 ) + dind (F1+3 2 ) = dimT*(K\G/H)

and, therefore, the geodesic flows of dsg o n the bi-quotient K \ G / H is completely integrable (in non-commutative sense). 6. Complete commutative algebras on T * ( G / W ) 6.1. Mishchenko-Fomenko conjecture Mishchenko and Fomenko stated the conjecture that non-commutative integrable systems x = XH(Z)are integrable in the usual commutative sense by means of integrals that belong to the same functional class as the original non-commutative algebra of integrals [ 55, 311. Note that, locally, noncommutative integrability always implies commutative integrability. For example, we can take commuting functions { 11, . . . ,IT,p:' q:' , . . . ,p z q:}.

+

+

When B = spanw{fi,.. . ,f i } is a finite-dimensional Lie algebra, this conjecture is proved for compact manifolds M by Mishchenko and Fomenko and for non-compact manifolds (under the assumption that all isoenergy levels H-'(h) are compact) by Brailov. Then the commuting integrals can be taken as polynomials in f 1 , . . . ,f i (see [ 56, 21, 31 1).

78

Recently, the conjecture has been proved in Coo-smooth case. This means that the r-dimensional invariant isotropic tori T ' can be organized into larger, n-dimensional Lagrangian tori T" which are the level sets of the commutative algebra of smooth integrals (91,.. . ,gn}. The point is that local commuting integrals { I l ,.. . ,I , , p : + g f , . . . ,p z + q ; } defined on different toroidal domains can be glued smoothly on the whole manifold [ 161. If 3 is any algebra of functions on symplectic (or Poisson) manifold, then we shall say that A c 3 is a complete subalgebra if ddim A

+ dind A = ddim 3+ dind 3.

Thus, the conjecture can be stated as follows. Let 3 be a complete algebra on a symplectic (or Poisson) manifold. Then one can find a complete commutative subalgebra A of 3,i.e., commutative subalgebra with differential dimension 1 ddim A = - (ddim F dind 3). 2

+

Here instead of 3 we usually have to consider its functional extension 3, that is, the algebra formed by the functions h = h(f1,f2,. . . ,fi), fi E 3, where h is polynomial, real-analytic or Cm-smooth depending on the class of functions we want to work with. 6.2. Integrable pairs

We turn back to the geodesic flows on homogeneous spaces. We have shown that the non-commutative integrability implies the classical commutative integrability by means of Coo-smooth integrals [ 161. Thus, a more delicate problem remains: the construction of complete commutative algebras of integrals of dsg that are polynomial in momenta. Let A be a complete commutative algebra on generic orbits in @ ( T * ( G / H ) ) and let B be a complete commutative subalgebra of 3 2 . Then, according to Corollary 5.1, A + I3 is a complete commutative algebra on T * ( G / H ) . There is a well known construction, called the argument shift method [ 541, which allows us to obtain a complete commutative family of polynomials on every coadjoint orbit of a compact group. For regular orbits this is proved by Mishchenko and Fomenko [ 541. For singular orbits there are several different proofs: by Mikityuk [ 571 , Brailov [ 21 ] and Bolsinov [ 91 (note that iP(T*(G/H))can be often a subset of singular set in g*). Thus, to construct a complete commutative algebra of functions on T * ( G / H ) we

79

need to find a complete commutative subalgebra 23 of G-invariant functions on T*(G/H). For symmetric and weekly symmetric spaces (spherical pairs) the algebra F2 is already commutative. In a neighborhood of a generic point x E T*(G/H) each G-invariant function f can be expressed as f = h o CP and thus we can use just functions from Fl to get the integrability of any G-invariant geodesic flow on G/H. Spherical pairs for G simple and semisimple are classified in [ 481 and [ 591, respectively. To discuss the general case, one first needs to describe the structure of the algebra 3 2 . Let us fix some bi-invariant metric on G, i.e., Adc-invariant scalar product ( ) on g. We can identify g* and g by ( ., - ) and T*(G/H) and T(G/H) by the corresponding normal metric dsg. Let g = lj D be the orthogonal decomposition of g, where lj is the Lie algebra of H . Then G-invariant functions on T*(G/H) are in one-to-one correspondence with AdH invariant polynomials on D. Within this identification, the Poisson bracket on T*(G/H) corresponds to the following bracket on R[ D ] ~ s l .

+

where R[DIH denote the algebra of AdH-invariant polynomials on D (see Thimm [ 781). We have ddim R[ D ]

= dim D

-

dim lj + dim anng (v),

dind R[oIH = dimpr,(ann,(v))

(6.2)

= dimann,(v) - dimanng(v)

(6.3)

for generic v E b [ 151. Here ann,(v) and anng(v) denote the annihilators of v in g and g respectively: anng(v>={rIEEl,[77,~1=O},

anng(v)={VE b,[77,v] = O } .

By genericity of v E D we mean that the dimensions of anng(w) and anng (v) are minimal. From (6.3), the condition that a commutative subalgebra 23 complete can be rewritten in the following form:

c R[ aIH

is

1 1 d d i m B = -(ddimR[aIH + d i n d R [ b J H ) dim^- - d i r n O ~ ( v ) , (6.4) 2 2

for a generic v E D, where Oc(v) is the adjoint orbit of G.

Remark 6.1. Every Casimir function of the algebra of G-invariant functions 3 2 in a neighborhood of a generic point x E T*(@/fj)is of the form

80

h o a, where h is a local invariant of the (co)adjoint representation. Therefore dind

(F1+F2)= dind F1 = dind 7

2 = ddim

(FI n F2)= dind R[ o I H .

In particular, the phase space of the geodesic flow of a normal metric is foliated by dimpr, (ann, (v))-dimensional isotropic tori.

Remark 6.2. There is a nice geometrical description of the algebra R[ D IH. We can pass from o to the orbit space b/H, with respect to the natural adjoint action of H on o. Denote by [ w ] the orbit of the AdH-action through v. It is clear that one can consider R[ oIH as the algebra R[ o/H] of functions on the orbit space, with respect to {., .}' - the reduced bracket of (6.1). Note that o/H is not smooth. However, in a neighborhood of a generic point, it is a smooth manifold of dimension (dimo - dim b dimanng(v)) (minimality of dimensions of anng (w)and anng(w) means exactly that [ v ] is a smooth point and the bracket {., .}' has maximal rank at [ v]). Now, the completeness of B as a subalgebra in F2 is equivalent to the condition that B is a complete algebra on the "singular" Poisson manifold (o/H, {-,.}').

+

Remark 6.3. The number (ddim R[ oIH - dind R[ t ) I H ) / 2is equal to the codimension S(G@,H') of maximal dimension orbits of the Bore1 subgroup B c G@in the complex algebraic variety G@/H@(see [SO]). The number b(G", H") is called the complexity of G@/H@[ 661. Example 6.1. Following [61, 601, we shall say that (G,H) is an almost spherical pair if ddim R[ oIH = 2

+ dind R[ o l H ,

(6.5)

or equivalently, if the complexity of G'/H' is equal to one. They are classified, for G compact and semisimple in [ 66, 61 1. As complete commutative algebra on T*(G/H) we can take arbitrary complete commutative subalgebra A c 3 1 and one G-invariant function functionally independent of FI(see Mikityuk and Stepin [ 61 I). The examples of Thimm [ 781 ( ( S O ( n )SO(n , - 2))) and Paternain and Spatzier [ 691 ( ( S U ( 3 )T2)) , are almost spherical pairs. In our notation, as a complete algebra B c R[ 0 I H we can take all the Casimir functions and an arbitrary non Casimir function in w[D]*.

Definition 6.1. We will call (G,H) an integrable pair, if there exists a complete commutative subalgebra f? in R[ b I H .

81

We can summarize the above considerations as follows. Theorem 6.1. If (G,H ) is an integrable pair then the geodesic flow of a normal metric dsg i s completely integrable in the commutative sense by means of analytic integrals, polynomial in velocities. Therefore, the Mishchenko-Fomenko conjecture for the non-commutatively integrable geodesic flow of the metric dsg on G / H can be stated as follows. Conjecture 6.1. All pairs (G,H ) are integrable. Spherical and almost spherical pairs (G, H ) are simplest examples of integrable pairs. In the section 8, following [ 15, 171, two natural methods for constructing commutative families of AdH invariant functions, namely the shift-argument method and chain of subalgebras method, are presented. In many examples (Stiefel manifolds, flag manifolds, orbits of the adjoint actions of compact Lie groups etc.) we have proved that those methods lead to complete commutative algebras. 7. Integrable deformations of normal metrics

Besides relation with the Mishchenko-Fomenko conjecture, we shall explain how, with a help of commuting integrals, one can construct new integrable geodesic flows on homogeneous spaces. Submersion metrics. Let A be a complete commutative algebra on a generic orbit in (a(T *( G / H ) )Then, . according to Corollary 5.1, A 3 2 is a complete algebra on T * ( G / H ) .Let h c ( e ) = ( C ( 5 ) , 5 ) / 2be a quadratic positive definite polynomial in A. Then h c o (a is the Hamiltonian of the geodesic flow of a certain metric that we shall denote by dsg. The metric d s g has the following nice geometrical meaning. This is the submersion metric of the right-invariant Riemannian metric on G whose Hamiltonian function is obtained from h c ( c ) by right translations. The geodesic flow of d s g is completely integrable since hc o CP commutes with every function from A 3 2 . The dimension of invariant tori is equal to

+

+

dind ( A

+ 3 2 ) = dim OG( w ) + dim pr, (anng( w ) ) ,

for a generic w E t~ (see [ 151). Notice that the argument shift method [ 541 (see below) always allows us to construct a commutative subalgebra A c 3 1 which contains non-trivial quadratic functions. Thus some integrable (in non-commutative sense) deformations of d s i always exist.

82

This construction for symmetric spaces is done by Brailov [ 21, 22 1. G-invariant metrics. Let B be a complete commutative subalgebra of .F2. Let hG E B be a G-invariant function, quadratic in momenta and positive definite. Then hG can be considered as the Hamiltonian of the geodesic flow of a certain G-invariant metric ds:. The geodesic flow of ds; is completely integrable since it admits the complete algebra of first integrals .Fl + B (see Corollary 5.1). The dimension of invariant tori is equal to dind (F1

+ B) = dind B = dim

1 2

- - dim OG(v),

for a generic v E b. Non-invariant metrics. Let h c o @ and hG be as above and let HA,,xz = A1 hc o @ A2 h~ be positive definite. Then HA,,xz is the Hamiltonian of the metric which we shall denote by ds:l,xz. The family of metrics ds:l,xz has completely integrable geodesic flows, but with no obvious symmetries. The phase space T*(G/H) is then foliated by invariant Lagrangian tori that are level sets of the complete commutative algebra of functions A B.

+

+

Example 7.1. The most natural and simplest example of integrable deformations is as follows. Consider a compact Lie group G as a homogeneous space. The geodesic flow of the biinvariant metric dsg is completely integrable in non-commutative sense and the dimension of invariant isotropic tori in T"G is equal t o indG. The first integrals of the flow are all leftand right-invariant functions on T*G = TG. To obtain integrable deformations of dsg one should consider a complete commutative subalgebra B c F2,where 3 2 is the algebra of left-invariant functions. Such a subalgebra can be constructed by the argument shift. Moreover one can describe all quadratic functions in B (see [ 541) in the following way. Consider the h e a r operator Ca,b,D: g --$ g defined by

+

where IG = 2 1 ~ 2 5 ,2 E t, x1 E t', t c g is a Cartan subalgebra, a , b E t and a is regular, D : t --t t is an arbitrary operator. Then we consider the quadratic form ( C a , b , ~ ( z ) , x on ) / 2g and extend it to the whole tangent bundle TG by left translations. As a result we obtain a quadratic function hf,& which can be considered as the Hamiltonian of a left invariant metric on G. Its geodesic flow will be integrable and the algebra

83

of integrals consists of two parts: B (commutative part) and 3 1 (noncommutative part which consists of all right-invariant functions). For such flows, the dimension of invariant isotropic tori in T* = T G will be equal to (dim G ind G)/2. But we can continue this deformation procedure by choosing a complete commutative subalgebra A c Fl. This can be done just in the same way as for F2.As a result we shall obtain a right-invariant right quadratic function ha,, which also gives an integrable geodesic flows

+

+

right

on G. But now we can take the sum h::; ha,,,,,,, which also gives an integrable geodesic flow whose algebra of integrals A B is commutative and, consequently, invariant tori are Lagrangian, i.e., of dimension dim G. 1

,

+

8. Methods and examples

8.1. Argument shij3 method Let R[ gIG be the algebra of Adc-invariant polynomials on g. Mishchenko and Fomenko showed that the polynomials

A, = {pt = p ( . + xu), A

E R, p E R[gIG)

obtained from the invariants by shifting the argument are all in involution [ 54 1. Furthermore, for every adjoint orbit in 8,one can find a E g, such that A, is a complete involutive set of functions on this orbit. For regular orbits it is proved by Mishchenko and Fomenko [ 541. For singular orbits there are several different proofs by Mikityuk [ 571, Brailov [ 211 and Bolsinov [ 91. Thus, as was already mentioned, the argument shift method allows us to construct a complete commutative subalgebra in 3 1 . Now we want to use it to construct such a subalgebra in Fz. By B, denote the restriction of A, to 0:

B a = { p , X ( v ) = p ( v + A a ) , X E R , P E I W [ ~ ]ED) ~,

It can be easily checked that if H is the subgroup of the isotropy group G,: H c G, = { g E G, Ad,(a) = a } , (i.e., c anng(a)) for some a E g then B, will be an algebra of AdHinvariant polynomials. Also, we have the following simple observation. If f l and f 2 are in involution {fl,fi}, = 0 and their restrictions to D: pl = fllo, p2 = fito are AdH-invariant; then {p1,p2}o = 0. Thus, B, is a commutative subalgebra of R[ D ] ~ . In order to estimate the number of independent functions, obtained by shifting the argument, we look at the algebra B, from the point of view of

84

the bi-Hamiltonian system theory. This approach gives us a possibility to use, in particular, the completeness criterion proved by the first author in

[91. Let {., .}I and {., .}2 be compatible Poisson structures on a manifold N . Compatibility means that any linear combination of {., .}I and {-,-}2 with constant coefficients is again a Poisson structure. So we have a family of Poisson structures: R={X1{-,.}1+X2{.7.}2,

X17X2

ER, X q + X i # o } -

By r denote the rank of a generic bracket in A. For each bracket {., .} E A of rank r , we consider the set of its Casimir functions. Let be the union of these sets. Then 3 A is an involutive set with respect t o every Poisson bracket from A. Together with A, consider its natural complexification A'= { X 1 { ~ , . } l + X 2 { . , . } 2 , X1,Xz E @, IX112+IX212 # 0). Here, for {-,.}in A', we consider {., .}(x) as a complex valued skew-symmetric bilinear form on the complexification of the co-tangent space (Tz")@.

Theorem 8.1. (Bolsinov, 1989[9]) Let {.,.} E A and rank {.,.}(x)= r . is complete at x with respect to {., .} i f and only i f Then rank {., -}'(x) = T for all {., .}' E A'. Note that if 3 A is complete at x then it is complete in some neighborhood U of x, or, if functions FA are analytic, in N . Now, let us pass from D t o the orbit space D/H (see remark 6.2). It is easy to see that the algebra W[oIH is closed with respect to the a-bracket defined by

{f(x),dx)}a = ( a , [ V d x ) ,V f ( x )I ). That is why {., .}a induces a Poisson bracket {-,.}'a on the orbit space. Moreover, it is well known that the Lie-Poisson brackets and a-brackets are compatible on g [9, l o ] . Thus, the brackets {.,-}' and {.,.}; are also compatible. Notice that p ( .+A a), where p is an Adc-invariant, is a Casimir function of the Poisson bracket {., .}' A{., .}'a. Using the above criterion (Theorem 8.1) we have found the following conditions sufficient for Ba to be complete.

+

Theorem 8.2. ( [ 1 7 ] ) Let H = G, for some a E g. Suppose that there exists generic u E D@ such that:

(Cl) (C2)

+

dim anngc(u Xu) = dim anngc(u), for all X E @, dimanngcprbc([ad;'u,u]) = dimann,c(u).

85

T h e n B, i s a complete commutative subalgebra in (G,G,) i s a n integrable pair.

R [ D ] ~In. particular

The homogeneous space GIG, is the adjoint orbit O G ( a ) of the G-action on g. If a is regular in g then G, is a maximal torus and GIH is usually called a flag manifold. In this simplest case conditions (Cl) and (C2) can be easily verified.

Corollary 8.1. (Bordemann 1191,see also 1151) Let H be a maximal tom s in a compact connected Lie group G. If a € g i s regular, then B, i s a complete commutative subalgebra in R[ D ] ~ .I n particular (G,H ) i s an integrable pair. We think that conditions (Cl) and (C2) hold for all compact Lie groups and for each a E g. In [ 171 we have verified them for U ( n ) ( S U ( n ) ) and , then (joint work with E. Buldaeva [as])for SO(n) and Sp(n).

Theorem 8.3. Let G be a classical compact simple Lie group (SO(n), S U ( n ) or S p ( n ) ) and O G ( a ) be a n arbitrary adjoint orbit of G. T h e n (G,G,) i s an integrable pair and B, i s a complete commutative subalgebra in R[ w I H .

-- -

For example, if we take G = U ( n ) and a E u(n) of the form

a

= diag(a1,. . . , a,) = diag(cr1,. . . , 1x1, 1x2,.. . ki

then O v ( , ) ( a ) = U ( n ) / U ( k I )x is an integrable pair.

k2

,1x2,.. . ,a,,. . . ,a,) kv

. . . x U ( k , ) and ( U ( n ) ,U ( k l )x . . . x U ( k , ) )

+

Pairs ( U ( n ) ,U ( k 1 ) x U(k2) x . . . x V ( k , ) ) are integrable for kl . . +k, < n as well. Indeed, there are commutative subalgebra t c D and diagonal matrix a such that u(lc1) . .+u(k,) t = annu(,)(a). Let R[ t ] be the algebra of polynomials on t considered as functions on D. Simple calculations show that B,+R[ t ]is a complete commutative algebra of AdU(kl)x...xU(k,) invariants. Just in the same way we can prove that if the pair (G,G,) is integrable and if H c G, is a normal subgroup such that G,/H is commutative, then ( G ,H ) is an integrable pair as well.

+.

+

8.2. Chains of subalgebras Trofimov and Thimm devised a method for constructing functions in involution on a Lie algebra g by using chains of subalgebras [ 80, 781. The

86

idea is very simple, but important. The invariant polynomials commute with all functions on g. If we have some subalgebra g1 c g, then invariant polynomials on g1 (naturally extended to the whole g) commute between themselves, but also with invariant polynomials on g. By induction, we come to the following construction. Suppose we are given a chain of connected compact subgroups

G1 c Gz

c ... c G,

= G,

and the corresponding chain of subalgebras in g: 81 c 82

c ... c g7l = g

(8.1) Let Ai be the algebra of invariants on gi considered as a subalgebra in R[ g 1. Then A1 . . . A, is a commutative subalgebra of polynomials on g [ 80, 781.

+ +

Example 8.1. The natural filtrations so(2) c so(3) c . . . c 5 0 ( n ) and u(1) c 4 2 ) c ... c u ( n ) lead to complete commutative algebras on 50(n) and u(n)respectively (see Thimm [ 781).

+

+

Now, if we have H C GI then polynomials in A1 ... A, are AdHinvariant. Therefore B = ,131 . . . B, is a commutative subalgebra of W[t l I H , where Bi is restriction of Ai to tl. With respect to (8.1) we have the orthogonal decomposition

+

tl = tll

such that gi

= Ij

+

+ tl2 + . . . + on,

+ tl1 + ... + tli. Let us define T I , . . .

,T,

by:

max dimpr,,span{Vp(v), p E R[giIGi}. vEtll+...+ai Let R = T I . . T,. It is clear that the number of independent functions in B is greater or equal to R. Hence, if R = dim tl - dim 0 ~ ( v then ) B is complete. An algorithm for computing numbers ~i is given by Bazaikin ~i

=

+. +

3

[71. In the next theorem we give examples of some integrable pairs with complete algebras B c R[ t l I H obtained by using chains of subalgebras. Theorem 8.4. (1171)

( S O ( n )W , k l ) x SO(k2)), ( U ( n ) ,U ( 1 P x U ( k 2 ) x U ( k 3 ) ) , ( U ( n ) ,S O ( k ) ) , ( S O ( n 1 )x S O ( n z ) ,W k ) ) , ( U ( n 1 )x U ( n 2 ) ,U ( k ) ) ,

2 0 , kl + k2 5 n kl,k 2 , k 3 2 0, kl + kz + k3 5 n k l , k2

k s n k I n1,n2

k I n1,722

87

are integrable pairs. In the last two examples S O ( k ) and U ( k ) are diagonally embedded into SO(n1) x SO(n2) and U(n1) x U(n2) respectively.

Example 8.2. As an example we shall indicate the chain for Stiefel manifold SO(rn)/SO(k)(see [ 151):

+

61 = 50(k 1) c 6 2 = 50(k . . ri=z, z = l,...,k,

+ 2) c . . .

6m-k

c 50(m),

Example 8.3. In Theorem 8.4, we consider naturally embedded subgroups (as block matrices). However, the same construction can be applied to some other embeddings. As an example, let us consider the so-called AloffWallach spaces M ~ =J sU(S)/Tk,I [ 31, where

Tk,l=

{(

e2~ikB 0 0 e27ri10

0

0

i

0 0

e E R IC,I E Z, 1 1 ~ 1 + Iq # 0 , ki > 0.

e-27ri(k+I)B

),

Among the spaces Mk,[ there are infinitely many with different cohomological structures: if k,1 are relatively prime, then H4(Mk,1,Z) = Z / r Z , with r = Ik2 + l2 kll (see [ 31).

+

Consider the following chain of subgroups Go = Tk,lC G I C G2 = S U ( 3 ) , where

GI = Let tt = 01

{ (;

de;g-l)

E SU(3), g E

1=

U(2)

U(2).

+ t ~ 2be the orthogonal decomposition a s above ia -Z12

0

-213

0 ib 0 0 - i ( a + b)

212

-223

, ka + lb + (k + l ) ( a+ b) = 0

0

The algebra of functions B = B1 +B2 is complete. Indeed, ,131 is generated by f i = a+b and f2 = (WI,W I ) (r1 = a), t32 is generated by f 3 = (w1+w2, wl+v2) and f 4 = t r ( q w ~ (7-2) =~ 2 ) and 2 2 = 4 = dim o - 0su(3) (w)/2 = 7 - 3.

+

+

88

Aloff and Wallach proved that the SU(3)-invariant Riemannian metrics ds: on M ~ ,obtained J from the quadratic forms

B,(v7W)=(1+t)(vl,vl)+(v2,v2),

v=v1+v2, O i E D i ,

-1 < t < O

have positive sectional curvature. Since the Hamiltonian functions of metrics ds: belong to a, the geodesic flows of ds: are completely integrable.

8.3. Generalized chain method

+

Let 8 be a Cartan involution on g and let g = I tu be the corresponding orthogonal decomposition into the eigenspaces of 8. Then (g, I)is called a symmetric pair and the following relations hold: [I,I] c [,

[[,tu]c m,

[tu,m]

c I.

Consider the following algebra of polynomials on g:

A,,r={f(z)=p(XI+w),

P E W B I G , XER),

(8.2)

+

where z = 1 w is the orthogonal decomposition (1 E I,w E tu). Let R[I] be the algebra of polynomials on Iconsidered as a subalgebra in R [ g ] .

Theorem 8.5. ( i ) A,,[ is a commutative algebra of polynomials in involution with polynomials f r o m R[ I],i.e., commutative subalgebnz in R[gIL. (ii) A,,[+&%[ I]i s a complete algebra of polynomials o n g. In particular, i f A[ i s a complete commutative subalgebra of R[ I],then A g , [+ A[ will be a complete commutative algebra o n g. This result was first proof by Mikityuk [ 581. Also, Theorem 8.5 is a special case of Theorem 1.5 [ 91. It is related to the compatibility of the LiePoisson bracket {., .), and the &bracket defined by:

i f,g>e(z>= ( z71Vg(z>,Of(.)

le >,

where [ ., is a new operation on g which differs from the standard one [ ., . ] by the only property that tu is assumed to be commutative

[ l i + W 1 , 1 2 + w 2 ] ~ = [ 1 1 , ~ 2 ] + [ 1 1 , W 2 ] +['W,b] (for more details and related references see [ 9, 101). Suppose we are given a chain of connected subgroups GI c ... c G, and the corresponding chain of subalgebras g1 c . . . c g,. Furthermore,

89

suppose that either (gi,gi-l) is a symmetric pair or ~i is a subalgebra of gi,for all i = 2 , 3 , . . . , n. Here gi = gi-1 wi, i = 2 , 3 , . . . ,n.

+

Let Al be an arbitrary complete involutive set of polynomials on 81. For symmetric pairs (gi,gi-l), let Ai be the algebra AO.,O.-l considered on g. Otherwise, if oi is a subalgebra, then let Ai be an arbitrary complete involutive set of polynomials on gi lifted t o g n (it is clear that in this case Ai commute with A l + - . - + A i - l ) . By induction, using Theorem 8.5, we get that A1 A1 .. A, is a complete commutative algebra of polynomials on Bn. Now, if we want to apply the above construction to the our problem, we have to prove "an inductive step", analogous to Theorem 8.5. Suppose we are given an integrable pair (L, H ) . Let L be a subgroup of G such that (g, K) is a symmetric pair. Let

+ +. +

g=h+D,

K=t)+Di,

D=Di+D2

be the orthogonal decompositions. Let ,131 be a complete commutative algebra of AdH-invariants on 01 lifted to D and ,132 be the restriction of algebra AO,[to D: ,132={f(vl+v2)=P(Xvl+vz),

From Theorem 8.5 we get that

P E R [ B I G I XER), ,131

v1 € 0 1 ,

U2ED2.

+ ,132 is a commutative subalgebra of

R[ D ] H . Question 8.1. Is ,131

+ ,132 complete in R[ D ] ~ ?

The following particular result holds.

Theorem 8.6. ([17]) Suppose that (L, H ) i s a n integrable pair, GIL i s a maximal rank symmetric space and a generic v1 E 01 i s a regular element of I. T h e n (G,H ) i s a n integrable pair and B = ,131 ,132 is a complete commutative subalgebra of R[ aIH.

+

The conditions of Theorem 8.6 are not necessary conditions (for example, consider ( H ,L, G) = ( S O ( n ) SO(n , l),SO(n 2))). It would be interesting to prove that algebra B is always complete. Then, if we have a chain of subgroups as above, such that H is a subgroup of GI and that ( G I , H ) is an integrable pair, we would get that (G,,H) is an integrable pair as well. Simply, in this case we can take complete algebra ,131 . . Bn, where Bi is the restriction of Ai t o D, for i = 2 , 3 , . . . , n, and ,131 is a complete commutative algebra for (GI , H ) considered as a subalgebra of R[ D ] H .

+

+

+. +

90

Example 8.4. We can use Theorem 8.6 for a maximal rank symmetric space S p ( n ) / U ( n )and integrable pairs ( U ( n ) , U ( k l )x - . -x U ( k , ) ) with ki 5 [ ( n + l ) / 2 ] , i = 1,.. . ,T (this guaranties the regularity conditions from the theorem) to obtain the integrability of pairs (Sp(n),U(k1)x - . . x U ( k , ) ) . Example 8.5. Suppose that ( S O ( n l ) H , I ) and (SO(n2),H2) are integrable pairs, where H I = SO(k1)x . . . x SO(kr,) and H2 = SO(l1)x . . . x SO(lT2). Suppose that generic u1 E u1 and u 2 E u2 are regular in so(n1) and m ( n 2 ) respectively. Here ui is the orthogonal complement of hi in so(ni),i = 1 , 2 (for example we can take integrable pairs (SO(2k + n ) , S O ( k ) ) and (S0(21+m), SO(21)) from Theorem 8.4). If S O ( n l + n ~ ) / S O ( nxl S) O ( n 2 ) is a maximal rank symmetric space (n1 = 722 f 0, l), then (SO(nl + n2), SO(k1) x . . . x SO(k,,) x SO(l1) x . . . x SO(lv2))is an integrable pair. Example 8.6. Take an integrable pair ( S O ( n ) , S O ( k l x) ... x SO(k,)) obtained by the above construction. Then, since S U ( n ) / S O ( n )is a maximal rank symmetric space, ( S U ( n ) SO(k1) , x . . . x SO(k,)) is an integrable pair as well. 9. Integrability and reduction

Submersions. Suppose we are given a compact Riemannian manifold (Q,g) with a completely integrable geodesic flow. Let G be a compact connected Lie group acting freely on Q by isometries. The natural question arises: will the geodesic flow on QIG equipped with the submersion metric be integrable?

+

Let 6, 7-1, = TxQ be the orthogonal decomposition of TxQ, where Gx is the tangent space to the fiber G . x. By definition, the submersion metric gsub is given by

( G , G ) p ( x=) ( C i , F 2 ) 2 , ti E Tp(x)(Q/G),Ci E %, ti = d p ( C i ) , where p : Q Q/G is the canonical projection. The vectors in Sx and --f

-

H ' , are called vertical and horizontal respectively. g* be the moment map of the natural Hamiltonian Let @ : T*Q G-action on T*Q. It is well known that the reduced symplectic space (T*Q)o = @-'(O)/G is symplectomorphic to T*(Q/G). If H is the Hamiltonian function of the geodesic flow on Q then H I + - I ( ~considered ) on the reduced space will be the Hamiltonian of the geodesic flow for the submersion metric. If we identify T*Q and T Q by the metric g, then W 1 ( 0 )will be the set of all horizontal vectors l-t. Moreover, i f f and g are G-invariant

91

and { f , g } l ~= 0 then f and g descend to Poisson commuting functions on the reduced space (see [ 691). Thus, the base space of the submersion has completely integrable geodesic flow if enough G-invariant commuting functions descend to independent functions. Paternain and Spatzier proved that if the manifold Q has geodesic flow integrable by means of S1-invariant integrals and if N is a surface of revolution, then the submersion geodesic flow on Q X ~ N I = (Q x N ) / S ' will be completely integrable [ 691. The connected sum CP"N - CPn is an example (by a different method they also constructed integrable geodesic flows on @p27~+1# (cp212+1

)*

Bi-quotients. Combining submersions and chain of subalgebras method, Paternain and Spatzier [69] and Bazaikin [ 7 ] proved integrability of geodesic flows of normal metrics on certain interesting hi-quotients of Lie groups. The idea is as follows. Consider a bi-quotient G//U endowed with the normal metric dsi. Let

4 : TG -g@g be the moment map of the G x G action on G: (g.1,92) . g = 91 g g y l , (gl,92) E G x G, g E G. Here we identified dual spaces by the bi-invariant metric on G. Let u c g @ g be the Lie algebra of U and let tu = u' be its orthogonal complement. Also, let U c g 2 T,G be the vertical space at the neutral element of the group. Then the horizontal space 'H at the neutral element is the orthogonal complement of U in g. Let [ E: 'H be the horizontal vector in the neutral of the group. Then d+(TC'H) is the vector suhspace of g @ g equal to:

d+(TC'H) = Q

=

(Aann,([))l

fl m,

(9.1)

where A denotes the diagonal embedding (see [ 69, 71). Suppose that F = {hl,.. .,h,} is a commutative family of Adu-invariant polynomials (with respect to the Lie-Poisson bracket) on g @ g. Then 3 = {+ o hl, . . . , 4 o h,} will be a commutative family of U-invariant functions on TG. Since we deal with analytic functions, it is clear from (9.1) that the number of independent functions on the reduced space is greater or equal to the number

Bazaikin developed the method of computing numbers (9.2) for the families of commuting Adu-invariant polynomials obtained by using chain of

92

subalgebras [ 71. In particular, he proved the complete integrability of the geodesic flows of normal metrics on the 7-dimensional and 13-dimensional bi-quotients of groups G = U(3) x U ( 2 ) x U(1), G = U ( 5 ) x U(4) x U(1) with strictly positive sectional curvature [ 71.

General approach. Now we will show how some of these results, as well as some of the mentioned results on the integrability of geodesic flows on homogeneous spaces, can be obtained directly, considering the relationships between symplectic reductions and the integrability of Hamiltonian systems. Let G be a compact connected Lie group with a free Hamiltonian action on a symplectic manifold ( M ,w ) . Let Q, : M -+ g* be the corresponding equivariant moment map. Let G, be the coadjoint action isotropy group of 77 E @ ( M )c g*. By (M,,w,) we denote the reduced symplectic space

-

= Q,a-'(q)/G1),

where

K

: Q,-'(Q)

W,(dT(<1),dK(<2))

= w(<17<2)7

c1,12

T~@-l(q),

Mq is the natural projection.

Suppose we are given an integrable G-invariant Hamiltonian system j: = X H ( Z ) with compact isoenergy levels Mh = H - l ( h ) . Then M is foliated by invariant tori in an open dense set that we shall denote by reg M . The following theorem is recently proved in [ 381 and in a slightly different version, independently, in [ 871.

Theorem 9.1. Ifreg M intersects the submanifold Q,-'(q) in a dense set then the reduced Hamiltonian system o n (M,,wq) will be completely integrable. Note that we do not suppose that integrals of the original system are Ginvariant. Also, the general construction used in the proof of the theorem leads to smooth commuting integrals on M,. In particular, if the Hamilte nian system x = X H ( Z )is completely integrable by means of G-invariant first integrals, then G is a torus and we can use original integrals to prove the integrability of the reduced system [ 381.

Remark 9.1. It is obvious that reductions by a discrete group C of symmetries have no influence on the integrability: quasi-periodic motions on M go to the quasi-periodic motions on M / C . Namely, suppose that the trajectory y fill up densely an r-dimensional torus T ' c M . Then " ( 7 ) c M / C will fill up densely a torus of the same dimension: C(T')/C.

93

Therefore we get

Theorem 9.2. (1381) Let a compact connected Lie group G act freely by isometries o n a compact Riemannian manifold ( Q , g ) . Suppose that the geodesic Bow of g is completely integrable. If reg T*Q intersects the space of horizontal vectors H ' E @ - l ( O ) in a dense set then the geodesic flow o n Q/G endowed with the submersion metric gsub i s completely integrable. Example 9.1. Eschenburg (for details see [ 291) constructed bi-quotients MZ,l,p,q= S U ( ~ ) / / T ~endowed , J ~ , ~with the submersion metrics d.s:7sub with strictly positive sectional curvature. Here Tk,l,p,q T' C T 2 x T 2 , where T 2 is a maximal torus and ds: is a one-parameter family of leftinvariant metrics on S U ( 3 ) . One can prove that the geodesic flows of the metrics ds? are completely integrable and that we can apply Theorem 9.2 to get the integrability of the geodesic flows of the submersion metrics on M:,l,P,P.

Remark 9.2. It is very interesting that on all known manifolds which admit metrics with strictly positive sectional curvatures (see Wilking [ 821) one can find (positive sectional curvature) metrics with completely integrable geodesic flows. Here is a simple construction that gives examples satisfying the hypotheses of Theorem 9.2. Suppose we are given Hamiltonian G-actions on two symplectic manifolds ( M 1 , w l ) and ( M 2 , w z ) with moment maps @ M and ~ @ M ~ . Then we have the natural diagonal action of G on the product ( M I x M2, w 1 @ w z ) , with moment map @M1xMz = @MI

+ @M2.

(9-3)

If ( Q 1 , g l ) and (Q2,92) have integrable geodesic flows, then (Q1 xQ2,g1@g2) also has integrable geodesic flow. Using (9.3), one can easily see that if the G-actions on Q1 and Q2 are almost everywhere locally free, and free on the product Q 1 x Q 2 then a generic horizontal vector of the submersion Qi x Q 2 Q1 X G Q2 = ( Q I x Q2)/G belongs to reg T*(Q1x Q 2 ) . Thus the geodesic flow on Q1 X G Q 2 , endowed with the submersion metric, is completely integrable. -+

Example 9.2. Suppose a compact Lie group G acts freely by isometries on ( Q , g ) . Let G1 be an arbitrary compact Lie group, which contains G as a subgroup. Let ds; be some left-invariant Riemannian metric on GI with integrable geodesic flow. Then G acts in the natural way by isometries on

94

(GI,ds;). Therefore, if the geodesic flow on Q is completely integrable, then the geodesic flows on Q X G Q and Q X G GI endowed with the submersion metrics will be also completely integrable. 10. Geodesic flows on the spheres In this section we shall list some of the known interesting integrable geodesic flows on the spheres.

Submersion metrics. The sphere Sn-' and the cotangent bundle T*Sn-l are given by the equations sn-1 =

T*$n-1

{.

E

R",

( q , q ) = 1)

-

{(Z,Y) E R2"{4,P), ( 4 7 4 ) = 1 , ( 4 , P ) = 01. Following section 7 to construct integrable geodesic flows on the sphere we can use the structure of a homogeneous space. This structure is not unique but one can start from the simplest one SnP1= S O ( n ) / S O ( n- 1). The moment map @ : T*S"-l so(n) 2 so(n)* of the natural SO(n) action is then given by

-

@(4,P) = 4 A P = QPt -p q t . To construct an integrable geodesic flow on S n - l we can use an arbitrary integrable system on the Lie algebra so(n) with quadratic positive definite hamiltonian (speaking more precisely such a system must be integrable on singular orbits lying in the image of the momentum mapping). There are several series and some exceptional examples of such systems (see, for instance, [51, 1, 2, 541) The most famous among them is the Manakov integrable case [ 511. Using it, Brailov obtained the following integrable geodesic flow [ 21, 221.

Let a1

> a2 > ... > a, > a,+l, b,+l > bl > b2 > ... > b,.

Theorem 10.1. (Brailov, 1983) The geodesic flow of the metric o n S"-l obtained by submersion f r o m the right-invariant Manalcov metric o n S O(n ) with Hamiltonian function

is completely integrable. Moreover, the deformation of the metric given by the Hamiltonian function

95

is also completely integrable. The later has integrals

It can be checked that the metric which corresponds to Ha,b with bi = - a i l is

ds: =

1

(A-lq, Q)

(Adx,dx),

A

= diag(a1,...,an).

(10.1)

For n = 3, the metric (10.1) is proportional to the metric on the Poisson sphere S2, i.e., t o the metric obtained after S O ( 2 ) reduction of the free rigid body motion around a fixed point with inertia tensor I = A-l (e.g., see [Sl) Let us note that in the above construction one can use the other representations of the sphere 2P-l as a homogeneous space, for example, S4n-1 = S p ( 2 n ) / S p ( 2 n - 1) and S6 = G2/SU(3). It is an interesting question whether one really can construct in such a way new examples of integrable geodesic flows or the flows so obtained will be reduced in some sense to the above case S O ( n ) / S O ( n- 1)?

The Maupertuis principle and Neumann system. Consider the natural mechanical system with Hamiltonian

on a compact Riemannian manifold ( M , d s 2 ) . Here gaj is the inverse of the metric tensor and V(z) is a smooth potential on the configuration space M . Let h be greater than maxV(rc). By the classical Muupertuis principle the integral trajectories of the vector field X H coincide (up to reparametrization) with the trajectories of another vector field XH,, with Hamiltonian

on the fixed isoenergy level & = { H ( x , p ) = h } = ( H h ( 2 , P ) = 1) ([ 4, 111). The Hamiltonian flow of Hh is the geodesic flow of the Riemannian metric

d s i = ( h - V(ic)) ds2, conformally equivalent to the original one ds2. Now, it is clear that if we start with an integrable system such that Eh is almost everywhere foliated

96

by invariant tori, the geodesic flow of the metric ds; will be completely integrable. This idea can be used to construct non-trivial integrable geodesic flows on S" starting from integrable potential systems on the standard sphere, ellipsoid or Poisson sphere. Such a potential systems have been studied, in particular, in [ 8, 83, 27, 39, 281. For example, consider the Neumann system on the sphere [ 651, i.e., the motion of a mass point on the sphere S"-' under the influence of the force with potential V(q)= ( A q , q ) / 2 (we take A as above): 1

1

H(Q,P)= 5 ( P l P ) + 5 ( A 4 , q ) . The algebraic form of the integrals is found by K. Uhlenbeck [ 62, 631:

Therefore, the geodesic flow of the metric

ds; = ( h - ( A q , q ) ) ( d q , d q ) l s f l - 1 is completely integrable. The integrals are given by (see [ 111)

Note that there is a remarkable correspondence between the Neumann system and the geodesic flow on the ellipsoid via the Gauss mapping (Knorrer [441).

Geodesical equivalence. After changing the coordinates xi = &qi,

the

metric (10.1) take the form (10.2)

conformally equivalent to the standard metric ds2 = ( d s , d s ) of the ellipsoid Q = { ( A - l z , s ) = l}. There is an interesting relation between these metrics from the point of view of geodesic equivalence. Namely, the standard metric is geodesically equivalent to the metric (see [ 52, 791):

Let g and tj be the corresponding metric tensors. Then one can define the operator

97 1

detij

z.

and metrics g k = gsk,g k = ijSk, k E It appears that metrics g k and g k are geodesically equivalent [ 791. They are all separable in elliptic coordinates and have integrable geodesic flows. Explicit calculations shows

S =(A-

X@X)

IQ

therefore, the metric & is (10.2). It is interesting that the Beltrami-Klein metric of the Lobachevsky space

can be seen as a limit of the metric da2 as the smallest semiaxis a, of the ellipsoid Q tends to zero [ 281.

Kovalevskaya and Goryachev-Chaplygin metrics on the sphere S2. We already know that on the sphere S2 we can find metrics with integrable geodesic flows by means of an integral polynomial in momenta of the first or second degree. The natural question is the existence of metrics with polynomial integral which can not be reduced t o linear and quadratic ones. The positive answer for additional integrals of 3-th and 4-th degrees is given by Bolsinov and Fomenko with two examples: the Kovalevskaya d s g and Goryachev-Chaplygin ds& metrics (see [ 11I). The motion of a rigid body about a h e d point in the presence of the gravitation field admits SO(2)-reduction (rotations about the direction of gravitational field). Taking the integrable Kovalevskaya and GoryachevChaplygin cases we get the integrable systems on T*S2. The metrics d s g and ds& then can easily be constructed by means of the Maupertuis principle. They are the restrictions of the metrics h-41 ( A d z , d a : ) dsg = 2 (A-'4,4) '

A = diag(1, 1,2),

to the unit sphere. The geodesic flow of d s k and ds& admits the first integrals of degree four and three in velocities, which can not be reduced to lower degrees (for more details see [ 111). New families of metrics with cubic and fourth degrees integrals are given by Selivanova [ 73, 741. Just recently K. Kiyohara has constructed integrable

98

geodesic flows with polynomial integrals of arbitrary degree k > 2 (see [ 421). The idea of his construction is the following. First take the constant curvature metric dsi. Its geodesic flow admit three independent linear integrals. Take two of them fi and fi and consider the polynomial integral P= of degree k = m 1. It appears one can perturb the metric dsg in such a way that its geodesic flows remains integrable and, moreover, the first integral P preserve its form, that is, P = where and are linear functions (but not integrals anymore). However, the geodesic flow still has the property that all geodesics are closed with the same period.

fi fr

+

fi f2m

fi

fi

Acknowledgments We are grateful t o the referee for various very useful suggestions which improved the exposition of the paper. Also, we would like t o use this opportunity t o thank the organizers of the workshop Contemporary Geometry and Related Topics for their hospitality. The first author was supported by Russian Found for Basic Research (grants 02-01-00998 and 00-15-99272). The second author was supported by the Serbian Ministry of Science and Technology, Project MM1643 (Geometry and Topology of Manifolds and Integrable Dynamical Systems).

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additional first integral that i s polynomial in the velocities, IZV.Akad. Nauk SSSR, Ser. Matem. 46,No 5, 994-1010, (1982), (Russian); English translation: Math. USSR Izv. 21,291-306, (1983). 46. V. V. Kozlov, Integrability and non-integrability in Hamiltonian mechanics, Uspekhi Mat. Nauk. 38, No 1, 3-67, (1983), (Russian); English translation: Russian Math. Surv. 38,No 1, 1-76, (1983). 47. V. V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 31,Springer Verlag, Berlin, 1996. 48. M. Kramer, Spharische untergruppen in kompakten zusammenhangenden Liegruppen, Compositio Math. 38,129-153, (1979). 49. T. Levi-Civita, Sulle trasformazioni delle equazioni dinamiche, Ann. Mat. 24,255-300, (1986). 50. P. Libermann and C. M. Marle, Symplectic Geometry and Analytic Mechanics, Math. and its Appl. 35,Reidel, Boston, 1987. 51. S. V. Manakov, Note o n the integrability of the Euler equations of ndimensional rigid body dynamics, F'unkts. Anal. Prilozh. 10, No 4, 93-94,

(1976), (Russian). 52. V. S. Matveev and P. Topalov, Quantum integrability f o r the Beltrami-Laplace opemtor a s geodesic equivalence, Math. 2. 238,No 4, 833466, (2001). 53. A. S. Mishchenko, Integration of geodesic flows o n symmetric spaces, Mat. zametki 31,No 2, 257-262, (1982), (Russian); English translation: Math. Notes. 31,NO 1-2, 132-134, (1982). 54. A. S. Mishchenko and A. T. Fomenko, Euler equations o n finite-dimensional Lie groups, Izv. Akad. Nauk SSSR, Ser. Matem. 42,No 2, 396-415, (1978), (Russian); English translation: Math. USSR Izv. 12,No 2, 371-389, (1978). 55. A. S. Mishchenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, F'unkts. Anal. Prilozh. 12, No 2, 46-56, (1978), (Russian); English translation: Funct. Anal. Appl. 12,113-121, (1978). 56. A. S. Mishchenko and A. T. Fomenko, Integration of Hamiltonian systems with noncommutative symmetries, Tr. Semin. PO Vekt. Tenz. Anal. 20,5-54, (1981), (Russian). 57. I. V. Mikityuk, Homogeneous spaces with integrable G-invariant Hamiltonian flows, Izv. Akad. Nauk SSSR, Ser. Matem. 47,No 6, 1248-1262, (1983), (Russian).

58. I. V. Mikityuk, Integrability of the Euler equations associated with filtrations of semisimple Lie algebras, Matem. Sbornik 125 (167),No 4, (1984), (Russian); English translation: Math. USSR Sbornik 53,No 2, 541-549, (1986). 59. 1. V. Mikityuk, O n the integrability of invariant Hamiltonian systems with homogeneous configuration spaces, Matem. Sbornik, 129 (171), No 4, 514534, (1986), (Russian); English translation: Math. USSR Sbornik 57, No 2, 527-547, (1987). 60. I. V. Mykytyuk, Actions of Bore1 subgroups o n homogeneous spaces of reductive complex Lie groups and integrability, Composito Math. 127,55-67, (2001). 61. I. V. Mykytyuk and A. M. Stepin, Classification of almost spherical pairs

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of compact simple Lie groups, Poisson Geometry, Banach Center Publ. 51, Polish Acad. Sci., Warsaw, 231-241, 2000. 62. J. Moser, Various aspects of integrable Hamiltonian systems, Proc. CIME Conference. Bressanone, Italy, 1978, Prog. Math. 8, 233-290, (1980). 63. J. Moser, Geometry of quadric and spectral theory, Chern Symposium 1979, Berlin- Heidelberg- New York, 147-188, (1980). 64. N. N. Nekhoroshev, Action-angle variables and their generalization, Tr. Mosk. Mat. Obshch. 2 6 , 181-198, (1972), (Russian); English translation: Trans. Mosc. Math. SOC. 26, 180-198, (1972). 65. C. Neumann, De probleme quodam mechanico, quod ad p r i m a m integralium ultra-ellipticoram classem revocatum, J. Reine Angew. Math. 56, 46-63, (1859). 66. D. I. Panyushev, Complexity of quasiafine homogeneous varieties tdecompositions, and a f i n e homogeneous spaces of complexity 1, Adv. Soviet Math. 8, 151-166, (1992). 67. G. P. Paternain, Entropy and completely integrable Hamiltonian systems, Proc. Amer. Math. SOC.113, No 3, 871-873, (1991). 68. G. P. Paternain, O n the topology of manifolds with completely integrable geodesic flows Z, ZZ,Ergod. Th. & Dynam. Sys. 12, 109-121 (1992); J. Geom. Phys. 13,289-298, (1994). 69. G. P. Paternain and R. J. Spatzier, N e w examples of manifolds with completely integrable geodesic flows, Adv. Math. 108, 346-366, (1994); arXiv: math.DW9201276.

70. A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Birkhauser, 1990. 71. A. G. Reyman M. A. and Semonov-Tian-Shanski, Group theoretical methods in the theory of finite dimensional integrable systems, Dynamical Systems VII, (Eds.: V. I. Arnold, S. P. Novikov), Springer, 1994. 72. J. Morales Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems, Progress in Mathematics 179, Birkhauser Verlag, Basel, 1999. 73. E. N. Selivanova, New examples of integrable conservative systems o n S2 and the case of Goryachev-Chaplygin, Commun. Math. Phys. 207, 641-663, (1999). 74. E. N. Selivanova, New families of conservative systems o n S2 possessing a n integral of fourth degree in momenta, Ann. Global Anal. Geom. Math. 17, 201-219, (1999). 75. R. J. Spatzier, Riemannian Manifolds with Completely Integrable Geodesic Flows, Proceedings of Symposia in Pure Mathematics, Vol. 54, Part 3, 599608, (1993). 76. I. A. Taimanov, Topological obstructions to integrability of geodesic flows o n non-simply-connected manifolds, Izv. Akad. Nauk SSSR, Ser. Matem. 51,No 2,429-435, (1987), (Russian); English translation: Math. USSR Izv. 30,No 2, 403-409, (1988). 77. I. A. Taimanov, T h e topology of Riemannian manifolds with integrable geodesicflows, Tr. Mat. Inst. Steklova 205,150-164, (1994), (Russian); English translation: Proc. Steklov Inst. Math. 205,No 4, 139-150, (1995).

103 78. A. Thimm, Integrable geodesic flows o n homogeneous spaces, Ergod. Th. & Dynam. Sys. 1,495-517, (1981). 79. P. Topalov, Integrability Criterion of Geodesical Equivalence. Hierarchies, Acta Appl. Math. 59, 271-298, (1999). 80. V. V. Trofimov, Euler equations on Bore1 subalgebras of semisimple Lie groups, Izv. Akad. Nauk SSSR, Ser. Matem. 43, No 3, 714-732, (1979), (Russian).

81. E. B. Vinberg, Commutative homogeneous spaces and co-isotropic symplectic actions, Uspekhi Mat. Nauk 56, No 1, 3-62, (2001), (Russian); English translation: Russian Math. Surv. 56, No 1, 1-60, (2001). 82. B. Wilking, Manifolds with positive sectional curvature almost everywhere, Invent. Math. 148,117-141, (2002). 83. S. Wojciechowski, Integrable one-particle potentials related t o the Neumann system and the Jacobi problem of geodesic motion on a n ellipsoid, Phys. Lett. A 107,107-111, (1985). 84. H. Yoshida, Necessary condition f o r the existence of algebraic first integrals, Celestial mechanics 31,363-399, (1983). 85. H. Yoshida, A criterion for the non-existence of an additional analytic integral in Hamiltonian systems with n degrees of freedom, Phys. Lett. A 141,No 3-4, 108-112 (1989). 86. S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I, 11, F'unkts. Anal. Prilozh. 16,No 3, 30-41, (1982); 17,NO 1, 8-23, (1983). 87. N. T. Zung, Reduction and integrability, arXiv: math .DS/0201087.

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CR SUBMANIFOLDS OF MAXIMAL CR DIMENSION IN COMPLEX SPACE FORMS AND SECOND FUNDAMENTAL FORM *

MIRJANA DJORIC + Faculty of Mathematics, University of Belgrade, Studentski trg 16, pb. 550, 11000 Belgrade, YUGOSLAVIA E-mail: mdjoricOmatf. bg.ac.yu

MASAFUMI OKUMURA Department of Mathematics, Saitama University Shimo-Okubo Urawa 338, JAPAN E-mail: [email protected]

We study m-dimensional real submanifolds of codimension p with (m - 1)dimensional maximal holomorphic tangent subspace in complex space forms. Consequently, on these manifolds there exists an almost contact structure ( F ,u,ZJ, g) naturally induced from the ambient space. In this paper we study certain conditions on the almost contact structure and on the second fundamental form of these submanifolds.

1. Introduction

a

Let M be a real submanifold of a complex manifold and J be the complex structure of %. If the maximal J-invariant tangent subspace of M , called the holomorphic tangent space, has constant dimension over M , then M is called a Cauchy-Riemann submanifold, or briefly a CR submanifold, and the constant complex dimension is called the CR dimension of M [ 181. This paper is devoted to the study of such submanifolds. A real hypersurface is a typical example of CR submanifold and its CR dimension is ( m- 1)/2, * MSC2000: 53C15, 53C25, 53C40. Keywords: CR submanifold, complex space form, almost contact structure, second fundamental form. t Work partially supported by the Serbian Ministry of Science, Technologies and Development under contract No MM1646.

105

106

where m is a dimension of M . In a real hypersurface case and in particular when is a Kahler manifold, many results have been obtained. See, for example, [ 171 for the fundamental definitions and results and for further references. Therefore, the generalization of some results which are valid for real hypersurfaces to CR submanifolds of CR dimension ( m - 1)/2 may be expected. Above all, the second author of this paper and S. Montiel and A. Romero gave a geometric meaning of the commutativity of the second fundamental tensor of the real hypersurface of a complex space form and its induced almost contact structure. The purpose of the present paper is to recall a corresponding result to these ones for certain CR submanifolds of the complex space forms. Namely, in [ 81 we studied CR submanifolds of maximal CR dimension in complex space forms which satisfy the condition h ( F X , Y ) h ( X , F Y ) = 0 , where F is the induced almost contact structure and h is the second fundamental form of M . We proved that under this condition there exists a totally geodesic complex space form M’of such that M is a real hypersurface of M’. Therefore, it was possible to apply the results of real hypersurface theory and prove some classification theorems for CR submanifolds of maximal CR dimension in complex projective space, complex hyperbolic space and complex Euclidean space. In Section 2 we recall some general preliminary facts concerning these submanifolds and derive useful formulas for later use. Further, if the CR dimension of the CR submanifold is ( m - 1)/2, then there exists a unit vector field [ normal to M such that J T ( M ) c T ( M )@span{[}. We prove our results in the case when the distinguished normal vector field [ is parallel with respect t o the normal connection. In Section 3 we recall some results from the real hypersurface theory and we construct several standard examples of real hypersurfaces of complex space forms which satisfy the commutativity condition on the second fundamental tensor and almost contact structure. Moreover, we state the classification theorems proved in [8]. Finally, in Section 4, we prove that if CR submanifolds of maximal CR dimension in complex space forms satisfy the condition h ( F X ,Y )- h ( X ,F Y ) = 0 , then the holomorphic sectional curvature of the ambient manifold is non-positive.

z

+

z

107

2. CR submanifolds of maximal CR dimension of complex space forms

+

Let % be an (m p)-dimensional Kahler manifold with Kahler structure (J,g) and M an m-dimensional real submanifold of % with the immersion z of M into %, that is, such that the Riemannian metric g of M is induced from the Riemannian metric g of % in such a way that g ( X ,Y ) = ij(zX,zY), where X , Y E T ( M ) . We denote by T ( M ) and T ' ( M ) the tangent bundle and the normal bundle of M , respectively. Also, we denote by z the differential of the immersion, or we omit to mention z for brevity of notation. Next, we assume that, for any IC E A4 the maximal holomorphic subspace H,(M) of T,(M) is (m - 1)-dimensional. In general, the dimension of H,(M) varies with IC, but if the subspace H,(M) has constant dimension for any IC E M , the submanifold M is called the Cauchy-Riemann submanifold or briefly CR submanifold and the constant complex dimension of H,(M) is called the CR dimension of M [ 181, [ 271. It is well-known that a real hypersurface is one of the typical examples of CR submanifolds whose CR dimension is (m - 1)/2, where m is the dimension of a hypersurface. We note that in the case of maximal CR dimension the above-given definition of CR submanifolds coincides with the definition of CR submanifolds given by Bejancu in [ 11. We refer to [ 61 for more details and examples of CR submanifolds of maximal CR dimension. In the sequel we consider CR submanifolds of maximal CR dimension, that is, dimR H,(M) = dimR(JT,(M) fl T,(M)) = m - 1 . Then it follows that M is odd-dimensional and that there exists a unit vector field J normal to M such that JT,(M) c T,(M) @span{<,}, for any IC E M . Hence, for any X E T ( M ) , choosing a local orthonormal basis J, Jl, . . . , & - I of vectors normal to M , we have the following decomposition into tangential and normal components:

Here F and P are skew-symmetric endomorphisms acting on T ( M ) and T I ( M ) ,respectively, U , Ua, a = 1, . . . ,p - 1,are tangent vector fields and u is one form on M . Furthermore, using (2.1), (2.2) and (2.3), the Hermitian

108

property of J implies

g ( U , X ) = .(X) , Ua = O F 2 X = -X + u ( X ) U , u(FX)=O,

(U

= I , . .. , p - I ) ,

(2.5)

PJ=O.

FU=0,

(2.4) (2.6)

Hence, relations (2.2) and (2.3) may be written in the form

Jc = -zU

7

Jea = Pea

(U

= 1,.. . , p - 1).

(2.7)

Moreover, these relations imply that ( F ,u,U,g ) defines an almost contact metric structure on M (see [ 261). Since {q E T’(M),q I5 ) is J-invariant, from now on let us denote the orthonormal basis of T* ( M ) by (1, . . . ,Eq, (I*, ... En* , where (a* = JEa and q = ( p - 1)/2. Next, let US denote by 7 and V the Riemannian connection of and M , respectively, and by D the normal connection induced from 7 in the normal bundle of M . They are related by the following well known Gauss equation [)

)

a

-

V*XZY = ZVXY+ h ( X ,Y ),

(2.8)

where h denotes the second fundamental form, and by Weingarten equations Q

-

VaxE = -%AX + C { S a ( X ) E a+ sa* ( x ) E ~ * ) ,

(2.9)

a=l 9

V a X Q = -2AaX

+ x { s a b ( X ) c bf Sab+( X ) [ b * } ,

- Sa(X>E

(2.10)

b=1

9

vEXEa* =

-2Aa*X

-

+

Sa*(X)cf x { S a * b ( X ) c b Sa*b*(X)cb*}, (2.11) b=1

ca*,

where A, A,, A,* are the shape operators for the normals (, Q , respectively, and s’s are the coefficients of the normal connection D. Since the ambient manifold is a Kaehler manifold, it follows

Aa*X = F A a X - sa(X)U,

(2.12)

Sa*(X)= u ( A a X ) = g ( A a X , U ) = g(AaUi X ) , Sa*b* = S a b , sa*b = -sub* ,

(2.13)

+

4

h(x,Y ) = g ( A X ,Y ) E C { g ( A a X ,y ) t a u=l

for all X , Y tangent to M .

+ g(Aa*X,Y)Q*>,

(2.14) (2.15)

109

Moreover, assuming that the vector field E is parallel with respect t o the normal connection D ,it follows DxE = CQ,=1{Sa(X)Ea sa*(X)e,a-} = 0 , from which sa = sa* = 0 , a = 1 , . . . ,q. Next, differentiating relations (2.1) and (2.2) and comparing the tangential and normal parts, we get

+

(VyF)X = u(X)AY - g(AY, X ) U , (VYU)(X) = S(FAY, X) VxU = F A X .

(2.16) (2.17)

7

(2.18)

Now, using relations (2.12) and (2.13) we obtain

A,*

= FA,

A,U = 0

( a = 1 , . . . ,q ) , ( a = 1,.. . , q ) .

(2.19) (2.20)

Since the second fundamental form h ( X , Y ) is symmetric with respect to X , Y , (2.15) and (2.19) imply that FA,*, a = 1 , . . . ,q are symmetric and hence it follows

FA,

+ A,F

= 0,

FA,*

+ A,* F = 0

( a = 1 , . . . ,q ) .

(2.21)

a

Finally, we suppose that the ambient manifold is a complex space form, i.e. a Kahler manifold of constant holomorphic sectional curvature 4k. Then, using relations (2.1) and (2.4), the Codazzi equation for the unit vector E becomes

( V y A ) X - (VxA)Y = k {u(Y)FX - u ( X ) F Y

+2g(FX,Y) V } ,

(2.22)

for all X , Y tangent to M ([12]). 3. CR submanifolds satisfying h ( F X , Y )

+ h ( X , F Y ) = 0.

Many authors have dealt with real hypersurfaces of complex space forms [ 41, [ l o ] , [ 131, [ 151, [ 161, [ 211, [ 201, [ 251. In general, a real hypersurface M of a complex space form %f has two geometric structures: namely, an almost contact structure q5 induced from the complex structure J of %f, and a submanifold structure represented by the second fundamental tensor H of M in %f. Moreover, some authors have studied real hypersufaces of complex projective and hyperbolic spaces P"(C) and H"(C) from some different point of view [ 201, [ 151, [ 161. The common basis of these works is to construct, following Lawson's method in [ 141, a hypersurface M' of the sphere S2"+' (anti-De Sitter space , respectively) which is principle

110

S1-bundle over M compatible with the Hopf fibration. Consequently, conditions imposed on M can be translated into those on M' and the known papers concerning the hypersurfaces of a real space form can be utilized [ 171. Therefore, it is interesting to relate both structures, almost contact structure and a submanifold structure, and to study the consequences that are derived for the principal S1-bundle M' corresponding to M . In this sense, the second author of this paper in [ 201 and S. Montiel and A. Romero in [ 161, studied and classified the real hypersurfaces M of a complex projective space P"(C) and of a hyperbolic space H" (C), respectively, which satisfy the commutativity condition

q5H = Hq5. (3.1) First of all, we shall describe the distinguished examples of real hypersurfaces in spaces of constant holomorphic sectional curvature which satisfy the condition (3.1). We start with real hypersurfaces in complex projective space Pn(C). For z = (zo,z1,. . . , z"), w = (wo, w1,. . . , w"),z , w E en+',let us write n k=O

and let

< z , w >= Re G ( z ,w), the real part of G ( z ,w). The (2n+ 1)-sphere

S 2 n + 1 ( ~of ) radius

T

is defined by

S 2 " + l ( T ) = ( 2 E @"+I[

Let

7r

< z , z >= 7 - 2 ) .

be the canonical projection of S2"+'(r>to complex projective space

P" (@.> , n. : S2"+l

-

P"(C).

(3-2)

Further, we write Cn+l = C=P+l~ C Q + ~ , w h e r e p , q L O a n d p + q = n ->l O and we choose b so that 0 < b < T . Also, let

MA,q = {Z

= ( ~ 1 ,~

2 E)

C"+'(Gl(z1,zl) =

-

b2,

G 2 ( 2 2 , ~ 2= ) b2},

where G1 and G2 are the restrictions of G to CP+l and @ q + l respectively. Then Mi,q,called the generalized Clifford hypersurface, is the Cartesian product of spheres whose radii have been chosen so that lies in S'"+~(T), i.e.

Therefore, we get the fibrations

111

compatible with the standard fibration (3.2). The surfaces MEq are called "generalized equators" [ 141. The second author of this paper proved that the condition (3.1) for M occurs if and only if the second fundamental tensor of M' is parallel [Theorem 4.3, 201. From this result and Ryan's paper [ 241, he obtained

Theorem 3.1. ([Theorem 4.4, 201) MEq are the only complete, real hypersurfaces of a complex projective space in which the second fundamental tensor H commutes with the fundamental tensor q5 of the submersion T. Next, we introduce the complex hyperbolic space H n ( C ) . The construction is parallel to that of P"(C) with some important differences. For z = (20,z1,. . . ,zn), w = (wo, 201,. . . ,wn),z , w E Cn+l, let us write n

G(Z,W)

= -20

UIo

+ C zk k=l

and let < z , w >= Re G ( z ,w). The anti-De Sitter space of radius T in Cnfl is defined by

IHI?+~ = { z E cn+lI < z , z >= -r Further, we denote by H"(C) the image of tion 7r to complex projective space, 7r

:H p + l

-

H"(C)

2

1.

by the canonical projec-

c P"(C).

(3.3)

Thus, topologically, H"(C) is an open subset of P n ( C ) . However, as Riemannian manifolds, they have quite different structures. It is well-known that Hq"" is a principal S1-bundle over H"(C) with projection 7 r , which is a Riemannian submersion in the sense of O'Neill [ 231 with fundamental tensor J and time-like totally geodesic fibres. Now, let

ML

= { z E Cn+l I

< z , z >= -r 2 , Izo - zll = t } .

MA is a Lorentz hypersurface of Using fibration (3.3), we define the so-called "horospheres" ME as ME = 7rf(Mn)where 7r' is the submersion which is compatible with the fibration 7r in (3.3). Further, for given integers p , q with p + q = n - 1 and r E R with 0 < r < 1, we denote by M 4 p + l , 2 q + l ( the ~ ) Lorentzian hypersurface of H?+' defined by the equations

112

where GI and G2 are the restrictions of G to CP+' and @ q + l respectively. Then M4p+l,2q+lis the Cartesian product of an anti-De Sitter space and a sphere whose radii have been chosen so that it lies in WT+'(r) and each factor is embedded in in a totally umbilical way, i.e. Mip+l,2q+l - W 2 P + l ( ( b 2 r 2 ) l I 2 )x S2q+1 (b). Since M4p+1,2q+1(r) is S1-invariant, M f q ( r )= ~ ' ( M i ~ + ~ , ~is~ a+real ~(r)) hypersurface of H n ( C ) which is complete and satisfies the condition (3.1). In [ 161 S. Montiel and A. Romero studied the converse problem and classified the real hypersurfaces M of a complex hyperbolic space H"(C) satisfying the commutativity condition (3.1) by using the S1 fibration T given by (3.3). They constructed a Lorentzian hypersurface M' of which is a principal S1-bundle over M with time-like totally geodesic fibers compatible with the fibration T. Although the second fundamental form H' of M' is not necessarily diagonalizable, they proved the same geometric characterization as in the case of a complex projective space. Furthermore, after studying local geometric properties of certain family of Lorentzian hypersurfaces of IHI?" which have parallel second fundamental form, they determined the complete list of real hypersurfaces of H"(C) satisfying the condition (3.1). Namely, they proved

+

WT+l

Theorem 3.2. ([Theorem 5.1, 161) Let M2n-1 be a complete real hypersurface of the complex hyperbolic space H"(C) whose second fundamental form and induced almost contact structure commute. Then, we have the following possibilities: (a) M2n-1 is congruent to Mg+l,2q+l(r), (b) M2n-1 is congruent to M:. Finally, in order to classify all complete real hypersurfaces of the complex Euclidean space which satisfy the condition (3.1), we first recall that the induced almost contact structure of a hypersurface of a Kahlerian manifold is normal if and only if its second fundamental form and induced almost contact structure commute ([ ZO]), and then bring into use

Theorem 3.3. ([Theorem 4.2, 211) A normal almost contact hypersurface M2n-1 in Euclidean space &lznis locally isometric with one of the following: ~2n-1 ~ 2 n - 1 1

where r is an odd number and r

, S'

x

E",

+ s = 2n - 1.

113

Now, let M be a complete m-dimensional CR submanifold of CR dimension ( m - 1 ) / 2 of a complex space form J ) , with almost contact structure ( F ,u,U,g ) and such that the distinguished normal vector field E is parallel with respect t o the normal connection. In [ 81 we studied these manifolds satisfying the condition

(a,

h ( F X , Y )+ h ( X ,F Y ) = 0 ,

(3-4)

for all X , Y tangent t o M , where h denotes the second fundamental form of M . We proved that in this case there exists a totally geodesic complex space form M' of such that M is a real hypersurface of M'. Moreover, it follows,

4H'

= HI$,

where H' is the second fundamental tensor of the hypersurface M' and 4 is its almost contact metric structure. Therefore, it is possible to apply the above-mentioned results of real hypersurface theory, namely, the classification theorems of hypersurfaces of complex space forms satisfying the condition (3.1) and prove some classification theorems:

Theorem 3.4. ( [ 8 ] ) Let M be a complete m-dimensional CR submanifold of CR dimension (m - l ) / 2 of a complex Euclidean space C p . If the distinguished normal vector field E is parallel with respect to the normal connection and i f the condition (3.2) is satisfied, then M is congruent to

E", S",

SZPfl

x

p - Z P - 1

Theorem 3.5. (181) Let M be a complete m-dimensional CR submanifold of CR dimension (m - l ) / 2 of a complex projective space P* ( C ) . If the distinguished normal vector field E i s parallel with respect to the normal connection and if the condition (3.2) i s satisfied, then M is congruent to ME9, f o r some p , q satisfying 2 p 2 q = m - 1.

+

Theorem 3.6. ( [ 8 ] ) Let M be a complete m-dimensional CR submanifold of CR dimension (m - l ) / 2 of a complex hyperbolic space H*(C). If the distinguished normal vector field E is parallel with respect to the normal connection and if the condition (3.2) i s satisfied, then M is congruent to M* or MF9(r),f o r some p , q satisfying 2 p + 2 q = m - 1. 4. CR submanifolds satisfying h ( F X , Y ) - h ( X , F Y ) = 0 In this section we study CR submanifolds M" of maximal CR dimension of complex space forms whose distinguished normal vector field is parallel

114

with respect to the normal connection and which satisfy the condition

h ( F X ,Y )- h ( X ,F Y ) = 0 ,

(4.1)

for all X , Y E T ( M ) . Using relations (2.15) and (2.19), we obtain

h ( F X ,Y ) - h ( X ,F Y ) = { g ( A F X ,Y ) - g ( A X , FY)}E

(4.2)

9

+ C { [ s ( A a F X Y, )

-

g ( A a X ,F Y ) IQ

a=l

+ [ g ( F A J ' X , Y ) - g ( F A a X ,F Y ) ] L * } . Since F is skew-symmetric and using relation (2.21), we get

h ( F X , Y ) - h ( X , F Y ) = [ g ( A F X , Y )+ g ( F A X , Y ) ] [ .

(4.3)

Therefore, relation (4.1) is equivalent to

AF

+ F A = 0.

(4.4)

Now, using relations (2.6) and (4.4), it follows that U is an eigenvector of the shape operator A, namely, A U = a U . Differentiating this equation covariantly and using the Codazzi equation (2.22), we obtain

2kg(FX,Y)

+ 2 g ( A F A X , Y ) = ( X ~ ) U ( Y )( Y c I ) u ( X ) -

+ a g ( ( F A+ A F ) X ,Y ).

(4.5)

Since U is an eigenvector of A, when we put Y = U in relation (4.5), we get

X a = .(X)

ua.

(4.6)

Using relations (4.5) and (4.6), it follows

2 k g ( F Y , X ) + 2 g ( A F A X , Y )= a g ( ( F A + A F ) X , Y )

(4.7)

and now the relation (4.7) implies that k g ( F Y ,X ) = g ( A X , F A Y ) . Finally, if we put X = F Y in the last relation, we obtain

k g ( F Y ,F Y ) = g ( A F Y ,F A Y ) = - g ( F A Y , F A Y ) 5 0 , and we conclude k 5 0 , since rank F = m - 1. See also [ 51. Therefore, we can state

Theorem 4.1. Let A4 be an m-dimensional CR submanifold of CR dimension ( m - 1 ) / 2 of a complex space form. If the distinguished normal vector field E is parallel with respect to the normal connection and i f the condition (4.1) is satisfied, then the holomorphic sectional curvature of the ambient manifold is non-positive.

115

References 1. A. Bejancu, CR-submanifolds of a Kahler manifold I, Proc. Amer. Math. SOC.69, 135-142, (1978). 2. J. Berndt, Uber Untermanningfaltigkeiten von komplexen Raumformen, doctoral dissertation, Universitat zu Koln, 1989. 3. D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509, Springer, Berlin, 1976. 4. T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. SOC.269, 481-499, (1982). 5. 'M. Djorid and M. Okumura, CR submanifolds of maximal CR dimension of complex projective space, Arch. Math. 71, 148-158, (1998). 6. M. Djorid and M. Okumura, CR submanifolds of maximal CR dimension in complex manifolds, PDE's, Submanifolds and Affine Differential Geometry, Banach center publications, Institute of Mathematics, Polish Academy of Sciences, Warsawa 2002, 57, 89-99, (2002). 7. M. Djorid and M. Okumura, Certain application of a n integral formula to C R submanifold of complex projective space, Publ. Math. Debrecen 62/1-2, 213-225, (2003). 8. M. Djorid and M. Okumura, Certain CR submanifolds of maximal CR dimension of complex space forms, submitted. 9. M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. SOC.296, 137-149, (1986). 10. M. Kimura, Sectional curvatures of holomorphic planes o n a real hypersurface in P"(C), Math. Ann. 276, 487-497, (1987). 11. W. Klingenberg, Real hypersurfaces in Kahler manifolds, SFB 288 preprint No 261, Diferentialgeometrie und Quantenphysik. 12. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry 11, Interscience, New York, 1969. 13. M. Kon, Pseudo-Einstein real hypersurfaces in complex space forms, J. Diff. Geom. 14, 339-354, (1979). 14. H. B. Lawson, Jr., Rigidity theorems in rank-1 symmetric spaces, J. Diff. Geom. 4, 349-357, (1970). 15. Y. Maeda, On real hypersurfaces of a complex projective space, J. Math. SOC. Japan 28, 529-540, (1976). 16. S. Montiel and A. Romero, O n some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20, 245-261, (1986). 17. R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms, in Tight and taut submanifolds, (eds. T.E. Cecil and S. S. Chern), Math. Sciences Res. Inst. Publ. 32, Cambridge Univ. Press, Cambridge, 233-305, (1997). 18. R. Nirenberg and R. 0. Wells, Jr., Approximation theorems o n differentiable submanifolds of a complex manifold, Trans. Amer. Math. SOC.142, 15-35, (1965). 19. M. Okumura, Codimension reduction problem f o r real submanifolds of complex projective space, Colloq. Math. SOC.JBnos Bolyai 56, 574-585, (1989).

116

20. M . Okumura, O n some real hypersurfaces of a complex projective space, Trans. Amer. Math. SOC. 212, 355-364, (1975). 21. M . Okumura, Certain almost contact hypersurfaces in Euclidean spaces, K6dai Math. Sem. Rep. 16, 44-54, (1964). 22. B. O’Neill, Isotropic and Kahler immersions, Canad. J. Math. 17, 907-915, (1965). 23. B. O’Neill, The fundamental equations of a submersion, Mich. Math. J. 13, 459-469, (1966). 24. P. J. Ryan, Homogeneity and some curvature conditions for hypersurfaces, Tbhoku Math. J. 21, 363-388, (1969). 25. R. Takagi, O n homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10, 495-506, (1973). 26. Y . Tashiro, Relations between almost complex spaces and almost contact spaces, Sdgaku 16, 34-61, (1964). 27. A . E. Tumanov, The geometry of C R manifolds, Encyclopedia of Math. Sci. 9 V I , Several complex variables III, Springer - Verlag, 201-221, 1986.

PAPPUS-GULDIN'S FORMULAE VERSUS WEYL'S TUBE FORMULAE: OLD AND RECENT RESULTS *

M. CARMEN DOMINGO-JUAN Departmento de Matematica Economico-Empresarial, Universidad de Valencia, Valencia, SPAIN E-mail: [email protected] VICENTE M I Q U E L ~ Departamento de Geometria y Topologia, Universidad de Valencia Burjassot (Valencia), SPAIN E-mail: [email protected]

We survey recent results on Pappus-Guldin type formulae for volumes in real and complex space forms. This survey includes results in the published papers [ 6 ] , [ 161, [ 51, [ 121 and [ 11 (with many viewpoints of [ 14]), and also the announce of [ 21 and [ 31, which will be published elsewhere.

1. Introduction In 1939, in the curse of his study on certain statistical problems, H. Hotelling published a paper [ 151 giving the first terms of the asymptotic expansion, in function of the radius, of the volume of a tube around a submanifold of the n-dimensional sphere. In the same year and journal, H. Weyl [ 181 published his formulae on the volume of tube P, and a tubular hypersurface aP, of radius r around a q-dimensional submanifold P of R" or 5'". For R", the definitions and formulae are P, = {x E R"; there is p E P satisfying 1x - p i 5 T and (z - p ) l P } , aP, = {z E R"; there is p E P satisfying Iz - pl = T and (x - p ) l P } ,

* MSC2000: 53C20, 53C21, 53C55. t Work partially supported by a dgi grant no BFM2001-3548.

117

118

(*T2)("-d/2

vol(dP,)

=

c

[4'21

2c

k2c(P)T2c--1

(i(n - q ) ) ! c=O ( n - Q + 2 ) ( n - q + 4) . . .

(TI. -

4

+2 ~ ) '

kzC ( P )is an invariant of P which depends only on its Riemann curvature tensor (see the details of its definition after Theorem 3 . 2 ) . Then vol(P,) and vol(dP,) depend only on T and the intrinsic geometry of P,they are independent of the particular way in which the submanifold P is embedded in Euclidean space. As a first step in its proof, Weyl shows that vol(P,) and vol(dP,) depend only on the second fundamental form of P. It will be implicit in the results we shall see along this paper that even this first step fact can be considered of some interest. For submanifolds in the sphere S", H. Weyl obtained analog formulae with the same qualitative consequence. A much older result, discovered by Pappus (around 300), and rediscovered by Guldin (around 1600) states that, if D is the domain in Iw3 generated by the rotation of a plane domain DO (resp. curve Co) around an axis R contained in the same plane than DO (resp. CO), Vol(D) = Area ( D O )xLength (c(s)), Area(C) = Length (CO)xLength (c(s)), where c(s) is the circle described by the center of mass of DO (resp. C O ) .

DO plain domain, COplain curve co center of mass of DO or CO,c(s) circle

The domain D of Pappus-Guldin formula can be considered as a kind of tube around c(s), but having a not necessarily ball shaped section. Also C

119

can be considered as a tubular hypersurface around c ( s ) with non-spherical section. With this viewpoint, the objects of study of both Pappus-Guldin and Weyl’s formulae are part of a big family of “tubes of arbitrary section”. What is special of the tubes studied by Weyl is that the only allowed section is a ball (or a sphere). What is special in the tubes studied by PappusGuldin is that section has its center of mass on the submanifold (the curve) and that section moves along the curve in a very special way. Of course, both conditions are much weaker than those of Weyl: every ball or spherical section in the tubes considered by Weyl have their center of mass on the submanifold, and the question on motion has no sense when a ball or sphere are moved, because all the motions give the same result. The weak part of Pappus-Guldin formula is, of course, that it considers a very small family of submanifolds: circles. The aim of this paper is to give account on the research done in generalizing Pappus-Guldin formulae for wider classes of submanifolds: first curves in R3 (the works of Goodman and Goodman [ 61, Purse11 [ 161 and Flanders [ 51, X. Gual and the second author of this paper [ 14]),then on curves in simply connected spaces of constant sectional curvature and finite dimension (the works of A. Gray, X. Gual and the authors of this paper [ la], [ 11) and, finally, on arbitrary submanifolds in real and complex space forms of finite dimension (the works [ 21 and [ 31). Along the results which we shall expose here, the role of the shape of the section and its motion in the Weyl’s formulae will be elucidated. Moreover we will see there is a big difference between the cases of “domains obtained by a motion along a submanifold” (the concept generalizing that of tube) and “hypersurfaces obtained by a motion along a submanifold” (the concept generalizing that of tubular hypersurface). In this exposition, we shall give the concepts, ideas, and the statement of the theorems, but no proofs, but in each theorem, we shall indicate where the proof is given. In case a theorem was previously proved for R3 and, after, for a real space form, we’ll only give the more general statement, but the references will include both papers. Along this paper, all the domains D p , Do and hypersurfaces C,, Co considered in it are connected.

120

2.

Motion along a curve

2.1. Definitions

Notation M r : space form of dimension n and sectional curvature A. c : I = [O,L] MF: a curve parameterized by its arc-length t which is also an embedding from [O,L] into M p if c(0) # c(L) or induces an embedding from S1 into M n if c(0) = c(L). Ad:;': the complete totally geodesic hypersurface of MF through c(t) and orthogonal to c. D: the normal connection on the normal bundle T c = UtEI{c'(t)}L of the curve c. Pt: the D-parallel transport in T c along c(t). sx : 1R R:= the function solution of the differential equation s"+X s = 0 satisfying the initial conditions sx(0) = 0 and si(0) = 1. cx := s'x. cut(c(0)): the set of cut points of c(0) in Ad:;'.

-

-

-

Definition 2.1. A motion cp along a curve c of M n is a C" isomorphism cp : I x ( ~ ' ( 0 ) ) ~ %c of vector bundles along the curve c satisfying that, for every t 6 I, the map cpt : (~'(0))' { ~ ' ( t ) }defined ~ by cpt(e) = cp(t,[)is an isometry of Euclidean vector spaces, and cpo = Id. ---f

This cp defines a C" map

and the maps q5t ' ;::A {

-

cut(c(0))

4 : I x (Ad;;'

-

M:;'

-

-

cut(c(0)))

-

cut(c(t)) defined by

are isometries of Riemannian manifolds, and

$0

= Id.

M y by

& ( z ) = $(t ,z )

Moreover,

cpt = 4 t * c ( O ) .

Since q5 and cp are mutually defined through (2.1), we could equivalently to define a motion by the map 4, and we shall refer to a motion indistinctly defined by 4 or cp. If {e2, ...,en} is an orthonormal basis of {c'(O)}l, a motion cp along c defines a smooth orthonormal frame { E l ( t ) = c'(t), &(t) = cpt(ez),..., E,(t) = cpt(en)} along c(t), called a smooth orthonormal frame associated to the motion cp. In the Pappus formulae for curves having Frenet frames, two special types of motions play a fundamental role:

121

Definition 2.2. A Frenet motion is a motion pF along a curve c for which a Frenet frame { f l ( t ) = c'(t),f Z ( t ) , ...,f n ( t ) } of c is an orthonormal frame associated to c p F , that is, pC(fi(0)) = f i ( t ) for 2 5 i 5 n.

Definition 2.3. The parallel motion pp along a curve c is defined by ptp = Pt. The next figures illustrates three different motions along an spherical helix of the same orthonormal basis, starting from the top of the curve:

Fig.1: Frenet motion

Fig.2: Parallel motion

Fig.3: Arbitrary motion

2.2. Motions as curves in SO(n) and o ( n ) An interesting tool to understand the Pappus formulae is to consider motions along curves as curves in the group of rotations. In order to do it, first we need to choose an auxiliary model motion 4". Given any motion 4, for each t E I , we consider the maps

A"(t)

:= (cp, M ) -1

0

pt : T,(o)M?&l

-

T,(o)Mr&', t E I ,

(2.2)

which are isometries preserving the orientation (as follows directly from the definition of motion), then elements of SO(n - 1). Therefore, once 4" is fixed, we can identify a motion 4 along c ( t ) with a smooth curve A" : I SO(n - 1) such that A"(0) = Id. Moreover, since the exponential map exp : o(n - 1) S O ( n - 1) defined on the Lie algebra o(n - 1) of SO(n - 1) is a diffeomorphism in a neighborhood of 0 E o(n - 1) and exp(0) = Id = AM(0), there is an interval J = [0,b] c I on which lnAM : J + o(n - 1) (where In = exp-l) is well defined and satisfies lnA"(0) = 0, and a motion can be considered as a curve 1nA" : J + o(n - 1) satisfying lnA"(0) = 0.

-

-

In the next sections, we shall use, as models 4", a Frenet motion 4F (which will appear in a natural way when we consider volumes of domains) and the parallel motion q5p (which will appear in the formulae for the volume of a hypersurface), then the associated curve A" will be denoted by AF and AP respectively.

122

The next pictures illustrate the graphics of the curves AF and AP in SO(2) = S1 corresponding to the motions along an spherical helix described by figures 1 to 3 above (in all cases the motion starts from the right):

Fig.4: Graphics of AF : I

+ S'

corre-

Fig.5: Graphics of AP : I

sponding to Frenet and parallel motions

S' corre-

sponding to Frenet and parallel motions

Fig 6: The graphics of the curves AF(t) and AP(t) associated to the motion in Fig.3.

The curves lnAF,lnAP : I illustrated in the next pictures:

-

o(2) =

R

of the same motions are

14==\ . .

-1

-2

-2

-3

-3

Fig.7: Graphics of lnAF : I

--t

R corre-

sponding to Frenet and parallel motions

6

8

~ i 8: ~ hi^^ . of l n ~ P I: -----t R corresponding to Frenet and parallel motions.

123

+-

50

Fig 9: The graphics of the curves lnAF(t) and lnAP(t) associated to the motion in Fig.3

2.3. Moment respect to a geodesic hyperplane and center of mass

Notation: I? will denote an oriented totally geodesic hypersurface of MT-' with unit normal vector field <. Let A be the connected component of - I? where points to. Let E be the real function defined on MT-l

<

by E(Z) =

1 if z E A, -1 otherwise.

Definition 2.4. If 93 is a domain or a submanifold of Ad;-', l ( z )denotes the distance from z E 93 to r, and dx denotes the volume form of %, the moment Adr(93) of 93 respect to r is

For every o E r, denoting by T~ the arc-length parameterized geodesic from o to 2, from the law of sine, we obtain

Definition 2.5. A point o E Ad:-' is the center of mass of 93 if Mr(93) = 0 for every oriented totally geodesic hypersurface r through o.

124

The center of mass of % is also the point where the function

’’

{

JBdist(p,x)’ dx if X = 0 X JBcx(dist(p, x)) dx if X # 0 ’

attains its minimum, because, for every unit vector (d$),(~)= -

J’ sx(r(x))(grad.~(p),E)rmdx B

E =

E T,M;-l,

-Mrc(%),

(2.5)

where r,(p) = dist(p,x), and ?!< is the totally geodesic hypersurface of orthogonal to at p , with the orientation induced by E. By computing the (Hessian $), it can be shown ( if % is contained in a ball of radius 5 $6when X > 0) that there is only one critical point of 8, which is its minimum . This gives the existence and uniqueness of the center of mass of %.

<

2.4. The volume of a domain obtained b y the motion along a curve We shall consider only domains in M”;’ satisfying that DO c UO,where UO is an open set of M;;’ such that exp : U c ‘31c t U = UtEI4t(U0)is a diffeomorphism. We shall denote Dt = 4 t ( D 0 ) , D = 4 ( [ 0 , L ] x) D O ) = UtE[o,yDt(the domain obtained by the motion 4 of DO along c ) , rt the totally geodesic hypersurface of orthogonal t o fi(t) (the normal vector of the Frenet frame), and ot the volume element of Dt.

Theorem 2.1. ([12], and [6]) The volume of the domain D obtained by the motion 4 of DO along c is given by v O l ( ~= ) L

S,,

cx(r1.0

L

-

Ici(t)Mr,(Dt)dt.

This formula for vol(D can be rewritten (as was indicated in [ 141 for weighted motions in R’), using the curve A F ( t )associated t o the motion 4,under the form vol(D) = L

Lo

cx(r)ao -

.h

L

I c i ( t ) M ~ F ( t ) ~ l r , ( D o ) d t . (2.6)

Formula (2.6) for vol(D) has two summands. The extrinsic geometry of c ( t ) is present only in the second one, and only through the first curvature Icl (t). Also the motion 4 is present only on the second term of the sum, through

125 M A F ( t ) - i r o ( D O ) As .

a consequence, the following corollary (remarked in 14 for weighted motions in R3) holds Corollary 2.1. ([6].[12] and 1141) 1) vol(D) does not depend on k l ( t ) i f and only i f one of the following conditions hold: a ) 4 is a Frenet motion and Mr(D0) = 0 , b) c(0) is the center of mass of DO. 2) vol(D) does not depend on the motion 4 i f and only i f c(0) is the center of mass of DO. In all these cases

vol(D) = L

Lo

cX(r)oo.

2.5. The volume of a hypersurface obtained b y the motion along a curve Let CObe an embedded hypersurface of MC : 1 with Co c Uo, Ct = $t(Co) and let C = 4 ( [ 0 L , ] )x C O )= U t E [ o , q C t(the domain obtained b y the motion 4 of Co along c).

Theorem 2.2. ( [ I ] , and 1161) The volume of the hypersurface C obtained by the motion 4 of COalong c is given by VOl(C) =

I’(LtJ

DNX, (TtT(t),Et)’s?

+ (CX

-

s x N x t z ( t ) h ( t ) )dxt 2

)4

where dxt is the volume element of Ct, xt E Ct, N, is the unit normal vector field along c(t) defined b y N x ( s ) = (P~((P;’N,,), Nxt being the unit vector tangent to the unique minimizing geodesic from c(t) to x t , rt is the parallel transport along the above geodesic, Et is the unit vector an T,,M:;l orthogonal to Ct (then Et = 4t*,,Eo), Nxti(t)= ( N ( t ) f, i ( t ) ) ,sx = s x ( r ( z t ) ) , cx = c x ( r ( x t ) ) ,and r ( x t ) = dist(c(t),x t ) . Again, this formula for vol(C) can be rewritten (as was indicated in [ 141 for weighted motions in R3), using the curves A F ( t )and A P ( t )associated to the motion 4, under the form

126

Two consequences of this theorem are

Remark 2.1. In general, all the curvatures of c appear in the formula for vol(C) (they are hidden in T ( t )a)situation , very different from that of the volume of a domain. The reason for this difference can be understood by comparison of formulae (2.8) and (2.6): whereas the dependence of vol(D) on the motion is only through the curve A F ( t )vol(C) , depends on this curve and on (lnAP)’(t), and are the curvatures ICz, ...,k,-1 of c ( t ) the responsible of the difference between the curves A F ( t )and AP(t)corresponding to the same motion along c(t). Theorem 2.3. ( [ l ]and [16]) Let CObe a hypersurface with center of m a s s at c(0) and let C p be the hypersurfaces obtained by parallel m o t i o n of CO along c. For every hypersurface C obtained by a m o t i o n of COalong c, vol(C)

2 vol(Cp) = L

Lo

cx(r) dxo.

The surfaces in the pictures below illustrate how an ellipse moving along a spherical helix, with the center of mass of the ellipse on the helix, has more surface for Frenet motion ( picture on the right) than for parallel motion (picture on the left), as Theorem 2.3 says. However, as it follows from (2.7), these two surfaces enclose the same volume.

After Theorem 2.3, a problem of uniqueness arises: Is parallel m o t i o n the only one giving the minimum of vol(C) for a given CO2. Because of Weyl’s tube formula, the above question has to be modified by restricting Co not to be a geodesic sphere. In fact, not only spheres, but also hypersurfaces of revolution are involved in the study of this problem of uniqueness. First, we define these type of hypersurfaces:

Definition 2.6. A hypersurface Jj of Ad:;’ is called a revolution hypersurface around a totally geodesic submanifold TJ of dimension d < n - 2 if which has Jj is invariant by the natural action of S O ( n - 1 - d ) on the submanifold TJ as the set of fixed points.

127

The next theorem is the best possible answer to the uniqueness question. Before stating it, let us observe that every d-dimensional vector subspace V c %,(olP defines an embedding of SO(n - 1 - d) into SO(n - 1) as the group of rotations of %*P which has V as the set of fixed points. Again, it induces an embedding of the algebra o(n - 1 - d) as a subalgebra of o(n - l), and V is the set of points in %,P which, under the action of all the elements of o(n-1 -d), goes to zero. Then when we refer to o(n-1-d) in the next theorem, we are also implicitly doing reference to V.

Theorem 2.4. (on "uniqueness", [I] and [ 2 ] ) Let 4 be a motion along a curve c(t) satisfying that the curve lnAP(t) is not contained in any hyperplane of a subalgebra o(n - 1 - d) of o(n - l), and vol(C) = vol(Cp), then (a) when d = 0, Co is contained in a geodesic sphere of MTC'. If, moreover, Co is closed, then it is a sphere, (b) when d > 0, and CO i s closed (and dist(Co,exp,(,)V) < if X > O), Co i s a revolution hypersurface around exp,(,)V. 3.

Motion along a submanifold

3.1. Definitions New notation: P: a regular submanifold of dimension q of a Riemannian manifold M of dimension n. VP:the normal bundle of P in M , W: the maximal neighborhood of P in %P on which exp is a diffeomorphism. D: the normal connection on %P induced by the Levi-Civita connection V on M . Pt:the D-parallel transport from p E P to a ( t ) along a geodesic a ( t )of P satisfying a(O)= p . L N : the Weingarten map of P at x E P in the direction of a unit N E %,P. r z ( z ) = dist(x,z), where dist is the distance in M , x E P and z E exp,(W n %,P). C, : P R/!,(z) = distP(pl x).

-

-

Definition 3.1. A motion cp along a submanifold P of M from p E P is a C" isomorphism cp : P x T P P V P of vector bundles over P satisfying that, for every x E P , the map pa:: %,P V,P defined by (P,([) = p ( x , 5) is an isometry of Euclidean vector spaces, and cp, = Id.

-

128

Let W, an open set in exp,(W n %,P) such that 4,(z) = exp, 0 vX 0 exp;'(z) is well defined for every z E W, and every x E P. Then cp defines a C" map 4 :P x W, M by

-

4(x,z ) = # J x ( z ) ,

W, = 4,(Wp)carry geodesics through p on geodesics the maps 4, : W, through x, and they are related with cp, by (Pa: = 4 x * p ,

then, their differentials 4,*,are isometries. Like for curves, a motion along P from p will be indistinctly defined by 4 or 'p. If there is a motion along P , then %P must be a trivial bundle. This is not true for most submanifolds P , but there is always a closed set 2 c P of zero measure satisfying that the restriction of %P to P - 2 is a trivial bundle and P - Z is connected. Then, f r o m now on, when we consider a motion 4 or 'p along a closed (compact without boundary) P , we shall consider the maps 4 and 'p defined only o n ( P- Z ) x M:'-q or o n ( P - 2 )x %,P respectively, although we shall not write the set Z explicitly. Also, the integral along P will mean the integral along P - Z . These restrictions are of no importance for us, because we are interested on volumes. In this section we shall consider the case when M = MT. Then , if M:iq denotes the totally geodesic submanifold of dimension n - q through x E P and tangent to %,P, the maps 4, : MrpPq- cut(p) - cut(x) are isometries of Riemannian manifolds, and q5p = I d .

-

More notation: For every z E Mr,-q - cut(p), 7," will denote the Vparallel transport from x E P to +,(z) E along the minimal geodesic in - cut(z) joining these two points. N, will denote the unit normal vector field along P such that N,(x):= the unit vector at x E P tangent to the minimizing geodesic of from x to $,(z) E Mr,-'. Then N,(x)= 'px(Nz(p)). For y E - cut(x), Ny(x) := N4L~(y)(x). A special role will be plaid by the following special motions: The radial motion f r o m p , { ( ~ " , , ~ p ,is defined by (u)for every u E %,P. 'p$(u)= x) A parallel motion is a motion satisfying that for every x E P , every X E T,P and every z E A4:-', D x ( N , ) = 0. This motion exists if and only if the D-holonomy of the normal bundle is trivial, and, if so, it coincides with the radial motion.

129

Definition 3.2. If D, is a domain and C, is a hypersurface contained in the set D, where the 4x are defined, the domain D and the hypersurface C obtained by the motion 4 along P from p are =

u

4x(Dp) = 4(P x Dp),

c = (J4X(C,) = 4 ( P x C,).

XEP

XEP

3.2. q-center of mass

The property of the center of mass of minimizing some functional can be used to generalize the notions and define :

Definition 3.3. The q-center of mass of 93 c M:-,'

'

if X ) dx ~ cxq(dist(p, x)) dx if X

JB dist(p, x

{ sB X

is the point where =0

#0 '

attains its minimum. For every [ E T,M;,-', (d5),([) = -q X2 J, cI-lsx(gradrx(p),<)dx, which allows to define, by analogy with (2.5), the q-momentum respect to a totally geodesic hypersurface of Mf,-' through p and orthogonal to a unit vector [ E TPMrp-', as follows

Definition 3.4. The q-momentum of 23 respect to I'< c MF-'

M r F p ( %= ) 4

x2

s,

c:-'sx(gradrx(p),

EP~.

and p is (3.1)

Then p is the q-center of mass if all the moments Mrep= 0 for every [. Let us remark that 1-moment and 1-center of mass are just the moment and the center of mass defined in section 2.3. However, if q > 1, the q-moment for q = 1. depends on p and re, whereas it depends on

3.3. The volume of D Theorem 3.1. ([2]) If dx and dz are, respectively, the volume elements of P and D,, then vol(D) = vol(P)

(3.2)

130

where sx = sx(r,(y)), cx = cx(r,(y)), H i u ( , ) is the i-th mean curvature of P in the direction of N y ( x ) ,and H , and h, are, respectively, the norm of the mean curvature at x and the unit vector in its direction. An interesting consequence is that vol(D) depends only on the second fundamental form of P and not on the ( k 1)-th fundamental forms for k 2 (we call ( k + 1)-th fundamental form to the ( k + 1) symmetric covariant tensor sk defined in [ 171). The non triviality of this statement will be clear when we note in Remark 3.1 that vol(C) generally depends on all the ( k + 1)-th fundamental forms. The formula for vol(D) looks similar to that for curves in the first two summands, and, if p i s the q-center of mass of D,, then vol(D) does not depend on the mean curvature of P . The situation looks different from that of Weyl’s Theorem (and from that of Pappus formulae for motions along curves). Here, it is no longer true that vol(D) depends only on D, and the intrinsic geometry of P , even when p is the q-center of mass of D, (a counterexample is shown in [ 21). However, we still have an analogous of Weyl’s Theorem if D, has enough symmetries. We can write the formula for vol(D) under a form still more similar to that given in the case of curve by using the symmetric s-tensor field Q, defined

+

>

by

(-l)s(pGV(,, =~

If

S X ( ~ ~ ( X ) , . . ’ , ~ Y ( X ) ) .

Qlil...i, sx

are the components of qsx in an orthonormal basis {px(eq+l), . . . ,pz(e,)}, the formula (3.2) becomes

(3.3) where each summand of the integrand along P is like a “q-moment of order s” of D, weighted by the second fundamental form of P at x . Since it is not possible to choice a point in D, satisfying that all these “q-moments of order s 2 2” be 0, we look for bodies D , with enough “symmetries” for vol(D) being dependent only on the Riemannian geometry of P and on D,. First, we define the “basic moments of D, of order b” :

131

Proff. for a Proff. for a Proff. for a Proff. for a Proff. for a

132

3.4.

Examples of q-symmetric domains

Given an orthonormal basis {e4+l,...,en} of %,P, ni is the hyperplane through the origin orthogonal to ei, nil, is the subspace through the origin of dimension n - q - 2 and orthogonal to ei and ek. A rotation of axis is an isometry of %,P preserving the orientation and having IIik as the set of fixed points.

Proposition 3.1. (121) If a domain D, in Mrp-q satisfies that there is an orthonormal basis {eq+l, ..., e n } of %,P such that (i) exp;l(D,) is invariant by each one of the n - q symmetries with respect t o I I i , n - q + l < i < n , and (ii) expp'(Dp) is invariant by each one of the rotations with axis n i k and angle 7 r / ( a b ) , b = 1,..., [ q / 2 ] ,with n - q 1 5 i n, i # k , then D, i s q-symmetric with respect to p .

+

<

The next pictures show some of these examples 2 and 3-symmetric plane domains

convex

not convex

4 and 5-symmetric plane domains

convex

not convex

133

6 and 7-symmetric plane domains

convex

not convex

2 and 3-symmetric domains in R3

convex

not convex

3.5. The volume of C

Theorem 3.3. ( [ 2 ] ) If C is the hypersurface in M; obtained by the motion 4 of a hypersurface C, in MTPpqalong a submanifold P of dimension q and {e,}X,l is an orthonormal basis of TxPdiagonalizing the Weingarten map L N z ( x ) then, 7

) ~( , ~N ,k$l e) ,i denotes the restriction where 9, = det(cx1- s ~ L ~ ~ ( (~T ~) D to the subspace of TxPorthogonal to e,, and we take the convention that det(cx1- S X L N , ( ~ )=) ~1 ~zf ;q = 1.

134

Remark 3.1. In general, all the k + l-th fundamental forms of P appear in the formula for vol(C) (through ga), (unlike in vol(D)). This dependence can be checked taking two helices hl and h2 in R3with the same curvature and different torsion, and considering, in R4,the surfaces hl x R and hz x R. Corollary 3.1. ([2])

and we have the equality for a parallel motion (which perhaps does not exist).

Remark: For a hypersurface C, of M:p-q it is possible to define the notion of q-symmetry with respect to p , as we did with D, in Definition 3.5. In general, the lower bound given by Corollary 3.1 depends on the motion 4 and the second fundamental form of P , but, when C, is q-symmetric with respect t o p , there is the corresponding Weyl's type formula (like (3.4)) for the right hand of the inequality (3.6), then it does not depend o n the motion, and it is a n intrinsic invariant of P and C,. Like for motions along curves, the motions from p along P in a neighborhood of p can be considered as immersions of P in SO(n - q ) or in o(n- q ) , once a model motion has been chosen. If P b ] denotes the open set of all the points x E P which can be joined to p by a unique minimizing geodesic, and cp$ is the radial motion from p along P b ] ,to every motion cp from p along P we associate a C" map

A : Plp]

--f

SO(n - 4);

-

A(x) = ((p$)-' o pz,

and the corresponding C" map 1nA : Pb] J:

I+

o(n - q ) lnA(x) :rt,P-rt,P.

Using this way of looking at motions along a submanifolds, it is proved in [ 21 an analog of Theorem 2.4 which shows that, for most of the motions, if the lower bound of vol(C) given by Corollary 3.1 is attained, then C, must be a revolution hypersurface. The following consequence is interesting:

Theorem 3.4. ([2]) If n - q = 2 and cp is not a parallel motion, the equality in (3.6) holds if and only if C, is an arc of a geodesic circle of

135

4. Holomorphic motions along complex curves Analogs of Weyl's formula for the volume of a tube around a complex submanifold P of a complex space form were obtained by F. J. Flaherty [ 41, P. A. Griffiths [ 131 and A. Gray [ 7'8, 91. They show that vol(P,) has an even weaker dependence of the geometry of P (it depends only on the topology and the volume of P (see [ 91 and [ l o ] ) . In this section we report the work [ 3 ] , which is the first step in the comprehension of the corresponding Pappus formulae in the complex case. In it is study the case when the real dimension of the submanifold is 2 , and which has as remarkable facts: (a) a formula (4.2) for vol(D) very similar to that for curves in the real case, (b) In real codimension 2, vol(D) depends only on the volume and the topology of P (and, of course, of D p ) with no restriction on the symmetries of D,, (c) In general codimension the dependence indicated in (b) remains true if and only if D, has some integral symmetries (Proposition 4.2) which are interpreted as the invariance of the complex moments of D, respect to all the complex hyperplanes through p .

More notation for this section: @Ad?:= the simply connected complex space form of real dimension 2n and holomorphic sectional curvature 4X. P:= a complex submanifold of @My of real dimension 2. P will be called a complex curve of CM?. @Ad,",--':= the totally geodesic submanifold through 5 , of real dimension 2n - 2, tangent to %,P. That is, CM:z-l = exp,(%,P). a:= the second fundamental form of P. K:= the sectional (or Gauss) curvature of P. It is a real function of P . CP,"-2:= complex projective space whose elements are the complex lines of '31,P. rI:= any element of @P,"-'. p':= a point of P satisfying a,' # 0.

4.1. D e f i n i t i o n of holomorphic and Frenet holomorphic motion

Definition 4.1. ([ 31) A holomorphic motion cp along a complex curve P of @M? from p E P is a motion along P from p satisfying that every map (pa: : %,P %,P is an holomorphic isometry of hermitian vector spaces.

-

The C" map

4:P

x

(CM:,-'

- cut(p))

-

CM? defined by cp satisfies

136

that the maps

4, : CMTp-'

-

cut(p)

-

CMTx-l - c u t ( z ) defined by & ( z ) = 4 ( ~z ),

are holomorphic isometries of Kahler manifolds.

Definition 4.2. ( [ 3 ] ) A F'renet holomorphic motion is a holomorphic motion pF (or f ) along P from p satisfying that, for every p ' , E ~ P such that a, # 0 # a,,, 9," o p ~ - l ( N p ' ( T p / P , T , / F )= ) a,(TzP,TzP).

As a consequence, we have i) If n = 2 or P is totally geodesic, then any holomorphic motion is a Frenet holomorphic motion. ii) If a,(T,P,T,P) # 0, a holomorphic motion pF (or 4 F )is Frenet if and only if pf(ap(T,P,T,P))

= a,(T,P,T,P)

for every z E P such that a,

# 0.

Similarly to Frenet frames for real curves, a sufficient (but not necessary) condition for the existence of a F'renet holomorphic motion along a complex curve P is that a, # 0 for every z E P. 4.2. The volume of D for holomorphic motions

By analogy with the definition of moment respect to a hyperplane in a real space form given in Definition 2.3, we give the

Definition 4.3. ( [ 3 ] ) The complex moment Qo,(II) of a domain D, in CMT,-' respect to a complex hyperplane (totally geodesic complex hypersurface) exp, (II'-) of CM?,-' is Qo,(H)

=

L,

~;(W(Y))~Y,

where rn denotes the distance to exp,(II') element of D,.

in CM:z-l

and d y is the volume

As in the real case, the following formula is computationally more useful than the definition Lemma 4.1. ((31)

where rn denotes the orthogonal projection o n II.

137

This definition raises the definition of the map

Qo, : CP,"-z

--+

R;

II H QD,(II),

which is a C" map. Moreover, we shall define Qo,(O) = 0.

Theorem 4.1. ([3]) Let D be the domain in CMF obtained by the holomorphic motion 4 f r o m p E P along a complex curve P of a domain D, of CM;,-' . If dx and dz are, respectively, the volume elements of P and D,, then

In a Frenet holomorphic motion, ~'or,(T,P,T,P) ( P m z ( P I ) = 7r'orP' (T,,P,Tp, P ) (cp; "Z(P)), for every x,p' E P satthen Qo,(ax(T,P,T,P)) = QD,,(~,/(T,/P,T,~P)) isfying ax # 0 # a,!, and, if a, = 0, then also 4X - K ( x ) = 0, and we obtain

Corollary 4.1. ( [ 3 ] ) Let DF be the domain in CM? obtained by a Frenet holomorphic motion dF f r o m p E P along a complex curve P of a domain D, of CMFP-'. T h e n vol(DF) = vol(P)

Lp

cz(r,(z))dz

(

- QD,,(CX,,(T,,P,T~/P)) 2Xv0l(P) -

1s,

-

)

K ( x ) dx ,

(4.3)

for any p' E P satisfying a,! # 0. That is, vol(D) depends only o n the geometry of D,, the intrinsic geometry of P , and the complex line

a,' (T,, P, Tp,P).

Remark. When X > 0 and P is closed, J p K ( x ) d s = 27rx(P), where x ( P ) is the Euler characteristic of P. Then, when X > 0 and P is closed, vol(DF) depends only on the moment of Db respect to the complex line a,r(T,,P, T,,P) and the topology and the volume of P. When n = 2, ax(TxP,TxP)= %,P or zero at every point x E P , then any motion is Frenet holomorphic and QD,, (a,! (T,! P, Tp,P ) = JD,,

s;(rZ(p'))dz>

so

138

Corollary 4.2. ( [ 3 ] ) Let D be the domain in CM: obtained b y a holomorphic motion f r o m p E P along a complex curve P of a domain Dp of “Mi,.T h e n #J

vol(D) =vol(P)

jD c i d z P

4.3. Bounds for vol(D) when n motion.

>2

and P admits a Frenet

I n this section we shall consider only not totally geodesic complex curves admitting a Frenet holomorphic motion # J F . If #J is a holomorphic motion from p along P , let 4’ be the holomorphic motion from p‘ along P defined by

& ( x ,2’) = # J 0~ #~;’(z’) for every z’E CM;,;’. If Dp, = #Jp,(Dp),then D = # J ( Px D,) = #J’(Px Dp,). Since we are interested only on computing vol(D) and we suppose that P is not totally geodesic, we can suppose, without loose of generality, that a p # 0, and, in all the above formulae where p’ appears, p = p’. As we have seen, for n > 2 and general Dp, vol(D) depends on the motion. In the next theorem we obtain upper and lower sharp bounds for vol(D). These bounds depend, again, only on the intrinsic geometry of P and the maximum and minimum values of the function QD, : CPp”-’ +R defined by (4.1). We identify the group U ( n - 1) with the group of holomorphic isometries of @.M:,-’ having p as a fixed point by the usual action RexppX = exp,RX

for every R E U ( n - 1).

Theorem 4.2. ( [ 3 ] ) Let P be a complex curve in CM?,and let D, be a domain of @MTp-’. There are two unitary transformations R,, RM E U ( n - 1) such that, for every motion #J along P f r o m p ,

V O I ( ( R ~ D ) ~2) VOI(D)

2v~i((~,~)F),

(4.5)

where (R,D)F = #J”(Px (R,Dp)), a E { m , M } . Moreover, these bounds are sharp (that is, they give the supremum and the infimum for vol(D) among all the D obtained by a holomorphic motion of D, along P f r o m p ) .

139

Remark. In general, the infimum (respectively, the supremum) given by (4.5) is not a minimum (respectively a maximum). 4.4. Symmetries on D, which make (4.3) true f o r any motion Since for n = 2 we have the stronger result of Corollary 4.2, in this section we shall suppose that n > 2.

Definition 4.4. We say that a domain D, of CMF,-’ the function Qn, is constant.

is 1C-symmetric if

Proposition 4.1. ([3])Let us suppose that P admits a Frenet holomorphic motion. Let D be a domain in CM? obtained by a holomorphic motion along a complex curve P f r o m p E P of a domain Dp . T h e n vol(D) does not depend o n the holomorphic motion 4 i f and only i f D, is lC-symmetric. I n this case, vol(D) is given by formula (4.3). Let us remark that, without the hypothesis on the existence of Frenet motions, it is still true that if D, is le-symmetric, then the equality (3.7) holds for every motion. The 1C-symmetry is equivalent to a kind of integral symmetry like that of Definition 3.5. This is the contents of the following

Proposition 4.2. ([3]) Dp is 1Csyrnmetric if and only if there is a J orthonormal basis { e z , J e 2 , ...,e n , Je,} of 91pP satisfying that, if we denote N,(P)~ = (N,(P),e i ) and N Z ( p ) i *= (N,(P),J e i ) , then

and

A family of examples satisfying the hypotheses of the above proposition can be constructed as follows. Let Dz be a central symmetric domain (that is,

140

invariant by - I d ) of the complex line C.Let us identify Cn-' = C x ... x C with %,P = IIz @ ... @ II, (where IIk are complex lines of %,P) in such a way that to the basis {l,i} of the ( k - 1)-th factor C corresponds the orthonormal basis { e k , J e k } of IIk. Let D, be the image of D2 x ... x D2 by this identification. Then D, = exp,(D),) satisfies the conditions of Proposition 4.2, and they are not geodesic balls. References 1. M. C. Domingo-Juan, X. Gual and V. Miquel, Pappus type theorems for hypersurfaces in a space form, Israel J. Math. 128,205-220, (2002). 2. M. C. Domingo-Juan and V. Miquel, Pappus type theorems for motions along a submanifold, submited to J. Diff. Geom. Appl., 2003. 3. M. C. Domingo-Juan and V. Miquel, On the volume of a domain obtained by a holomorphic motion, preprint, 2002. 4. F.J. Flaherty, The volume of a tube in complex projective space, Illinois J . Math. 16,627-638, (1982) 5. H. Flanders, A further comment of Pappus, American Math. Montly 77, 965-968, (1970). 6. W. Goodman and G. Goodman Generalizations of the theorems of Pappus, American Math. Montly 76,355-366, (1969). 7. A. Gray, An estimate f o r the volume of a tube about a complex hypersurface, Tensor (N. S.) 39,303-305, (1982). 8. A. Gray, Volumes of tubes about Kahler submanifolds expressed in terms of Chern classes, J. Math. SOC.Japan 36,23-35, (1984). 9. A. Gray, Volumes of tubes about complex submanifolds of complex projective space, Trans. Amer. Math. SOC.291,437-449, (1985). 10. A. Gray, Tubes, Addison-Wesley, New York, Reading, 1990. 11. A. Gray, Modern Differential Geometry of Curves and Surfaces, Second Edition, CRC, Boca Raton, Ann Arbor, London, Tokyo, Reading, 1997. 12. A. Gray and V. Miquel, O n Pappus-Type Theorems on the Volume in Space Forms Annals of Global Analysis and Geometry 18,241-254, (2000). 13. P.A. Griffiths, Complex differential and integral geometry and curvature integrals associated to singularities on complex analytic varieties, Duke Math. J. 45, 427-472, (1978). 14. X. Gual and V. Miquel, Pappus-Guldin Theorems for Weighted Motions, preprint, 2002. 15. H. Hotelling, Tubes and spheres in n-space and a class of statistical problems, Amer. J . Math. 61,440-460, (1939). 16. L. E. Pursell, More generalizations of a theorem of Pappus, American Math. Montly 77,961-965, (1970). 17. M. Spivak, A Comprehensive Introduction to Differential Geometry, volume 4, Publish or Perish, Boston, 1975. 18. H. Weyl, On the volume of tubes, Amer. J. Math. 61,461-472, (1939).

NON-ARCHIMEDEAN GEOMETRY AND PHYSICS ON ADELIC SPACES *

BRANKO DRAGOVICH Institute of Physics, P.O. Box 57, 11001 Belgrade, YUGOSLAVIA

E-mail:dragovichaphy. bg. ac.yu

This is a brief review article of various applications of non-Archimedean geometry, p a d i c numbers and adeles in modern mathematical physics.

1. Introduction

It is well known that theoretical physics is strongly related to mathematics. Space, time and matter are basic concepts in all physical theories. They have become usually profound and gradually unified in new theories using more general mathematical tools. For example, transition from nonrelativistic to relativistic kinematics required to pass from Euclidean space and time to the Minkowski space. To describe phenomena in strong gravitational fields and accelerated frames, general theory of relativity was discovered, where space-time is described by pseudo-Riemannian geometry which is related to the distribution of matter. Dynamics in quantum mechanics can be regarded as motion of a particle in a phase space (x,k) with symplectic geometry and the Heisenberg uncertainty AxiAkj 2 ( t i / 2 ) b i j , where ti = h/(27r)is the reduced Planck constant. In recent years, noncommutative geometry based on relation [xi, xi] # 0 has attracted a significant interest in quantum theory. According to various considerations, which take together quantum and gravitational principles, there is a restriction on empirical accuracy of physical variables due to the relation

* MSC2000: 81T30, 12Y25, llE95, 51P05, llR56. Keywords : non-Archimedean geometry, adelic quantum mechanics, adelic quantum c o s mology, p-adic string.

141

142

where Ax is an uncertainty measuring a distance, Co is the Planck length, G is Newton’s gravitational constant and c is the speed of light in vacuum. The uncertainty (1.1) means that one cannot measure distances smaller than Co. Since this result is derived assuming that space-time consists of real points and has an Archimedean geometry, it becomes desirable to employ also non-Archimedean geometry based on padic numbers. Quite natural framework t o consider simultaneously real (Archimedean) and p adic (non-Archimedean) spaces is by means of an adelic space. In this paper, at an introductory level I briefly review some basic characteristics of nomarchimedean geometry, padic numbers and adeles,, as well as their use in some parts of modern mathematical physics. 2. Non-Archimedean Geometry and p-Adic Numbers Recall that having two segments of straight line of different lengths a and b, where a < b, one can overpass the longer b by applying the smaller a some n-times along b. In other words, if a and b are two positive real numbers and a < b then there exists an enough large natural number n such that na > b. This is an evident property of the Euclidean spaces (and the field of real numbers), which is known as Archimedean postulate, and can be extended to the standard Riemannian spaces. One of the axioms of the metric spaces is the triangle inequality which reads:

4 x 7 Y> 5 4 x ,z> + 4% Y),

(2.1)

where d ( s , y ) is a distance between points x and y. However, there is a subclass of metric spaces for which triangle inequality is stronger in such way that:

4 x 7 Y> I m a x { d ( z ,z ) , 4 z , Y>) P

+,

z)

+ 4 2 7 Y).

(2.2)

Metric spaces with strong triangle inequality (2.2) are called non-Archimedean or ultrametric spaces. Since a measurement means quantitative comparison of a given observable with respect to a fixed value taken as its unit, it follows that a realization of the Archimedean postulate is practically equivalent t o the measurements of distances. According to the uncertainty (l.l),it is not possible to measure distances shorter than 10-33cm and consequently there is no place for an Archimedean geometry beyond the Planck length. By this way, standard approach to quantum gravity, which is based only on Archimedean geometry and real numbers, predicts its own breakdown at the Planck scale. Hence, a new approach, which takes into account not only Archimedean

143

but also a non-Archimedean geometry, seems to be quite necessary. The most natural ambient to realize both of these geometries is an adelic space, which is a whole of real and all padic spaces. Any set with trivial metric ( d ( z ,y) = 1 if z # y and d(z, y) = 0 if z = y) is a simple example of the non-Archimedean space. Set of all real polynomials R[X],for which metric is defined by a suitable valuation, is another example of the non-Archimedean space. Namely, for a nonzero polynomial f E R[X] given by

f

= a,xn

+ . . . + a1X + ao,

a,

# 0,

one can define the degree of f as d(f) = n, and d(f) = -XI if f is the zero polynomial. Then a norm can be introduced as If1 = pd(f) if f # 0, and If1 = 0 if f = 0, where p is a real number greater than 1 . Since

If + 91 5 maz{lfl, 191) and lfgl = lflbl, where f , g E this is a non-Archimedean valuation which gives a non-Archimedean geometry by d(f,g) = If - 91. This can be extended to the field R(X) of rational functions. On the field of complex numbers C there exist infinitely many inequivalent non-Archimedean valuations which make C into complete valued fields (see [ 33., p. 461) with non-Archimedean geometries. Hyperreal numbers in nonstandard analysis also have non-Archimedean properties. In the sequel we will restrict exposition to the spaces of padic numbers as presently the most important class of non-Archimedean geometries. In the rest of this section, a brief review of some basic properties of padic numbers and their functions will be presented. There are physical and mathematical reasons to introduce padic numbers starting with the field of rational numbers Q and performing completions with respect to its non-Archimedean valuations. From physical point of view, numerical results of all experiments and observations are some rational numbers. From algebraic point of view, Q is the simplest number field of characteristic 0. Recall that any 0 # z E Q can be presented as infinite expansions into the two essentially different forms [ 331 :

i"],

c

--oo

z=

aklOk,

ak =0,1,'

' '

, 9 , a,

# 0,

(2.3)

b,#O,

(2.4)

k=n +a3

z= c b k p k ,

bk=0,1,"',p-1,

k=m

where (2.3) is the ordinary one to the base 1 0 , (2.4) is another to the base p ( p is any prime number), and n, m are some integers which depend on

144

z. The above representations (2.3) and (2.4) exhibit the usual repetition of digits, however the expansions are in the mutually opposite directions. The series (2.3) and (2.4) are convergent with respect to the metrics induced by the usual absolute value l-lm and padic absolute value (padic norm, padic valuation) I ., I respectively. Due to the Ostrowski theorem these valuations exhaust all possible inequivalent non-trivial norms on Q. Performing completions to all non-trivial norms, i. e. allowing all possible combinations for digits, one obtains standard representation of real and padic numbers in the form (2.3) and (2.4), respectively. Thus, the field of real numbers iR and the fields of padic numbers Q, exhaust all number fields which may be obtained by completion of Q, and consequently contain Q as a dense subfield. Since padic norm of any term in (2.4) is I b k p k J p= pPk if bk # 0, geometry of padic numbers is the non-Archimedean one, i e . strong triangle inequality 1% yl, 5 maz(lzl,, Iyl,) holds and 1z(, = P - ~ . iR and Qp have many distinct algebraic and geometric properties. Unlike the real case, there are different algebraic extensions for all orders of padic algebraic equations. Algebraic closure of Q, is an infinite Q,-vector space and it is not metrically complete. After completion it becomes the field of padic complex numbers C, which is also algebraically closed, and is a padic analogue of the ordinary C. This C, has much richer structure than C and offers many new possibilities in related analysis and possible applications. It is often of interest for applications the ring of padic integers Z,= {z E Q, : 1x1, 5 l}, i.e. padic integers have the representation

+

z=xo+z1p+zzp

2

+... .

pAdic numbers can be suitably visualized by means of trees [ 261 and as fractals in the Euclidean spaces [ 321. Z,has property

Z, 3 p Z ,

3 p 2 z p 3 p 3 2 , I...

being a natural mathematical tool to investigate physical structures with a hierarchy. Z, is topologically compact and complete, and Z is dense in Z,. Q, is locally compact, separable and totally disconnected complete topological space. Some padic spaces have rather exotic properties: (i) all triangles are isoceles and unequal side is the shortest one, (ii) two discs cannot partially intersect, and (iii) every point of a disc may be regarded as its center.

145

There is no natural ordering on Qp. However one can introduce a linear order on Qp in the following way: z < y if 1zIp< lylp, or if Iz(p= lylp then there exists such index T 2 0 that digits satisfy 20

= yoyo,

...

= Ylr

7

2,-1

=Yr-17

Xr

Here, zk and yk are digits related t o z and y, respectively, in their expansions of the form z = p m ( z o z l p z 2 p 2 . . . ). This ordering is very useful in calculation of padic path integrals by the time discretization method. There are primary two kinds of analysis on Qp which are of interest for physics, and they are based on two different mappings: Qp Qp and Qp @. We use both, in classical and quantum padic models, respectively. Elementary padic valued functions and their derivatives are defined by the same power series (i.e with the same rational coefficients but padic arguments) as in the real case. Their regions of convergence are determined by means of padic norm and they are usually restricted to lzlp < 1. It is worth noting that

+

+

+

-

-

n>O

where Pk(n) is a polynomial in n of degree k with integer coefficients, converges on lzlp 5 1 for all p and their rational summation is investigated in [ 111. As a definite padic valued integral of an analytic function f(z) = fo + f 1 z + f 2 z 2 . . one takes difference of the corresponding antiderivative in end points, ie.

+.

C00

[f(z)dz=

fn

n=O

n+l

( p + l - an+l>.

(2.5)

Usual complex-valued functions of padic variable, which are employed in mathematical physics, are: (i) an additive character x p ( z )= exp 2 7 r i { ~ }where ~, { x } is~ the fractional part of z E Q p , (ii) a multiplicative character 7r,(z) = IzI;, where s G @, and (iii) locally constant functions with compact support, like O( Izlp), where

146

There is well defined Haar measure and integration [ 371. So, we have

-

where Sp(u)is the padic Dirac S function. The number-theoretic function

A p ( z ) in (2.8) is a map X p : Q;

m XP(X) =

(F), i (F),

C defined as follows:

PZ2,

= 2j,

m=2j+1, m

=2j

+ 1,

p~l(mod4), p

E

where z is presented as z = pm (zo + Z I P + z2p2 is the Legandre symbol defined as

(5)

1, -1, 0,

if if if

(2.9)

3(mod 4),

+ -..), and m , j E Z.

a = y 2 (modp),

a g y 2 (modp), a = 0 (modp),

(2.11)

and Q; = Qp \ (0). It is often sufficient to use their standard properties:

Recall that the real analogues of (2.7) and (2.8) have the same form, ie.

147

where Q, E R , xm(x) = exp (- 2 7r i IC) is additive character in the real case and 6, is the ordinary Dirac 6 function. Function X,(IC) is defined as

A&)

=

sign x

J

-

5

i

E R* = R \ {0}

(2.15)

and exhibits the same properties (2.12). For a more information on usual properties of padic numbers and related analysis one can see [ 24, 33, 37, 30, 29, 211.

3. Adeles and Their Functions Real and padic numbers are unified in the form of adeles. An adele x is an infinite sequence (see [ 21, 40, 311) 2 =

(x,,

,

22,. .. Z p , . .

.) ,

(3.1)

where IC, E R and xp E Q p with the restriction that for all but a finite set 9 of primes p one has xp E Z,. Rational numbers are naturally embedded in the space of adeles. Componentwise addition and multiplication are ordinary arithmetical operations on the ring of adeles A, which can be regarded as

9

PES

A is a locally compact topological space. There are also two kinds of analysis over topological ring of adeles A, which are generalizations of the corresponding analysis over R and Qp. The first one is related to the mapping A A and the other one to A C. In complex-valued adelic analysis it is worth mentioning an additive character

-

x(x)=X

-

n

O o ( ~ ~ )

xp(xp)

1

(3.3)

P

a multiplicative character P

and elementary functions of the form

where +,(x,) Iz,

1% ,4

(x,)

-

PES

-

P@S

is an infinitely differentiable function on R such that 0 as 1x,l, 03 for any n E { 0 , 1 , 2 , . . . }, and 4p(xp)

148

are some locally constant functions with compact support. All finite linear combinations of elementary functions (3.5) make the set S(d) of the Schwartz-Bruhat adelic functions. The Fourier transform of 4(x) E S(d), which maps S(d)onto d,is

) defined by (3.3) and dx = d x , dx2dx3 ... is the Haar where ~ ( x y is measure on A. One can define the Hilbert space on A, which we will denote by L 2 ( d ) . It contains infinitely many complex-valued functions of adelic argument (for example, Q l ( x ) ,Q 2 ( x ) ,. . . ) with scalar product (Ql, Q2) =

J, Wl.)

Q2(z) dx

and norm 11Q11 = p,Q ) +

< co

A basis of L , ( d ) may be given by the set of orthonormal eigefunctions in a spectral problem of the evolution operator U ( t ) ,where t E A. Such eigenfunctions have the form

1Cls,a(z7t)= +i,)(xCo,tCo)

n

PES

+a(p,)(Zpr t P )

n

R(IxPlP> 7

(3.7)

P#S

$La)

where E &(R) and q!&) E L2(Qp)are eigenfunctions in ordinary and padic cases, respectively. Indices n,a 2 , . . . ,a p , . . are related t o the corresponding real and padic eigenvalues of the same observable in a physical system. R(lxplp) is an element of L2(Qp), defined by (2.6), which is invariant under transformation of an evolution operator Up( t p )and provides convergence of the infinite product (3.7). For a fixed S,states $ ~ , ~ ( z , t ) in (3.7) are eigefunctions of L Z ( d ( S ) ) ,where d(S) is a subset of adeles d defined by (3.2). Elements of L 2 ( d ) may be regarded as superpositions Q ( x )= C s C ( S )Qs(z), where Q s ( x ) E L z ( d ( S ) ) ,ie.

and

Cs IC(S)l& = 1.

Theory of padic generalized functions is presented in [ 371. Construction of generalized functions on adelic spaces is a hard task, but there is some progress within adelic quantum mechanics [ 121.

149

4. Quantum Mechanics on Adelic Spaces There are a number of reasons to use padic numbers and adeles in investigation of mathematical and theoretical aspects of modern quantum physics. Some primary of them are: (i) the field of rational numbers Q, which contains all observational and experimental numerical data, is a dense subfield not only in R but also in the fields of padic numbers Q p , (ii) there is an analysis [ 371 within and over Q p like that one related to R, (iii) general mathematical methods and fundamental physical laws should be invariant [ 381 under an interchange of the number fields and Q p , (iv) there is a quantum gravity uncertainty A x (see (l.l)),when measures distances around the Planck length l,, which restricts priority of Archimedean geometry based on real numbers and gives rise to employment of non-Archimedean geometry related to padic numbers, (v) it seems to be quite reasonable to extend standard Feynman’s path integral method to non-Archimedean spaces, and (vi) adelic quantum mechanics [ 91 is consistent with all the above assertions. In order to investigate adelic quantum theory in a systematic way it is natural to start by formulation of adelic quantum mechanics. According to [ 9 ] , adelic quantum mechanics contains three main ingredients (&(A), W ( z ) ,U ( t ) )where: (i) &(A) is an adelic Hilbert space, (ii) W ( z ) denotes Weyl quantization of complex-valued functions on adelic classical phase space, and (iii) U ( t ) is the unitary representation of an evolution operator on L2 (A). Canonical noncommutativity in padic case can be represented by the Weyl operators ( h = I)

in the form

150

It is possible to introduce the product of unitary operators 1

@P(4

=XP

(-5

z

k) k P < P ) Q P ( 4 ,

E Qp x Qp,

(4.4)

which is a unitary representation of the Heisenberg-Weyl group. Recall that this group consists of the elements ( 2 , a ) with the group product ( z ,a ) . (z’, a’) = ( z

+ z’, a + a’ + -21 B ( z ,z’)),

(4-5)

+

where B ( t , z ’ ) = - kq’ qk’ is a skew-symmetric bilinear form on the phase space. Dynamics of a padic quantum model is described by a unitary evolution operator U P ( t )in terms of its kernel Icp)(z,y)

/

U p ( t )$ ( p ) (=4

I c P ) ( Z , Y)

$(P’(Y)

dY.

Q,

(4.6)

In this way, padic quantum mechanics [ 351 is given by a triple ( L 2 ( Q P ) , wP(zP>7U P ( t P ) ) .

(4.7)

Keeping in mind that ordinary quantum mechanics can be also given as the analogue of (4.7), ordinary and padic quantum mechanics can be unified in the form of the above-mentioned adelic quantum mechanics [ 91. Adelic evolution operator U ( t ) is defined by

where w = c q 2,3,. . . ,p

,.. . . The eigenvalue problem for U ( t ) reads

U ( t )$a(.)

= X(Ea t )$a(z),

(4.9)

where $, are adelic eigenfunctions, E, = (Ern,EZ, ..., E p ,...) is the corresponding adelic energy. Note that any adelic eigenfunction has the form (3.7). A suitable way to compute padic propagator Kp(~’‘,t’‘;s’,t’)is to use Feynman’s path integral method, i.e.

It has been evaluated [ 3, 71 for quadratic Lagrangians in the same way for real and padic cases, and the following exact general expression for

151

propagator is obtained:

1

q x ” ,t”;x’,t’) = A,

a2S

1,-

1 a2S

(-5m G ) l h

xu (-

1 S(x”,t”;2’,t’)), (4.11)

where A, functions satisfy relations (2.12) and S(x”,t”;x’,t’))is the action for classical trajectory. When one has a system with more than one dimension with uncoupled spatial coordinates, then the total v-adic propagator is the product of the corresponding one-dimensional propagators. As an illustration of padic and adelic quantum-mechanical models, the following one-dimensional systems with the quadratic Lagrangians were considered: a free particle and a harmonic oscillator [ 37, 91, a particle in a constant field [ 41, a free relativistic particle [ 61 and a harmonic oscillator with time-dependent frequency [ 51. Adelic quantum mechanics takes into account ordinary as well as padic quantum effects and may be regarded as a starting point for construction of a more complete quantum cosmology and stringlhl-theory. In the lowenergy limit adelic quantum mechanics effectively becomes the ordinary one

[GI. 5 . Adelic Quantum Cosmology

The main task of quantum cosmology is to describe the very early stage in the evolution of the Universe. At this stage, the Universe was in a quantum state, which should be described by a wave function. Usually one takes it that this wave function is complex-valued and depends on some real parameters. Since quantum cosmology is related to the Planck scale phenomena it is natural to reconsider its foundations. We maintain here the standard point of view that the wave function takes complex values, but we treat its arguments in a more complete way. Namely, we regard space-time coordinates, gravitational and matter fields t o be adelic, i.e. they have real as well as padic properties simultaneously.

As there is no appropriate complex-valued padic Schrodinger equation, so there is not also padic generalization of the Wheeler - De Witt equation for cosmological models. Instead of differential approach, Feynman’s path integral method was exploited [ 10 ] and minisuperspace cosmological models are investigated as models of adelic quantum mechanics [ 16, 81. Adelic minisuperspace quantum cosmology is an application of adelic quantum mechanics to the cosmological models. In the path integral approach

152

to standard quantum cosmology, the starting point is Feynman’s path integral method, ie. the amplitude to go from one state with intrinsic metric h:j and matter configuration @ on an initial hypersurface C’ to another state with metric h:>and matter configuration 4“ on a final hypersurface C” is given by the path integral

over all four-geometries gpv and matter configurations a, which interpolate between the initial and final configurations. In (5.1) S m [ g p v ,a] is an Einstein-Hilbert action for the gravitational and matter fields. This action can be calculated using metric in the standard 3 +1 decomposition ” - ( N 2 - Ni ds = gclvdz’ d ~ =

Ni)dt2 + 2 Ni d z i dt + hij dzi d z j , (5.2)

where N and Ni are the lapse and shift functions, respectively. To perform padic and adelic generalization we make first padic counterpart of the action using form-invariance under change of real to the padic number fields. Then we generalize (5.1) and introduce padic complex-valued cosmological amplitude

Since the space of all three-metrics and matter field configurations on a three-surface, called superspace, has infinitely many dimensions, in computation one takes an approximation. A useful approximation is t o truncate the infinite degrees of freedom to a finite number q a ( t ) , ( a = 1,2, ...,n). In this way, one obtains a minisuperspace model. Usually, one restricts the four-metric to be of the form (5.2), with N i = 0 and hij approximated by q a ( t ) . For the homogeneous and isotropic cosmologies the metric is a Robertson-Walker one, which spatial sector reads

+

hij dzi drcj = a 2 ( t )d 0: = a 2 ( t )[dx2 sin2 X(d02

+ sin20dq2)],

(5.4)

where a ( t ) is a scale factor. If we use also a single scalar field 4, as a matter content of the model, minisuperspace coordinates are a and 4. More generally, models can be homogeneous but also anisotropic ones. For the boundary condition qa(t”) = q:, qa(t‘) = q; in the gauge N = 1, we have v-adic minisuperspace propagator

153

where

is an ordinary quantum-mechanical propagator between fixed minisuperspace coordinates (q&,q&1) in fixed times. Sw is the v-adic action of the minisuperspace model, i. e.

where fa@ is a minisuperspace metric ( d s g = fa p dqa d q p ) with an indefinite signature (-, . . . ). This metric includes spatial (gravitational) components and also matter variables for the given model. The standard minisuperspace ground-state wave function in the HartleHawking (no-boundary) proposal [ 25 ] is defined by functional integration in the Euclidean version of

+, +,

% & j

1=

%pdm

ID(@),

Xm(-S&7pu,

@I),

(5.8)

over all compact four-geometries gpu which induce hij at the compact three-manifold. This three-manifold is the only boundary of the all fourmanifolds. Extending Hartle-Hawking proposal to the padic minisuperspace, an adelic Hartle-Hawking wave function is the infinite product

where path integration must be performed over both, Archimedean and non-Archimedean geometries. If evaluation of the corresponding functional integrals for a minisuperspace model yields $ ( q a ) in the form (3.7), then we say that such cosmological model is a Hartle-Hawking adelic one. It is shown [ 101 that the de Sitter minisuperspace model in D = 4 space-time dimensions is the Hartle-Hawking adelic one. It is shown in [ 16, 81 that padic and adelic generalization of the minisuperspace cosmological models can be successfully performed in the framework of padic and adelic quantum mechanics [ 91 without use of the HartleHawking approach. The following cosmological models are investigated: the de Sitter model, model with a homogeneous scalar field, anisotropic Bianchi model with three scale factors and some two-dimensional minisuperspace models. As a result of padic effects and adelic approach, in these models there is some discreteness of minisuperspace and cosmological constant. This kind of discreteness was obtained for the first time in the context of the Hartle-Hawking adelic de Sitter quantum model [ 101.

154

6. Adelic String/M-theory

A notion of padic string was introduced in [ 391, where the hypothesis on the existence of non-Archimedean geometry at the Planck scale was made, and string theory with padic numbers was initiated. In particular, generalization of the usual Veneziano and Virasoro - Shapiro amplitudes with complex-valued multiplicative characters over various number fields was proposed and padic valued Veneziano amplitude was constructed by means of padic interpolation. Very successful padic analogues of the Veneziano and Virasoro- Shapiro amplitudes were proposed in [ 191 as the corresponding Gel'fand-Graev [ 21 ] beta functions. Using this approach, Freund and Witten obtained [ 201 an attractive adelic formula, which states that the product of the crossing symmetric Veneziano (or Virasoro - Shapiro) amplitude and its all padic counterparts equals unit (or a definite constant). This gives possibility to consider an ordinary four-point function, which is rather complicate, as an infinite product of its inverse padic analogues, which have simple forms. These first papers induced an interest in various aspects of padic string theory (for a review, see [ 2, 371). A recent interest in padic string theory has been mainly related to the generalized adelic formulas for four-point string amplitudes [ 34 ] , the tachyon condensation [ 22 ] , nonlinear dynamics [ 36 ] and the new attractive adelic approach [ 13 1. Like in the ordinary string theory, the starting point in padic string theory is a construction of the corresponding scattering amplitudes. Recall that the ordinary crossing symmetric Veneziano amplitude can be presented in the following forms:

= g2/DX exp

(-

27r

1

d

PX,

&Jp)fi/

(

d 2ujexp ikF)Xp) , (6.4)

j=1

where h = 1, T = 1/7r, and a = -a(s) = -1 - 52 , b = -a(t), c = -a(.) with the condition s t u = -8, i.e. a b c = 1. To introduce a padic Veneziano amplitude one can consider padic analogues of all the above four expressions. pAdic generalization of the first

+ +

+ +

155

expression was proposed in [ 191 and it reads

where 1. I p denotes padic absolute value. In this case only string world-sheet parameter IC is treated as padic variable, and all other quantities maintain their usual (real) valuation. An attractive adelic formula of the form

A,(a, b)

n

Ap(a,b) = 1

(6.6)

P

was found [20], where A,(a,b) denotes the usual Veneziano amplitude

(6.1). A similar product formula holds also for the Virasoro- Shapiro amplitude. These infinite products are divergent, but they can be successfully regularized. Unfortunately, there is a problem to extend this formula to the higher-point functions. pAdic analogues of (6.2) and (6.3) were also proposed in [ 3 9 ] and [ 11, respectively. In these cases, world-sheet, string momenta and amplitudes are manifestly padic. Since string amplitudes are padic valued functions, it is not so far enough clear their physical interpretation. Expression (6.4) is based on Feynman's path integral method, which is generic for all quantum systems and has successful padic generalization. pAdic counterpart of (6.4) is proposed in [ 131 and has been partially elaborated in [ 141 and [ 181. Note that in this approach, padic string amplitude is complex-valued, while not only the world-sheet parameters but also target space coordinates and string momenta are padic variables. Such padic generalization is a natural extension of the formalism of padic [ 351 and adelic [ 91 quantum mechanics to string theory. This is a promising subject and should be investigated in detail, and applied to the branes and Mtheory, which is presently the best candidate for the fundamental physical theory.

7. Concluding Remarks One of the very interesting and fruitful recent developments in string theory has been noncommutative geometry and the corresponding noncommutative field theory, which may be regarded as a deformation of the ordinary one in which field multiplication is replaced by the Moyal (star) product

156

where x = (x’,x2,.. . ,x d ) is a spatial point , and B i j = - f3ji are noncommutativity parameters. Replacing the ordinary product between noncommutative coordinates by the Moyal product (7.1) we have x i * j,

- xj

* xi

= ih @ ,

(7.2)

which resembles the usual Heisenberg algebra. It is worth noting that one can introduce [ 181 the Moyal product in p a d i c quantum mechanics and it reads

where d denotes spatial dimensionality, and j ( k ) , g(k’) denote the Fourier transforms of f ( x ) and g ( x ) . Some real, p a d i c and adelic aspects of the noncommutative scalar solitons [ 231 are investigated in Ref. [ 171. A natural extension of adelic quantum mechanics t o adelic field theory is considered [ 151, as well. pAdic quantum mechanics with p a d i c valued wave functions has been investigated and presented in [ 37, 27, 281. There have been also applications of p a d i c numbers and adeles relevant for some other directions of mathematical research in physics, but I restricted this review t o some aspects of quantum theory.

Acknowledgements The work on this paper was supported in part by the Serbian Ministry of Science, Technologies and Development under contract No F1426 and by RFFI grant 02-01-01084.

References 1. I. Ya. Aref’eva, B. Dragovich and I. V. Volovich, O n the adelic string amplitudes, Phys. Lett. B209, 445-450, (1988). 2. L. Brekke and P. G. 0. Freund, p-Adic numbers in Physics, Phys. Rep. 233, 1-63, (1993). 3. G. S. DjordjeviE and B. Dragovich, p-Adic Path Integrals f o r Quadratic A c tions, Mod. Phys. Lett. A12, 1455-1463, (1997). 4. G. S. Djordjevic and B. Dragovich, O n p-adic functional integration, Proc. of the I1 Mathematical Conference, Pristina, Yugoslavia, 221-228, (1997).

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5. G. S. Djordjevic and B. Dragovich, p-Adic and Adelic Harmonic Oscillator with a Time-dependent Frequency, Theor. Math. Phys. 124,1059-1067, (2000). 6. G. S. Djordjevi6, B. Dragovich, Lj. NeSi6, p-Adic and Adelic Free Relativistic Particle, Mod. Phys. Lett. A14, 317-325, (1999). 7. G. S. DjordjeviC, B. Dragovich and Lj. NeBiC, Adelic Path Integrals for Quadratic Lagrangians, Infinite Dimensinal Analysis, Quantum Probability and Related Topics 6, 179-195, (2003), arXiv: hep-th/0105030. 8. G. S. Djordjevic, B. Dragovich, Lj. Nesic and I. V. Volovich, p-Adic and Adelic Minisuperspace Quantum Cosmology, Int. J. Mod. Phys. A17, 14131433, (2002). 9. B. Dragovich, Adelic Model of Harmonic Oscillator, Theor. Math. Phys. 101, 1404-1412, (1994); Adelic Harmonic Oscillator, Int. J. Mod. Phys. A10, 2349-2365, (1995). 10. B. Dragovich, Adelic Wave Function of the Universe, Proc. of the Third A. F'riedmann Int. Seminar on Gravitation and Cosmology, Friedmann Lab. Publishing, St. Petersburg, 311-321, (1995). 11. B. Dragovich, O n some p-adic series with factorials, Lecture Notes in Pure and Applied Mathematics 192,95-105, (1997). 12. B. Dragovich, O n Generalized Functions in Adelic Quantum Mechanics, Integral Transforms and Special Functions 6, 197-203, (1998). 13. B. Dragovich, On Adelic Strings, arXiv: hep-th/0005200. 14. B. Dragovich, p-Adic and Adelic Strings, Proc. Int. Conference dedicated to the memory of Prof. E. F'radkin: Quantization, Gauge Theory and Strings, Scientific World, Moscow, 108-114, (2001). 15. B. Dragovich, O n p-Adic and Adelic Generalization of Quantum Field Theory, Nucl. Phys. B (Proc. Suppl.) 102-103,150-155, (2001). 16. B. Dragovich and Lj. Nesic, p-Adic and Adelic Generalization of Quantum Cosmology, Gravitation and Cosmology 5,222-228, (1999). 17. B. Dragovich and B. Sazdovic, Real, p-Adic and Adelic Scalar Solitons, Summer School in Modern Mathematical Physics, Institute of Physics, Belgrade, SFIN A3,283-296, (2002). 18. B. Dragovich and I. V. Volovich, p-Adic Strings and Noncommutativity, Proc. Workshop on Noncommutative Structures in Mathematics and Physics, NATO Science Series: 11. Mathematics, Physics and Chemistry - Vol. 22, Kluwer AP, 391-399, (2001). 19. P. G. 0. Freund and M. Olson, Non-Archimedean strings, Phys. Lett. B199, 186-190, (1987). 20. P. G. 0. Freund and E. Witten, Adelic string amplitudes, Phys. Lett. B199, 191-194, (1987). 21. I. M. Gel'fand, M. I. Graev and I. I. Piatetskii-Shapiro, Representation Theory and Automorphic Functions (in Russian), Nauka, Moscow, 1966. 22. D. Ghoshal and A. Sen, Tachyon Condesation and Brane Descent Relations in p-adic String Theory, Nucl. Phys. B584 (300), (2000), arXiv: bep-th/0003278.

23. R. Gopakumar, S. Minwalla and A. Strominger, Noncommutative Solitons,

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JHEP 05 (020), arXiv:hep-th/003160. Gouvea, padic Numbers : An introduction, Universitext, SpringerVerlag, 1993. 25. J. B. Hartle and S. W. Hawking, Wave function of the Universe, Phys. Rev. 28,2960-2075, (1983). 26. J. E. Holly, Pictures of Ultmmetric Spaces, the p-adic Numbers, and Valued Fields, Amer. Math. Monthly 108, 721-728, (2001). 27. A. Khrennikov, p A d i c Distributions in Mathematical Physics, Kluwer AP, Dordrecht, 1994. 28. A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer AP, Dordrecht, 1997. 29. N. Koblitz, p-adic numbers, p-adic analysis and zeta functions, London Mathematical Society Lecture Notes Series 46, Cambridge U. P., Cambridge, 1980. 30. K. Mahler, p-adic numbers and their functions, Cambridge tracts in mathematics 76, Cambridge U. P., Cambridge, 1980. 31. V. P. Platonov and A. S. Rapinchuk, Algebraic Groups and Number Theory (in Russian), Nauka, Moscow, 1991. 32. A. Robert, Euclidean models ofp-adic spaces, Lecture Notes in Pure and Applied Mathematics 192,349-361, (1997). 33. W. H.Schikhof, Ultrametric Calculus : an introduction to p-adic analysis, Cambridge U. P., Cambridge, 1984. 34. V. S. Vladimirov, Adelic Formulas for Gamma and Beta Functions of One-Class Quadratic Fields: Applications to ,$-Particle Scattering String Amplitudes, Proc. Steklov Math. Institute 228, 67-73, (2000), arXiv: math-ph/0004017. 35. V. S. Vladimirov and I. V. Volovich, p-Adic Quantum Mechanics, Commun. Math. Phys. 123,659-676, (1989). 36. V. S.Vladimirov and Ya. I. Volovich, On the Nonlinear Dynamical Equation in the p a d i c String Theory, arXiv: math-ph/0306018. 37. V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, World Scientific, Singapore, 1994. 38. I. V. Volovich, Number theory as the ultimate physical theory, CERN preprint, CERN-TH. 4781/87,(1987). 39. I. V. Volovich, p-Adic string, Class. Quantum Grav. 4, L83-L87, (1987). 40. A. Weil, Adeles and Algebraic Geometry, Progress in Mathematics 23, Birkhauser, 1982.

24.

F. Q.

LAGRANGIAN ASPECTS OF QUANTUM DYNAMICS ON A NONCOMMUTATIVE SPACE *

BRANKO DRAGOVICH Institute of Physics, P.O. Box 57, 11001 Belgrade, YUGOSLAVIA E-mail: [email protected]. y u

ZORAN RAKIC Faculty of Mathematics, University of Belgrade Studentski trg 16, P.O. Box 550, 11001 Belgrade, YUGOSLAVIA E-mail: [email protected]. ac. yu

In order to evaluate the Feynman path integral in noncommutative quantum mechanics, we consider properties of a Lagrangian related to a quadratic Hamiltonian with noncommutative spatial coordinates. A quantum-mechanical system with noncommutative coordinates is equivalent to another one with commutative coordinates. We found connection between quadratic classical Lagrangians of these two systems. We also shown that there is a subclass of quadratic Lagrangians, which includes harmonic oscillator and particle in a constant field, whose connection between ordinary and noncommutative regimes can be expressed as a linear change of position in terms of a new position and velocity.

1. Introduction Quantum theories with noncommuting spatial coordinates have been investigated intensively during the recent years. M(atrix) theory compactification on noncommutative tori, strings in some constant backgrounds, quantum Hall effect and IR/UV mixing are some of the most popular themes (for a review of noncommutative quantum field theory and some related topics, see e.g. [ 81. Most of the research has been done in noncommutative field theory, including noncommutative extension of the Standard Model [ 21. Since quantum mechanics can be regarded as the one-particle nonrelativistic sector of quantum field theory, it is also important to study its * MSC2000: 81T75, 81540 ; PACS: 03.65. Bz . Keywords : noncommutative geometry, quadratic Lagrangians, Feynman path integral. 159

160

noncommutative aspects including connection between ordinary and noncommutative regimes. Because of possible phenomenological realization, noncommutative quantum mechanics (NCQM) of a charged particle in the presence of a constant magnetic field has been mainly considered on twoand three-dimensional spaces (see, e.g. [ 31 and references therein). Recall that to describe quantum-mechanical system theoretically one uses a Hilbert space &(Itn) in which observables are linear self-adjoint oper-

ators. In ordinary quantum mechanics (OQM), by quantization, classical canonical variables xk,p j become Hermitian operators x?k, & satisfying the Heisenberg commutation relations [.;C,@j]=itibkj,

[&,x?j]=O,

[p”k,&]=O,

k , j = 1 , 2 ... n . (1.1)

In a very general NCQM one has that [fk,gj]= i t i b k j , but also [.;C,sj] # 0 and [&,&I # 0. However, we consider here the most simple and usual NCQM which is based on the following algebra:

[.;c,gjl= i t i d k j , where 0

= (&j)

[ . ; c , ~= I i t i e k j , [ g k , g j=] o ,

(1.2)

is the antisymmetric matrix with constant elements.

To find @ ( x , t )as elements of the Hilbert space in OQM, and their time evolution, it is usually used the Schrodinger equation d i ti - Q(2,t ) = A Q(2,t ) , at which realizes the eigenvalue problem for the corresponding Hamiltonian operator fi = H($, x , t ) , where gk = - i ti ( a / a X k ) and fi = h / ( 2 T ) . However, there is another approach based on the Feynman path integral method [91

where IC(x’’, t”;x’, t’) is the kernel of the unitary evolution operator V ( t ) , t” functional S[ q ] = L(q,q, t ) dt is the action for a path q(t)in the classical Lagrangian L(q,q, t ) ,and x” = q ( t ” ) , X I = q(t’) with the following notation x = ( X I , xz, . . . ,zn)and q = (q1, q 2 , . . . , qn). The kernel K(x”,t’’;x’, t’) is also known as quantum-mechanical propagator, Green’s function, and the probability amplitude for a quantum particle to come from position x’ at time t‘ to another point x” at t”. The integral in (1.3) has a symbolic meaning of an intuitive idea that a quantum-mechanical particle may propagate from x’ to x” using infinitely many paths which connect these two points

h,

161

and that one has to take all of them into account. Thus the Feynman path integral means a continual (functional) summation of single transition amplitudes exp ( i / hS [q ] ) over all possible continual paths q ( t ) connecting x’ = q ( t ‘ ) and x” = q(t”). In direct calculations, it is the limit of an ordinary multiple integral over N - 1 variables qi = &) when N + 00. Namely, the time interval t” - t’ is divided into N equal subintervals and integration is performed for every qi E (-00,+00) and fixed time t i . In fact, lc(x”,t”;d ,t’), as the kernel of the unitary evolution operator, can be defined by equation (1.4) and then Feynman’s path integral is a method to calculate this propagator. Eigenfunctions of the integral equation (1.4) and of the above Schrodinger equation are the same. Note that the Feynman path integral approach, not only to quantum mechanics but also to whole quantum theory, is intuitively more attractive and more transparent in its connection with classical theory than the usual canonical operator formalism. In gauge theories and some other cases, it is also the most suitable method of quantization. The Feynman path integral for quadratic Lagrangians can be evaluated analytically and the exact result for the propagator is

where S(x”,t”;x’,t’) is the action for the classical trajectory which is solution of the Euler-Lagrange equation of motion.

In this article we search the form of a Lagrangian which corresponds to a system with noncommutative spatial coordinates. This is necessary to know before to employ Feynman’s path integral method in NCQM. To this end, let us note that algebra (1.2) of operators ?k,& can be replaced by the equivalent one [@k,$j] =ZhSkj7

[@k,‘$j]

=O7

[$k,$j]

=o,

(1.6)

where linear transformation

is used, while f i k are remained unchanged, and summation over repeated indices is assumed. According to (1.2), (1.6) and (1.7), NCQM related to the classical phase space (x,p)can be regarded as an OQM on the

162

other phase space ( q , p ) . Thus, in q-representation, pi, = - i i i ( O / d q k ) in the equations (1.6) and (1.7). It is worth noting that Hamiltonians H($,?,t) = H ( - z h ( a / d q k ) , Qk Z h ( ( e k j / 2 ) . ( d / d q j ) , t ) , which are more than quadratic in 2, will induce Schrodinger equations which contain derivatives higher than second order and even of the infinite order. This leads to a new part of modern mathematical physics of partial differential equations with arbitrary higher-order derivatives (for the case of an infinite order, see, e.g. [ 121 and [ 131). In this paper we restrict our consideration to the case of quadratic Lagrangians in x and x (for an example with 2 , see [ 101).

+

2. Quadratic Lagrangians

2.1. Classical case Let us start with a classical system described by a quadratic Lagrangian which the most general form in three dimensions is:

+ + 011 i l x 1 +

L(?, X ,t ) = ~ ~ 1 21 2 1 a 1+2 i l X 2 f

a33

a13 k1 k3 0 1 2 21 2 2

+ a 2 2 2; + a 2 3 X2 $3

+ 013 +

klX3

+

p2l k 2

x1

(2.1)

+ 0 2 2 2 2 2 2 + p 2 3 x 2 x3 + p31 x3 21 0 3 2 23 2 2 + p33 23 x3 + 711 2: + 7 1 2 21 x2 + 713 x1 + 7 2 2 xz + 7 2 3 2 2 23 + 733 xi + 61 21 + 6 2 ? 2 + 63 23 + < I 2 1 + <2 2 2 + (3 23 + 4 , IC3

where the coefficients ayij = a i j ( t ) , P i j = & ( t ) , yyij= y i j ( t ) , di = 6,(t), = &(t)and 4 = 4(t) are some analytic functions of the time t .

If we introduce the following matrices,

assuming that the matrix a is nonsingular (regular) and if we introduce vectors

one can express the Lagrangian (2.1) in the following, more compact, form:

L(k,X,t)= ( a x , x ) + ( P . , . ) + ( ~ x , . ) + ( s , a : ) + ( < , x ) + 4 ,

(2.2)

163

where ( . , - ) denotes standard scalar product. Using the equations

dL

p . - -, -

axj

j = 1,2,3,

one can express x as 1 2

jl= - 0 - 1 (P - P

X

-6).

(2.4)

Then, the corresponding classical Hamiltonian

becomes also quadratic, i.e.,

where:

Here, P' denotes transpose map of p. Let us mention now that matrices A and C are symmetric (A' = A and C' = C), since the matrices a and y are symmetric. Also, if the Lagrangian L ( ~ , xt ), is nonsingular (det a # 0), then the Hamiltonian H ( p , X ,t ) is also nonsingular (det A # 0). The above calculations can be considered as a map from the space of quadratic nonsingular Lagrangians L to the corresponding space of quadratic nonsingular Hamiltonians 'FI. More precisely, we have 'p : L X, given by

-

From relation (2.5) it is clear that inverse of 'p is given by the same relations (2.7). This fact implies that 'p is essentially involution, i.e. 'p o 'p = id.

164

2.2. Noncommutative case In the case of noncommutative coordinates [ i k , 3j] = i ii 6 k j , one can replace these coordinates using the following ansatz (1.7),

where 2 = ( 2 1 , 3 2 , 2 3 ) 1

G = ( h l G 2 , d 3 ) , ?; = (fiIlfi21fi3) Q=

(

:::)

.

-i12 -613

and

-023

0

Now, one can easily check that cji for i = 1 , 2 , 3 are mutually commutative operators (but do not commute with operators of momenta, [ Gk 1 fij ] = fi. a k j ) . If we start with quantization of the nonsingular quadratic Hamiltonian given by (2.6), i.e., = H ( A ,B , C, D , E , F,fi, 2 ) and then apply the change of coordinates (2.9), we will again obtain quadratic quantum Hamiltonian,

He :

Be = ( A ~ l i , f +i )( Be rile)-I- (ced , d ) -k ( D e , f i )-k ( E e ld ) +FQ,

(2.10)

where

and s y m denotes symmetrization of the corresponding operator. Let us note that for the nonsingular Hamiltonian H and for sufficiently small 6 k j the Hamiltonian He is also nonsingular. In the process of calculating path integrals, we need classical Lagrangians. It is clear that to an arbitrary quadratic quantum Hamiltonian we can associate the classical one replacing operators by the corresponding classical variables. Then, by using equations

from such Hamiltonian we can come back to the corresponding Lagrangian

165

where 1 2

P=-&'(q-BQq-D@) is replaced in H e ( p ,q, t ) . In fact, our idea is to find connection between Lagrangians of noncommutative and the corresponding commutative quantum mechanical systems (with 8 = 0). This implies to find the composition of the following three maps:

LQ = ( ' P o $ V ) ( L ) ,

(2.12)

where L e = ' ~ ( H o )HQ , = $ ( H ) and H = p(L) (here we use facts that 'p is an involution given by formulas (2.7), and .1c, is given by (2.11)). More precisely, if

L ( k , z , t ) =( Q ~ , k ) + ( p z , k ) + ( Y z 1 z ) + ( 6 , k ) + ( ~ , z ) + ~ , and

It is clear that formulas (2.13) are very complicated and that to find explicit exact relations between elements of matrices in general case is a very hard task. However, the relations (2.13) are quite useful in all particular cases.

166

2.3. Linearization and some examples In this section we try to introduce the linear change of coordinates to gain L Q ( ~q, ,t ) directly from L ( i ,x , t ) for some simple examples. Let L ( i ,x , t ) be a nonsingular quadratic Lagrangian given by (2.2). If we make the following change of variables:

x = U(O)Q+V ( Q ) q +W ( Q )= U q + V q + W ,

j. = 4 ,

(2.14)

we obtain again quadratic Lagrangian L e ( q ,q, t ) ,where

w + 2 u' y w + 6 + u' =2v ' w + v (,

bQ = ,8 [Q

PQ = 2 u'

=("+~'y~+PU)syrn,

Q@

[,

'yQ =

v + P v,

v' v,

(2.15)

4e = ( r W + t , W ) + 4 ,

and U, V, W are matrices. It is clear that formulas (2.15) are simpler than (2.13). We will see that for some systems, by this linear change of the coordinates, it is possible to obtain connection between Lagrangians L ( i ,x,t ) and L@(q,q, t )

Example 1. Harmonic oscillator, n = 2. In this case we have m a=-Id, P=O, y=kId, @=OJ, 2 where k < 0 is related to an ordinary harmonic oscillator,

J

=

( ') , J 2 -10

= -Id,

6 =O,

[=

(2.16)

4 = 0,

and Id is 2 x 2 unit matrix. Using formulas (2.13), one can easily find (YQ

m Id7 2 - kmO2

=

PO

=

2kmB 2k Id. J , 70 = 2 - km02 2-km02

(2.17)

From (2.15) it follows (YQ

= (Y

+ U'

yU

=

+ k U'

U , and consequently

k m2 O2 (YQ-a= Id. 2(2-kmB2)

(2.18)

So, it is clear that matrix U is proportional to an orthogonal operator, i.e, U = Xo. Now, from (2.18) we obtain X=&X-

m9

1

4 d2-kmB2'

where

EX

= fl and 2 - k m 0 2 > 0 .

(2.19)

167

Similarly, from (2.17) and 70 = V'y V, we find that matrix V is also proportional to an orthogonal operator V = p where

v,

Jz

E~

= f l and 2 - k m 0 2 > O .

(2.20)

It is known that

v = R(w)=

cos w sin w

where 0 _<

w

I,!I~

(2.21)

-

- sin w

%

or V = R l ( w ) =

cos w

I

- cos w sin w

< 2 n-. From (2.15) and (2.17), we find

and using (2.19)-(2.21), we have

O r V = E~ E~

J.

(2.22)

(1 or - l),we have nrr w=$-or w=$+-. (2.23) 2 2 If E = 0, from the relation & = 0 , we obtain that W = 0 . It implies that in this case the transition from Lagrangian L ( k , x , t ) to the Lagrangian L e ( q ,q, t ) is given by a linear change of the coordinates. From (2.22), depending on

E X E~

Finally, let us show that in the case # 0, it is not possible to obtain the transition from Lagrangian L ( k ,5,t ) to the Lagrangian LQ(Q,q , t ) by a linear change of the coordinates (2.14). From (2.13) and (2.15), we have

6Q=2UTyW+U'( and consequently

1 UdQ - -( 2k = 2 V' y W + V' .$,one can find

1 w = -2Xk Similarly, from the relation

(Q

1 w=2Pk

1 V ( g - -(. 2k

(2.24)

(2.25)

The relations (2.24) and (2.25) imply

(2.26)

168

From the other side, the relations (2.13) and (2.17) give

be =

m0 2-km02

JE,

Ee=2

1 -kmO2 2-km02

(2.27)

Combining the relations (2.26) and (2.27), we have -

1 1- k m Q 2 and consequently 2 - k m e2 = 0 , 2-km02 2-km02

but it is impossible, since 2 - k m 0 2 > 0, according t o (2.19) and (2.20). In the first case

(c = 0), the corresponding Lagrangian L@(q,q, t) is

L @ = ( C Y Q 4 , ~ ) + ( p @ ~ , ~ ) + ( y Q ~ , q ) + ( b Q , q ) + ( ~(2.28) 0,Q)+~@ 1 - k 0 2 m (4; + 4); + 2 k (q1 q 2 2 ) + 2 k m 0 (41 42 - qi 42) .

[

]

+

From (2.28), we obtain the Euler-Lagrange equations,

mq1+ 2 m k042

-

2 k q l = 0 and m q 2 - 2 m k e g 1 - 2 k q 2 = 0 .

(2.29)

Let us remark that the Euler-Lagrange equations (2.29) form a coupled system of second order differential equations, which is more complicated than in commutative case (0 = 0).

Example 2 . A particle in a constant field, n = 2 . This example is defined by the following data: m a=--Id, p=O, y=O, b = O , S = ( t l , t 2 ) , 2

+=O.

(2.30)

Using the general composition formula (2.13), one can easily find

m a~ = - Id, 2 m0 S Q = - - -2J (

@Q

=o,

and

ye = 0,

Ee = E,

(2.31)

m O2

+e=---(8 E,E),

where J is the same as in the first example (see (2.16)). Now, using the method of the linear change of coordinates, from the relation (2.15), we have

169

From (2.31) and (2.32), it is easy to see that for

U=--

mc9 J, 2

V=Id

m e2 8

W=---(,

and

(2.33)

the linearization gives the same as the general formula. In this case, it is easy t o find the classical action. The Lagrangian L@(q,q, t ) is

L0

=

+ 4; - 6 ( J l 4 2 - 5 2 4 1 ) ) + 5 1 41 m O2 + J 2 q 2 + 8( 5 1 2 + 5 2 2 ) . 2

(4;

(2.34)

The Lagrangian given by (2.34) implies the Euler-Lagrange equations, mq1 =

&

=J2.

(2.35)

5 2 t2 q 2 ( t ) 2=m- + t ~ 2 + ~ l ,

(2.36)

and

mq2

Their solutions are: 5 1 t2 4 1 ( t ) = 2- +mt C : ! + C l

and

where C,, C2,D1 and D2 are constants which have to be determined from conditions:

After finding the corresponding constants, we have (Z’!- x’.)- 3 5 ’ T ), j= 1,2, Qj(t)= x; + 2m T 3 2m qj(q = + (Z’!- $.) - 3 5.T j =1,2. m T 2m’

G! 1

(2.38) (2.39)

Using (2.38) and (2.39), we finally calculate the corresponding action T

mc9 - 52

(G 41 -

T3

(51

2

mT02

+5 2 7 + (512 +522) . 8

(2.40)

170

3. Concluding Remarks Note that almost all results obtained in Section 2 for three-dimensional case allow straightforward generalization to any n 2 2 spatial dimensions. According to the formula (1.5), the propagator in NCQM with quadratic Lagrangians can be easily written down when we have the corresponding t” classical action Se(z”,t”;z’,t’)= L e ( q , q , t )d t , where q = q ( t ) is solution of the Euler-Lagrange equations of motion. As a simple example we calculated &(x”, T ;z’, 0) for a noncommutative regime of a particle on plane in a constant field.

s,,

Note that the path integral approach to NCQM has been considered in the context of the Aharonov-Bohm effect [ 51, Aharonov-Bohm and Casimir effects [ 41, a quantum system in a rotating frame [ 61, and the Hall effect [ 71 (see also approaches [ 11] and [ 1I). Our approach includes all systems with quadratic Lagrangians (Hamiltonians) and some new results on path integrals on noncommutative spaces will be presented elsewhere.

Acknowledgements The authors thank S. Prvanovib for discussions. The work on this article was partially supported by the Serbian Ministry of Science, Technologies and Development under contracts No F1426 and No MM1646. The work of B. D. was also supported in part by RFFI grant 02 - 01 - 01084.

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THE MOBIUS EQUATION: A LOCAL ANALYTICAL CHARACTERIZATION OF TWISTED PRODUCT STRUCTURES *

M. FERNANDEZ-LOPEZ + AND E.

GARCIA-RIO+

Department of Geometry and Topology, University of Santiago de Compostela, 15782 Santiago de Compostela, SPAIN E-mails: [email protected], xtedugrQusc. es

D. N.KUPELI Bosna-Hersek Caddesi, N0:35/B-10, Emek Mahallesi, 06510 Ankara, T U R K E Y . E-mail: [email protected]

The Mobius equation, a partial differential equation, is introduced in semiRiemannian geometry and shown that it completely characterizes the local twisted product structure of a semi-Riemannian manifold. Characterizations of warped product and direct product structures of semi-Riemannian manifolds are obtained from the Mobius equation and an additional partial differential equation.

1. Introduction

In analysis, mostly the existence of a nontrivial solution to a differential equation on a certain domain is argued. But in geometry, one can also argue the existence of a manifold structure for a differential equation to possess a nontrivial solution. This may be considered as an analytic characterization (or representation) of a manifold structure by a differential equation if this manifold structure serves as a unique domain structure for this differential equation to possess a nontrivial solution in a certain class of manifolds. ~~~

~

* MSC2000: 53C12, 58599. Keywords :Mobius equation, twisted product, warped product, Bochner formula. t Work partially supported by a project BFM2003-02949 (Spain). 173

174

In other words, we give local analytic characterizations of semi-Riemannian product, warped product and twisted product structures of semiRiemannian manifolds by differential equations, that is, by the existence of nontrivial solutions to some differential equations on certain semiRiemannian manifolds. Although we expect that every manifold structure cannot be characterized (or represented) by a differential equation, such manifold structures that can be characterized by differential equations may be a larger class of manifold structures than semi-Riemannian product, warped product and twisted product structures of semi-Riemannian manifolds. In any case, having a better knowledge about characterizations of manifold structures by differential equations may lead us a better understanding of a possible relation between differential equations and differential geometric structures. The paper is organized as follows. Section 2 covers some preliminaries on product structures and second fundamental form of a map needed to state the Mobius equation. Solutions of the Mobius equation, in connection to local decomposition of manifolds are studied in sections 3 and 4,where the desired analytic characterization is obtained. Finally, Section 5 is devoted t o investigate some relations between solutions of Mobius equation and affine maps.

2. Preliminaries In this section we recall some basic material needed in what follows.

2.1. Twisted and warped product structures

--

Let (N1,h l ) and

(N2, h2)

be two semi-Riemannian manifolds and let Ni be the canonical projections (i = 1,2). Also let ( 0 , ~be ) a smooth function. Then the twisted product of ( N I ,h l ) and (N2, h2) with twisting function X is defined t o be the product manifold M = N1 x N2 with metric tensor g = hl @ X2h2 given by g = 7rThl X27r;h2. For brevity in notation, we denote this semiRiemannian manifold ( M , g ) by N I X A N2. Moreover, if the function X only depends on the points of N1, then N1 X A N2 is called the warped product of (N1,h l ) and (N2, h2) with warping function A. The product structure is just called product or direct product if the function X is the constant X = 1. Warped and twisted product structures have shown to be very useful in understanding many geometric and physical problems (see, for example : N1 x N2 X : N1 x N2

7ri

+

175

[ 51, [ 71, [ 91, [ 111 and the references therein). A local characterization of direct product, warped product and twisted product structures can be stated in terms of the extrinsic geometry of the foliations C N and ~ C N of ~ the product manifold M = N1 x N2 as follows: Let g be a semi-Riemannian metric tensor on M = N1 x Nz and assume that the canonical foliations C N ~and C N ~intersect orthogonally everywhere. is the metric tensor of (see [ 111) a twisted product N1 X A N2 if and only if C N ~is a totally geodesic foliation and C N is ~ a totally umbilic foliation, a warped product N1 x x N2 if and only if C N ~ is a totally geodesic foliation and C N is ~ a spherical foliation, a direct product if and only if C N ~and C N ~ are totally geodesic

foliations. Warped product structures can be detected among twisted product structures also from the intrinsic geometry of ( M , g ) . Indeed, it was shown in [ 51 that a twisted product N1 X A N2 is a warped product if and only if the Ricci tensor of ( M ,g ) satisfies Ric(X, V) = 0 for every X E r T N 1 and

V E rTN2. 2.2. Second fundamental f o r m of a m a p Let ( M I ,91) and (M2,92) be semi-Eemannian manifolds of dimensions

-

v1

2 v,

nl and n2 with Levi-Civita connections and respectively, and let f : ( M I ,91) (Mz,g2) be a map. We denote the set of vector fields on MI by r T M 1 and the set of vector fields along f by r f T M 2 . We also denote 2

2

the pullback of V along f by V. Then, the second fundamental form of f is the map

V f*: rTM1 x r T M 1 -

rfTM2

It is easy to check that Vf+ is symmetric and linear in its arguments (see, for example [ 71, [ S ] ) . Moreover, note that since Vf* is a bilinear map, the value of (Vf*) (X, Y ) at a point p l E M I only depends on the values of X and Y at the point p l E M I . Then, for z,y E TplM1 we define (Vf,)(z,y) by means of (Vf*)(z,y) = (Vf*)(X,Y ) ( p l ) ,where X, Y E FTM1 are vector

-

fields such that X ( p 1 ) = z and Y ( p 1 ) = y. A map f : (Ml,g1) (M2,gz) is said to be ufine or totally geodesic if (V f*) 0. Equivalently, affine maps are characterized by the property of

176

being geodesic-preserving [ 121. For our purposes in this paper, it is necessary to consider the second fundamental form of a composition as follows. Let ( M I ,g l ) , (M2,92) and (M3,93)

-

-

1 2 v, v

3 v,

be semi-Riemannian manifolds with Levi Civita connections and respectively. Let f 1 : (M1,gl) (M2,g2) and f 2 : ( M 2 , g 2 ) (M3,g3) be smooth maps. Then, the second fundamental form of the composition f 3 = f 2 o f l , satisfies (see, for example, [ I ] ) .

( V f 3 * ) ( X , Y )= ( V f 2 * ) ( f 1 * Xfl*V , +f2*((Vf1*)(X,Y)), for all X , Y E r T M 1 . The tension field, T ( f ) , of a map f : (M1,gl) (M2,92) is the vector field along f defined by the trace of the second fundamental form of f with respect to the metric 91, i.e., if { E l ,. . ,En}is an orthonormal local frame of T M 1 , then ~ ( f = ) C~~lgl(Ei,Ei)(Vf,)(Ei,E where i ) , dim M I = nl. A map f is said to be harmonic if ~ ( f=) 0. Note here that although the

-

composition of afine maps is affine, the corresponding result does not hold for harmonic maps [ 11. However, for our purposes it suffices to consider a special case as follows: let f 1 : (M1,gl) (M2,92) be a semi-Riemannian submersion and f 2 : (M2,9 2 ) (M3,93) a smooth map. Then the tension field of f 3 = f 2 o f 1 satisfies .(f3) = .(f2) f2*(7(f1)), [ 11.

- +

3. Mobius equation associated to a map In literature, a function f on a semi-Riemannian manifold ( M ,g ) is said to satisfy the Mobius equation if 1 Hf - df c3 df - [ A f - d V f ,O f )I9 = 0 n on ( M ,g ) [lo]. Here recall that the Hessian tensor of a map f is defined by h f ( X )= V x V f , where X E I'TM and O f denotes the gradient of f . Moreover, the Hessian form H f is given by H f ( X , Y ) = g ( h f ( X ) , Y )for all X , Y E I'TM. Also, by making the transformation t = e-f , the Mobius equation becomes

At Ht=-g

n on ( M ,g), which is called the localized Mobius equation. Although the time functions of physically realistic spacetimes do not satisfy localized Mobius equation, they satisfy

Af g ( X , Y ) H f ( X , Y )= n-l

and

H f ( X ,V ) = 0

177

for all X , Y E r k e r f , and U E r ( k e r f , ) l , which is called the local Mobius equation in [ 2 ] and [4],where applications of local Mobius equation in relativity theory are explored. Next we state the following definition, as a generalization of the local Mobius equation discussed in [ 21 and [ 41.

-

Definition 3.1. Let f : ( M 1 , g l ) (M2,92) be a smooth map. We say that f satisfies the Mobius equation if

( V f * ) ( XY , ) = g1(X,Y E ,

and

(Vf*)(X,V ) = 0,

for all X , Y E I ' K e r f , , V E r ( k e r f * ) l . We say that f satisfies the special Mobius equation if for all p E M I , where r, = dim kerf*, 2 1.

tP= ( l / r p )~(f)(p),

Remark 3.1. Note that solutions of the M6bius equation need not to have constant rank. Indeed, consider the product manifold (M1,gl) = (R2 x N , d x 8 dx d y 8 dy g N ) and the harmonic map i : R2 R2 defined by i(x,y) = (x2-y2, 2xy), and put f = i o ~where , T is the projection onto the first factor in the product above. It is clear that f does not have constant rank. Next we will show that f satisfies the Mtibius equation. First of all, note that the projection T is an affine map and thus it satisfies the Mobius equation. Then (V.rr,)(X,Y ) = ( 1 / 2 )g1(X, Y )T ( T ) for all X , Y E r k e r T * , and hence

+

-

+

1 (vf*)(x,y) = i*(vT*>(x, y ) = 5 gl(x,y)i * T ( T ) for all X , Y E I'kerr,. Now, since ~ ( f=) ~ * T ( T )at , those points where ~ ( p #) 0, then we have that ( V f , ) ( X , Y ) = ( 1 / 2 ) g l ( X , Y ) ~ (for f ) all X , Y E r k e r n , = I'kerf,. Moreover, if n ( p ) = 0, then ~ ( f=) i,~(.rr)= 0 and since (V~,)O= 0, we have ( V f * ) ( X , Y )= 0 for all X , Y E I'TM,, which shows that f satisfies the Mobius equation.

A sufficient condition for a map satisfying the special Mobius equation to have constant rank can be given as follows

-

Proposition 3.1. Let f : ( M 1 , g l ) ( M 2 , g z )be a smooth map satisfying the special Mobius equation. If M I is connected and T ( f ) does not vanish at any point then f has constant rank.

Proof. Let p be a point in MI such that ~ ( f ) ( p#) 0 and suppose that the rank of f is not constant in any neighborhood of p . Then, there exists a sequence (p,) of points in M I which converges to p such that

178

dim kerf,,, = r, < r = dim kerf,,, for all n E N. Now, let X E I'kerf, be a vector field such that X, # 0. Then,

1

(Vf*)(XPnrXP,)= rn g1(X,,,Xpn)T(f)(p,) converges to (Vf*)(xp,xp) = (l/r)gi(X,, x,) ~(f)(p), which is a contradiction. Thus f has constant rank in a neighborhood of p and the result follows from the connectivity of M. 0

Remark 3.2. An analogous result to that of previous proposition cannot be expected for solutions of the (non-special) Mobius equation. In fact, Q # 0 is not a sufficient condition to guarantee that f has constant rank in a neighborhood of p . Consider the warped product (R2 x R2, drcl @I dzl + X ( q , ~ 2 ) ~ ( d@I z d22 2 d~ @ d23 d24 @ d24)) and let n be the projection onto the first factor. Let i : (R2,dzl @ dzl x ( q , ~ 2 ) ~ d 8 2 dz2) 2 ( R ~dzl , @I d s l +dz2 @I dx2) be defined by i(z1,2 2 ) = (51, 2 ; ) and put f = i o n. Further, specialize the warping function X such that (aA/ax1)(0,2 2 ) # 0 for all 2 2 E R. If Xi denote the coordinate vector fields, Xi = then

+

+

-

+

vx,x3 = VX,X,

= -A

from where we get

(Elx1 + -

Next, let p be a point with n(p) # 0. Then kerf,,

Moreover, if T(P) = 0, then kerf,, Thus we have

(Vf*)(X3,X,)

=

),

-x2 ax2

= kern,,

and thus

(X2,X3, X,) and kern,, = (X3,X,).

= (Vf*)(X4,X4)=

x ax a x XI # 0

(Vf*)(x27x2) = (Vi*)(x2,x2) = -i*lJVxzx2. Now, after some calculations in (R2,dsl @ d z l has dX ax Vx,X2 = -A (-XI -X2), 8x1 3x2

+

+ X ( r c 1 , ~ 2 ) ~ d r c@I2 d22) one and hence

179

Thus,

Therefore, we only have t o show that ( V f * ) ( X l , X 2 )= 0. But, since V x , X 1 = X (aX/az2)X 2 , we get dX = 0, ( V f * ) ( X , ,X I ) = ( V i * ) ( X 2X, , ) = -i*X-X2 ax2 which is the desired result.

4. Decomposition results

We begin with the following result, which provides a wide family of maps satisfying the (special) Mobius equation (see also [ 31).

-

+

Theorem 4.1. Let (M1,gl) = ( N I x N2, hl X2h2) be a twisted product (M2,92) an immersion. Then, f = i o 7r satisfies the and i : (NI, h1) Mobius equation. Moreover, f satisfies the special Mobius equation if and only zf the immersion i is harmonic. Proof. Let PI E M I , X p l E TplMI and Y E r k e r 7r*. Then, 2

( V n * ) ( X p 7, Y p , )

=vxpl7r*Y - T * p l - -=*p1

1 VXp1y

1

V X p 1 y = -..*p,B(Xp,,Yp,),

where B denotes the second fundamental form of the leaf of ker T* containing the point p l . Now, since M I has a twisted product structure, we have that ker 7r* is a totally umbilical distribution and ker T,' is totally geodesic distribution (cf [ 111). Therefore B ( X p l,Y p l )= g l ( X p l,Y p l ) u p l , and then ( V ~ * ) ( X p l , Y p=l )~ i ( ~ p l , ~ p l ) ( -=~g~l ( p X pl l ~ , Ypp ll) V) p lwhere , VPl = - r * P l V P l .

Next show that qPl = (l/rpl)T ( T ) ~where ~ , rpl = dim ker 7r*pl. For that, let { X I ,. . . ,xkl, Y I ,. . . ,Yk,} be an orthonormal basis for TplM I , such that { X I , . . . , xkl}is a basis for ker niPl and { Y I , . . ,Y k z } is a basis for lcerr,,,. From [ 121 we have that (V7r*)(Xi,X i ) = 0 , i = l , . . .,k l , and thus

i=l

i=l

180

Next, show that (V7r,)(X, V ) = 0 for every X E r k e r 7r* and V E r k e r 7r:. 1

Since ker7r: is integrable arid totally geodesic, we have V V W E rker7r: 1

for every V,W E r k e r 7r:.

Hence we have g1(vv X , W ) = 0 , for each 1

X E r k e r r , , and thus V v X E I'kerr,, from where it follows that 1

-

( V n , ) ( X , V ) = -7r* V V X = 0 , which indeed shows that the projection 7r = 7rl : M I = N1 x N2 N1 satisfies the special Mobius equation. Now we show that f = i o r satisfies the Mobius equation. Now, the second fundamental form of the composition is given by

Hence ( V f , ) ( X , Y ) = ( l / r ) g 1 ( X , Y ) i , ~ ( 7 r for ) , all X , Y E r k e r f , , and ( V f * ) ( X , V )= 0 for all X E r k e r f , and V E rkerf?. This shows that f satisfies the Mobius equation with 6 = (l/r)Z,T(T). Finally note that since 7r is a semi-Riemannian submersion, the tension field satisfies T (f ) = Z,T(T) ~ ( i )and , thus (1/r) T (f ) = ((= ( 1 / r )~ , T ( T ) ) if and only if ~ ( i=)0, that is, f satisfies the special Mobius equation if and only if the immersion i is a harmonic map. 0

+

-

Remark 4.1. Let us mention here, that for any twisted product space ( M , g ) = (N1 x N2, hl + X2h2),the projection 7r : M = N1 x N2 N1 satisfies the special Mobius equation.

A converse of previous theorem allows us t o obtain a characterization of local twisted product structures as follows.

-

Theorem 4.2. Let f : ( M 1 , g l ) (M2,g2) be a m a p of constant rank satisfying the Mobius equation and suppose that kerf, and kerf? are nonsingular distributions. Then, we have the following (local) decomposition

(Ml,91) = ( N l x N2, hl

+ X2h2)

as a twisted product, where N1 and Nz verify that T N 1 = kerf: T N 2 = kerf,. Moreover, the map f can be locally factorized as

and

181

where

IT

denotes the projection and i a immersion.

Proof. First of all, note that the distribution (Kerf,)' is integrable and totally geodesic. Indeed, let X E r k e r f , and V E I'kerfb. Since f satisfies the Mobius equation, we have that (Vf*)(X, V) = 0 and thus 2

0 = (Of*)(X, V )= v v 1

This shows that

VV X

1

f*X- f* v v

E r k e r f , for all

E

1

vv

rkerf,,

x. V

E

rkerfk.

1

gl(W,X) = g1(Vv X,W )= 0, we have that 1 W,X).Therefore v v W E rkerf,' for all V,W E rkerf,',

Also, if W E r k e r f b , since 1

X

x = -f*

0 = g1(vv which shows that kerf,' is an integrable and totally geodesic distribution. Next, let X,Y E r k e r f , . Then, 2

1

1

(Of*)(X,Y) = v x f*Y- f* v x Y = -f* v x Y

= -f*B(X,Y),

where B denotes the second fundamental form of the leaves of the foliation defined by ker7r*. Now, since f satisfies the Mobius equation we have that gl(X,Y)( = -f,B(X,Y). Moreover, if 2 E r k e r f , is a unit vector field and put q = B ( Z , Z ) , then 6 = -f*q, from where it follows that f,(B(X, Y )-g1(X,Y ) q )= 0. This proves that B ( X ,Y)= gl(X, Y ) v ,and thus k e r f , is a totally umbilical distribution. Now, it follows from [ 111 that M1 is (locally) a twisted product. Moreover, since f has constant rank, it factorizes as in the following

Nl where i is defined by i : x i ( z )= f(x,yo), being yo E N2. Finally note that the definition of i is independent of the choice of yo since the map fz : N2 M2,fz(y) = f ( x , y ) has constant zero rank and thus is locally a constant map. 0

-

y-)

Assuming that f satisfies the special Mobius equation allows us t o be more precise on the properties of the immersion in previous factorization:

182

-

Theorem 4.3. Let f : ( M 1 , g l ) ( M 2 , g z ) be a map of constant rank satisfying the special Mobius equation. Then, we have the following (local) decomposition (Mi,g1) = ( N i x N z , h l + X2h2)

as a twisted product, where N1 and N2 verify that TN1 = kerf: TN2 = kerf,. Moreover, the map f admits a (local) factorization

and

Nl where T is the projection and i a harmonic immersion.

Proof. Since T is a semi-Riemannian submersion, ~ ( f = ) Hence, for all X, Y E r k e r f*,we have

(Vf*)(x,y = 9) 1 ( X , Y ) E

Z,T(T)

+~ ( i ) .

1 r

= - gl(X,Y)i*T(r).

Then, = ( l / r ) i * T ( r )from , where 6 = ( l / r ) T ( f )if and only if ~ ( i=)0. This proves that f satisfies the special Mobius equation if and only if i is a harmonic immersion. 0

Remark 4.2. The twisted product structure in Theorem 4.2 can be specialized to be warped in two different ways. Based on the results in [ l l ] , one may add an extra equation to the Mobius one in order to ensure that the leaves of kerf* are not only totally umbilical but spherical. This approach was considered in [ 31 for submersions. More generally, local warped product structures are characterized by the existence of constant rank solutions of the Mobius equation satisfying VV f * ( X ,Y,Z ) = 0 f o r every X , Y,Z E r k e r f * . The other approach is based on the geometry of the manifold, since warped products are characterized as those twisted products whose Ricci tensor is mixed Ricci-flat in [ 51. Therefore, a manifold ( M 1 , g l ) is locally a warped product if and only if there exists a map f : ( M 1 , g l ) (M2,92) of constant rank satifying the Mobius equation and R i c ~(,X ,U ) = 0 for all X E I'kerf,, U E I ' ( k e r f * ) L , where R ~ c Mdenotes ~ the Ricci tensor of (M1,gd.

-

183

5. Mobius equation and affine maps

Next we will derive some sufficient conditions for a solution of the Mobius equation to be totally geodesic. Equivalently, to ensure that a twisted product structure is indeed, a product structure. In order to make use of Bochner methods, manifolds are assumed to be Riemannian and compact in the next theorem (see [ 71 and the references therein for more information on Bochner formulae). Theorems 5.2, 5.3 and 5.4 are also valid when the manifold M is not necessarily compact. We begin with the following

-

Theorem 5.1. Let f : (M1,gl) (M2,gz) be a m a p of constant rank satisfying the Mobius equation. Let suppose that ( M I ,gl) is orientable, compact and has non-negative Ricci curvature and that (M2,92) has nonpositive sectional curvature. If

92([, E ) where r

=

dim k e r f ,

2 2, T h e n f

f

div(T(f)) 2 0 ,

is a n a f i n e map.

-

Proof. As a matter of notation, the adjoint map of a given linear map L : (V1, ( , ) 1 ) (V2, ( , ) 2 ) between two inner product spaces, denoted by * L is the linear map * L : (V2, ( , ) 2 ) (V1, ( , ) I ) satisfying

-

(%,*

LY)l =

m,

Y)2

for every couple of vectors z E V1 and y E V2. Now, let { X I , . . . ,Xn,} be an orthonormal basis for (TplMl,gl(pl))consisting in unit eigenvectors of the map * f*pl o f*,, with the corresponding set of (non-negative) eigenvalues {XI,. . . ,An,}. Obviously, we can admit that { X I , .. . ,Xr} is a basis for kerf,,, . Then, using the Bochner identity (see [71)

184

That is, l l f t ( 1 2 is a subharmonic map. Since MI is compact we have that llf*1I2 is harmonic and thus

c 121

0

=

9 2 ( ( V f * ) ( X i , X j ) (, 0 f * ) ( X i , X j ) )

i,j=r+l 121

RMz(f*Xi~f*Xj,f*Xj,f*Xi)

i,j=l

+

c 711

(xi,xi)+ r g2(<,<) + d i u ( T ( f ) ) .

xi ~ i c M ,

i=l

Now, from the assumptions,

<

2

g 2 ( ( V f * ) ( X i Xj), r ( V f * ) ( X iX, j ) ) = 0,

i,j=r+l

and hence, = (l/r)~ ( f ) . Next, it follows from the generalized semi-Riemannian divergence theorem (see [ 71) that ~ ( f=)0 since r 2 2. Then, f is an affine map. 0 Next we will consider a special class of functions called Riemannian maps [ 61. These maps, being as “isometric as they can be”, are specially suited to compare geometrical properties of manifolds. Let f : M1 M2 be a smooth map and let f*,, : TplM1 + Tp2M2 be the tangent map of f at p l , where p2 = f ( p 1 ) . We will call f Riemannian at p l E MI if

-

- -

f*,, : ( k e r f*,,

( r a n g e f*,,

is isometric. We will call a smooth map f : M I M2 Riemannian if f is Riemannian at each p l E M I . Note that Riemannian maps have constant rank and moreover AM^ llf*112 = 0, [ 71.

-

Theorem 5.2. Let f : ( M 1 , g l ) (M2,92) be a Riemannian m a p of rank greater or equal than two satisfying the Mobius equation. Let suppose that M I is connect, R i C M , 2 A , K M 5~ B and A 2 ( r a n k ( f ) - 1)B. If cg2(6,<)

+ diu(T(f))2

f o r some constant C < r, where r = d i m k e r f * , then f i s a n a f i n e map. Proof. Let { X I ,. . . , X,, Xr+ll . . . , Xnl} be an orthonormal basis for (TplM1,g l ( p 1 ) ) such that { X I , . . . , X,} is a basis for Kerf,,,. Then we have -

2

i , j.--r+ 1

~ M 2 ( f * X fi *, X j , f*Xj, f*&)

2 4 r a W . f ) )( r a n k ( f )- 1)B ,

185

Proff. for a Proff. for a Proff. for a Proff.- for a Proceeding in an analogous way, one has the following.

Theorem 5.3. Let f : ( M 1 , g l ) ( M z , g z ) be a Riemannian m a p satisfying the Mobius equation. Let suppose that M I is connected, R ~ c M2, A, S C M5~ B and n2A 2 B. If C92(E,0

f o r some constant C

+ di'U(T(f)) 2 0

< r, where r = d i m k e r f * , then f i s a n a f i n e map.

If rank of f is assumed to be one in the previous theorem, we have.

186

Theorem 5.4. Let f : ( M 1 , g l ) -+ (M2,92) be a map of rank one satisfying the Mobius equation. Suppose that M I is connected and that R i c ~ > , 0.

If

+

C92(J,J) diW(T(f))

for some constant C

< (nl - 1) then f

>0

is a n a f i n e map.

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SYMPLECTICALLY ASPHERICAL MANIFOLDS WITH NONTRIVIAL r2 AND WITH NO KAHLER METRICS *

M.FERNANDEZ~ ,

v.MUNOZS

AND J. A. SANTISTEBAN~

t, Departamento de Matemciticas, Facultad de Ciencias, Universidad del Pals Vasco, Apartado 644,48080 Bilbao, SPAIN E-mails: mtpferolQlg.ehu. es, mtpsaeljQ1g.ehu.es % Departamento de Matemciticas, Facultad de Caencias, Universidad Autdnoma de Madrid, 28049 Madrid, SPAIN

E-mail: vicente.munozQuam.es

In a previous paper, the authors show some examples of compact symplectic solvmanifolds, of dimension six, which are cohomologically Kahler and they do not admit Kahler metrics because their fundamental groups cannot be the fundamental group of any compact Kahler manifold. Here we generalize such manifolds t o higher dimension and, by using Auroux symplectic submanifolds [ 31, we construct four-dimensional symplectically aspherical manifolds with nontrivial ~2 and with no Kahler metrics.

1. Introduction During the last years, the study of symplectic manifolds has been of much interest. These manifolds appeared first in mathematical physics, but they * MSC2000: 53D05, 55P62. Keywords : symplectically aspherical manifolds, Auroux symplectic submanifolds, ho-

motopy groups, formality, hard Lefschetz theorem. tPartially supported through grants MCYT (Spain) Project BFM2001-3778-c03-02 and UPV 00127.310-e-14813/2002. Also partially supported by The European Contract Human Potential Programme, Research Training Network HPRN-CT-2000-00101. $Supported by MCYT (Spain) Project BFMZOOO-0024. Also partially supported by The European Contract Human Potential Programme, Research Training Network HPRNCT-2000-00101. §Partially supported through grants MCYT (Spain) Project BFM2001-3778-c03-02 and UPV 00127.310-e-14813/2002. 187

188

are now of independent interest due to their relationship to differential and algebraic geometry. A symplectic manifold is a pair ( M , w ) where M is a 2n-dimensional differentiable manifold and w is a closed non-degenerate 2-form on M . The form w is called a symplectic form. Darboux's theorem states that any sufficiently small neighborhood in a symplectic manifold is symplectomorphic to an open set in R2" with the canonical skew-symmetric bilinear form

2d x i

A

dxn+a.

i=l

Any symplectic manifold ( M ,w ) carries an almost complex structure J compatible with the symplectic form w, which means w ( X , Y ) = w ( J X , J Y ) for any X , Y vector fields on M (see [ 2 2 ] , [23]). In particular, if ( M ,w ) possesses an integrable almost complex structure J compatible with the symplectic form w , such that the Riemannian metric g given by g ( X , Y ) = - w ( J X , Y ) is positive definite, then ( M , w ,J ) is said to be a Kahler manifold with Kuhler metric 9 . Thus, one can think of a symplectic manifold as a generalization of a Kahler manifold, and it is natural to ask: Which manifolds carry symplectic forms but not Kahler metrics? Several geometric methods to construct symplectic manifolds were given by different authors (see for example [ 31, [ 41, [ 61, [ 91, [ 131, [ 161, [ 201, [ 211). Many of the symplectic manifolds there presented do not admit a Kahler metric since either they are not formal or do not satisfy the hard Lefschetz theorem, or they fail both properties of compact Kahler manifolds. In order to find more classes of symplectic manifolds, especially some with no Kahler metric, we generalize the construction of [ 111. There the authors show examples of compact symplectic solvmanifolds M 6 ( k ) ,of dimension six, each one of which is formal and hard Lefschetz, but it does not possess Kahler metrics because its fundamental group cannot be the fundamental group of any compact Kahler manifold according to the results given by Campana in [ 51. In Section 3 we present the compact symplectic manifolds M 2 ( " + l ) ( k )as a generalization to higher dimension of M 6 ( k ) and, in Proposition 3.1, we prove that each manifold M2("+')(k) is formal and hard Lefschetz. Again, each one of the manifolds M 8 ( k ) does not have Kahler metrics since it fails the properties of the fundamental group of a compact Kahler manifold proved by Campana in [ 51. But, we do not know

189

whether or not M2("+l)(Ic),for n 2 4, admits Kahler metrics. However we show that, when n is even, all of them have indefinite Kahler metrics. On the other hand, a symplectic form w on M is said to be symplectically aspherical if the restriction [w]la2(M) = 0, that is,

-

f*w = 0

for every map f : S 2 M. In this case, the symplectic manifold (M,u) is said to be symplectically aspherical. Such manifolds have been very relevant in the study of the Arnold conjecture [ 121. Clearly, any symplectic manifold (M,u) with second fundamental group n 2 ( M ) = 0 is symplectically aspherical. Examples of Kahler and non-Kahler 4-dimensional symplectically aspherical manifolds with nontrivial 7r2 were obtained by Gompf in [ 141. There, it is mentioned that J. Kollb produced, in an unpublished paper, another construction of symplectically aspherical Kahler manifolds with 7r2 # 0. Recently in [ 19] examples of symplectically aspherical symplectic manifolds are given by using Donaldson symplectic submanifolds [ 91. In Section 4 we construct compact symplectically aspherical symplectic manifolds of dimension 4 with n2 # 0 by using the symplectic submanifolds obtained by Aurow in 131 as an extension to higher rank bundles of the symplectic submanifolds constructed by Donaldson in [ 91. In Theorem 4.2 we prove that any 4-dimensional Auroux symplectic submanifold of the manifolds M2("+l)(Ic) is a symplectically aspherical manifold with nz # 0 and does not admit Kahler metrics for n 5 3.

2. Preliminaries In this section, we recall some definitions and results about formal manifolds and the hard Lefschetz property, that we will need in the next sections. A dzflerential algebra ( A , d ) is a graded commutative algebra A over the real numbers, with a differential d which is a derivation, i.e.,

d ( a . b) = ( d a ) . b

+ (-l)deg(a)

a . (db),

where deg(a) is the degree of a. A differential algebra ( A ,d) is said to be minimal if it satisfies: a ) A is free as an algebra, that is, A is the free algebra A V over a graded vector space

190

V = BVZ, and b ) there exists a collection of generators { a , , ~E I } , for some well ordered index set I , such that deg(a,) 5 deg(a,) if p < T and each da, is expressed in terms of preceding up ( p < T ) . This implies that da, does not have a linear part, i.e., it lives in A VO ’ . A VO ’ c A V. Morphisms between differential algebras are required t o be degree preserving algebra maps which commute with the differentials. Given a differential algebra ( A , d ) ,we denote by H * ( A ) its cohomology. A is connected if H o ( A )= R, and A is one-connected if, in addition, H 1 ( A )= 0.

A differential algebra ( M , d ) is said t o be a minimal model of the differential algebra (A,d) if ( M , d ) is minimal and there exists a morphism of differential graded algebras p : ( M ,d ) ( A ,d ) inducing an isomorphism p*: H*(M) H * ( A ) on cohomology. Halperin in [ 171 proved that any connected differential algebra (A,d) has a minimal model unique up to

-

-

isomorphism.

-

A minimal model ( M ,d ) is said to be formal if there is a morphism of dif( M ,d ) ( H * ( M )d, = 0) that induces the identity ferential algebras on cohomology.

+:

A minimal model of a connected differentiable manifold M is a minimal (A V, d ) for the de Rham complex (RM, d ) of differential forms on M . If M is a simply connected manifold, then the dual of the real homotopy vector space .rri(M)8 R is isomorphic t o V ifor any i. We shall say that M is formal if its minimal model is formal or, equivalently, the differential , = 0) have the same minimal model. (For algebras ( G M ,d ) and ( H * ( M ) d details see [ 151 for example.) model

-

An algebraic-topological condition for the formality of a manifold M is the existence of a morphism p : ( H * ( M ) , d= 0) (A* V,d)of differential algebras inducing the identity on cohomology. Consider a map p defined by choosing closed forms representatives for each cohomology class of M . But notice that, in general, the map p is not a morphism of algebras. In [ 101 the condition of the hard Lefschetz property for a symplectic manifold is weakened t o the s-Lefschetz property as follows.

Definition 2.1. Let ( M ,w)be a compact symplectic manifold of dimension

191

2n. We say that M is s-Lefschetz with s [w

in-i

is an isomorphism for all i

:H

~(M)

f

< (n - 1) if ~ 2 n - i

(MI

< s.

Note that M is (n-1)-Lefschetz if and only if M satisfies the hard Lefschetz theorem.

3. The manifolds M 2 ( n + 1 ) ( k ) Let G2n+' ( k ) be the connected completely solvable Lie group of dimension 2n 1 consisting of matrices of the form

+

; ;),

E2n t 0 2 n A2n

.=(a:

where z E R, Ozn is the 1 x 2n matrix with all the entries equal to zero, t 0 2 n denotes the transposed matrix of 0 2 n , Aan is the 2n x 1 matrix ( x 1 , y 1 , ~ 2 , y 2 , . . ., z n , y n ) with z i , y i E R (1 i n), and E2,, is the diagonal 2n x 2n matrix whose principal diagonal is the vector ( e k z , e - k z , e k z , e - k z ,.. . ,e k z ,e - k z ) , of length 2n, being k a real number different from zero. Then a global system of coordinates xi,y i , z (1 5 i n) for G2"+l(k) is given by xi(.) = xi, y i ( a ) = yi, .(a) = z. A standard calculation shows that a basis for the right invariant 1-forms on G2n+1(k) consists of

< <

<

{ d x i - k xi d z , dyi

+ k yi d z , d z 1 1 5 i 5 n} .

Alternatively, the Lie group G2"+l(k) may be described as a semidirect product G2"+'(k) = R K + Ran, where $ ( z ) is the linear transformation of W2" given by the diagonal matrix Ezn for any z E R. Thus, G2"+'(k) has a discrete subgroup l?2n+1(k) such that the quotient space ~ 2 n + l ( k ) = r2n+l (k)\G2"+l ( k ) is compact. Therefore the forms d x i - k xi d z , dyi k yi d z , d z (1 5 i 5 n) descend to 1-forms ailpi, y (1 5 i 5 n) on N2"+l(Ic) satisfying

+

192

where 1 5 i I n, and such that at each point of N2*+'(k), the collection { a i , & , y I 1 5 i 5 n } is a basis for the 1-forms on N2"+l(k). Using Hattori's theorem [ 181 we compute the real cohomology of N2"+l(k) :

HO(N2"fl

( k ) ) = (I), H1( j p + l ( k ) ) = ([YI), H2(N2"+1

( k ) ) = ([aiA P j 1 1 1 I i , j 5 4, H3(N2"+'(k)) = ( [ a (A p j A y ] 11 5 i , j 5 n ) , ~4 ( ~ 2 " f (l k ) ) =

([ai A/$ A a k A D r ] 11 5 i

In general for p 2 2 we have

H 2 P ( N 2 n + 1 ( k= ) ) ([ailA pj, A aiz A pjz A . . . A ~

i ,A

pj, ] I

l < i l
H2p+1(N2"+1(k)) = ([ailA pj,

A

aiz A pjz A . . . A ai, A pj, A y] I

1 I i l
Next let us consider the manifold M 2 ( , + l ) ( k ) = N2"+1(k)x S1. Hence there are 1-forms a*,pi, y,r] on M2("+l)(k)such that

dai = - k a i A 7,

dpi = k p i A 7,

dy = d v = 0,

<

where 1 i I n, and such that at each point of M2("+')(k),{ai,pi,y,1711 i n } is a basis for the 1-forms on M 2 ( " + l ) ( k ) .

<

<

Proposition 3.1. T h e manifold M2(*+l)(k)is formal, and has a symplectic f o r m w such that (M2("f1)(k),w ) satisfies the hard Lefschetz property. Proof. We define a morphism p : (H*(M2("+')(k)),d= 0 ) --t (!2*(M2("+1) (k)),d)

by linearly choosing closed forms representatives for each cohomology class; that is, p( [ y 1) = y,p ( [ 771) = r], etc. One can check that p is multiplicative and then it is a homomorphism of differential graded algebras which induces the identity on cohomology. Therefore, the manifold M2("+l)(k)is formal.

193

The collection {ai A P j , y A v 11 5 i , j 5 n} is a basis for the closed 2-forms on M2(n+1)(k).Thus the 2-form w on M2("+l)(k)defined by n

i=l

is a symplectic form on M2(n+l)( k ) .

-

Now, a straightforward calculation shows that the map [W]n+l--i:Hi(M2("+1)@))

is an isomorphism for any 1 5 i 5 n the hard Lefschetz property.

H2(n+l)-i(M2(n+l)

(k))

+ 1, and so ( M 2 ( " + l ) ( k )w, ) satisfies 0

Remark 3.1. We must notice that the formality of the manifolds M 2 ( n + 1 ) ( kmust ) be understood only in the sense of existence of the morphism p : ( H * ( M 2 ( n + 1 ) ( k ) )= , d0 ) (C2*(M2(n+')(k)),d), defined in the previous Proposition, such that p induces an isomorphism on cohomology, but it does not directly relate t o rational homotopy theory.

-

Theorem 3.1. M2("+')(k) does not admit Kahler metrics for n 5 3. Proof. It is similar to that given in [ 111 for the manifolds M 6 ( k ) . In fact, t o show that M 8 ( k ) does not admit Kahler metric, recall that r 8 ( k )= 7r1(M8(k)) is a semidirect product Z2K Z6. Moreover, its abelianization is H1(M8(k);;Z)and thus it has rank 2. We shall see that r8(k) cannot be the fundamental group of any compact Kahler manifold. The exact sequence

o -+ z6

- rs(k)

z2-+

0,

(3.1)

shows that I " ( k ) is solvable of class 2, i.e., D 3 r 8 ( k )= 0. Moreover its rank is 8 by additivity (see [ 1] for details). Assume now that r8(k)= 7r1(X),where X is a compact Kahler manifold. According to Arapura-Nori's theorem (see Theorem 3.3 of [ 2 ] ) , there exists a chain of normal subgroups

o = 03r8(k)c Q c P c r 8 ( k ) ,

194

such that Q is torsion, P / Q is nilpotent and r8(k)/Pis finite. The exact sequence (3.1) implies that r 8 ( k )has no torsion, and so Q = 0. As r8(k)/P is torsion, thus finite, we have r a n k P = rankr8(k) = 8. Now, the finite inclusion P c P ( k ) defines a finite cover p : Y + X that is also compact Kahler and it has fundamental group P . We show that P cannot be the fundamental group of any compact Kahler manifold. For this, we use Campana7s result (see Corollary 3.8, page 313, in [ 51) that states that if G is the fundamental group of a Kahler manifold such that G is nilpotent and non-abelian, then G has rank 2 9. Since P is the fundamental group of the Kahler manifold Y , P is nilpotent and has rank < 9, it has to be abelian. This is impossible since any pair of non-zero elements e E Z2 c r 8 ( k ) = Z2 K Z6, f E Z6 c r 8 ( k ) do not commute.

0

Remark 3.2. Notice that the previous proof fails for n 2 4 since we would have rank P 2 10, and so we cannot use the result of Campana mentioned before. For n 2 4 we do not know whether or not M2(n+1)(k)possesses Kahler metrics. In [ 71 DardiB and MBdina prove that any completely solvable, unimodular and Kahler Lie algebra is abelian. This fact implies that if w is an invariant symplectic form on M 2 ( n + 1 ) ( k(for ) arbitrary n) and J is an invariant integrable almost complex structure compatible with w , then ( J , w ) does not define a (positive definite) Kahler metric on M 2 ( * + l ) ( k ) . Next, we shall construct an indefinite Kahler metric on M 2 ( n + 1 ) ( kwhen ) n is even. Let { X i , y Z , 2,T 1 1 5 i 5 n} be a basis of (global) vector fields on M2(n+1)(k) dual to the basis of 1-forms { a i , p i , r ,I ~ 1 5 i 5 n}. Then

[yZIZ]= -Icy,,

[ X i , Z ]= k X i ,

(1 5 i 5 n ) ,

and the other brackets are zero. Define an almost complex structure J on M 2 ( " + l ) ( k )(recall that n is even) by JX2i-1

= X2i,

JY2i-1

= Y2i,

JZ

=T,

where 1 5 i 5 n. A direct computation shows that the Nijenhuis tensor of J vanishes. Consequently, J is complex. A basis (X2i-1, p2i-1, v I 1 5 i 5 n } for the forms of bidegree ( 1 , O ) is given by X2i-1

= a2i-1

+

d3a2i7

p22-1

= p2i--1

+

d3p2i7

v

=y

+ -77.

195

Thus we have

Define

and so the Then R is a symplectic form of bidegree (1,l)on M2(n+1)(k), metric g given by g(U, V ) = R(U, J V ) , for vector fields U, V on M 2 ( " + l ) ( k ) , is an indefinite Kahler metric. 4. Symplectically aspherical manifolds with nontrivial 7r2

and with no Kahler metrics In this section we show a method to construct symplectically aspherical manifolds. Those of dimension 4 have nontrivial 7r2 and do not admit Kahler metrics. For this, we use the symplectic submanifolds constructed by Auroux in [ 31. Let ( M , w ) be a compact symplectic manifold of dimension 2 n with [ w ] E H 2 ( M )admitting a lift to an integral cohomology class, and let E be any hermitian vector bundle over M of rank r . In [ 31 Aurow constructs symplectic submanifolds 2, L) M of dimension 2 ( n - r ) whose Poincarh dual are PD[Z,.] = k'[wl'+k'-'cl(E)[w]'-'+...+c,(E) for any integer number k large enough, where c i ( E ) denotes the ith Chern class of the vector bundle E . Moreover, these submanifolds satisfy a Lefschetz theorem in hyperplane sections, which means that the inclusion j : 2,. c-1 M is ( n-r)connected, i.e., the map there j*:Ha(M) Hi(&)is an isomorphism for i < n - r and a monomorphism for i = n - r .

-

Also Auroux proves (see Proposition 5 of [ 31) that the Euler characteristic of 2, is given by x(Z,.) = (-l)n-'(zI:) wnkn O(knv1). Therefore for k large enough, Hn-'(ZT) is of very large dimension. In particular Hn-'(M) H"-'(Z,) is not an isomorphism. The formality and the hard Lefschetz theorem for Auroux symplectic submanifolds were studied by the authors in [ 111. There it is proved the following theorem:

-

+

Theorem 4.1. (1111) If ( M , w ) is formal and/or hard Lefschetz, any Auroux symplectic submanifold 2, is formal and/or hard Lefschetz. Let

196

j : 2,

~f

M be the inclusion, f o r [ z ] E H P ( M ) with p 2. n - r

+ 1 and

d i m M = 272, we have j".]

= o e [ z ] U c , ( E K J L @=o. ~)

(4-1)

Regarding to the cohomology of 2, we have

Proposition 4.1. Let M be a compact symplectic manifold of dimension 2n, and let 2, L) M be an Auroux submanifold of dimension 2 ( n - r ) . Let us suppose that M is s-Lefschetz with s 5 ( n - r - 1). Then, for each p = 2 ( n - r ) - i with i 5 s, there is an isomorphism

Proof. From (4.1), we know that there is an inclusion

HP(M)

ker(c,(E KJ L e k ) : H p ( M )

-

L)

HP+2T(M))

HP(2,).

To prove the reverse inclusion, let us consider an arbitrary metric on H * ( M ) ;for example, the L2-metric on harmonic forms. Let S c H i ( M ) be the unitary sphere, and denote by K an upper bound of

{a u [ W ] n - i - 9 U c9(E)I a E S, q = 1,.. . , r } .

On the other hand, the s-lefschetz property of manifold M implies that su[ W]n--ic HSn-i ( M ) does not contain zero. Therefore, there is a lower bound c > 0 of the set { a u [w]"-Z I a E S } . Now, for any [ z ] E S , we obtain

[z]u[w]~-'-~U(kr[w]~+Ic~-~[W]T-~UC~(E)+... # O+ C , ( E ) ) taking Ic

> (r - 1)K/c.

-

The s-Lefschetz property guarantees an isomorphism [,]?-i

-

:Hi(M)

HPf2r

(M).

Suppose that c T ( E@ L@'"): H P ( M ) HP+2'(M) is not surjective. Then let a E H P f 2 ' ( M ) be an element of norm one in the perpendicular of its image. There exists ,f3 E H p ( M ) such that k'[w]' U p = a. So, the norm

197

the norm of c,(E @ Lmk),d- a is less or of ,8 is at most ~ - l k - ~ Then . equal than ( r - 1) K/ck. Choosing k large enough we see that this is a contradiction. Now computing dimensions, we have

P ( M ) - ( P ( M )- P+"(M))

= P+',.(M) = bi(M) = bi(ZT) = bp(Z,.),

which completes the proof.

0

-

Let us identify the de Rham cohomology group H 2 ( M ) with the group of the homomorphisms Hom(Hz(M), R), and let hM: 7rz(M) H 2 ( M ) be the Hurewicz homomorphism for M . Suppose that ( M , w ) is a compact symplectic manifold, and denote by [ w ] the de Rham cohomology class defined by the symplectic form w . We say that w is a symplectically aspherical form if [ w ] o hM = 0, i.e., [ w ] I T 2 ( = ~ )0, which means that

-

f*w =0 I

2

for every map f : S2 M . In this case, ( M , w ) is said to be a symplectically aspherical manifold.

Theorem 4.2. Let ( M ,w ) be a symplectically aspherical compact manifold. Then any Auroux symplectic submanifold 2,. L, M is also symplectically aspherical. Moreover any 4-dimensional Auroux symplectic submanifold v M2("+')(k) is formal, hard Lefschetz and 7r2(Zn-1) # 0, and the submanifolds 2 2 ~f M 8 ( k ) do not admit Kahler metrics.

Proof. First we note that any symplectic submanifold j : ( N , j * w ) ~f ( M , w ) is also symplecticaily aspherical. In fact, we have [ j * w ] I X 2 ( = ~ )0 since (j*w>(hiv(r2(W))= w(j*(hN(r2(N)))) = W h ( j * ( 7 r 2 ( " )

-

-

= 0,

where we denote with the same symbol j , the maps H 2 ( N ) H 2 ( M ) and 7rz(N) 7r2(M) induced by the inclusion j . In particular, if ( M ,w ) is a symplectically aspherical manifold, any Auroux symplectic submanifold 2,. L, M is also symplectically aspherical. (Notice that without loss of generality we can assume that w is an integral symplectically aspherical form since, according to Proposition 1.4 in [ 191, any compact symplectically aspherical manifold has an integral symplectically aspherical form.)

198

Next let us consider the compact symplectic manifolds ( M 2 ( " + l ) ( k )w, ) which are symplectically aspherical since 7r2(M2("+')(k))= 0. Now, from Proposition 3.1 and Theorem 4.1 it follows that any Auroux symplectic submanifold 2, L) M2("+')(k)is formal and hard Lefschetz. Consequently, any 4-dirnensional Auroux symplectic submanifold L) M2("+') (k) is formal and hard Lefschetz. Also it satisfies w2(Zn-1) # 0 since H z ( Z n - l ) and H2 (M2("+') (k)) are not isomorphic. Moreover, for any Auroux symplectic submanifold 2 2 c, M 8 ( k ) ,a similar argument to the one given in Theorem 3.1 proves that the fundamental group 7r1(22) = 7r1 ( M 8(k)) cannot be the fundamental group of any compact Kahler manifold, and so the submanifolds 2 2 ~f M 8 ( k ) do not admit Kahler metrics. This completes the proof. References 1. J. Arnorbs, M. Burger, K. Corlette, D. Kotschick, D. Toledo, findamental groups of compact Kahler manifolds, Math. Sur. and Monogr. 44, Amer. Math. SOC.,1996. 2. D. Arapura, M. Nori, Solvable fundamental groups of algebraic varieties and Kiihler manifolds, Compositio Math. 116, 173-188, (1999). 3. D. Auroux, Asymptotically holomorphic families of symplectic submanifolds, Geom. Funct. Anal. 7, 971-995, (1997). 4. C. Benson, C. S. Gordon, Kahler and symplectic structures o n nilmanifolds, Topology 27, 513-518, (1988). 5. F. Campana, Remarques sur les groupes de Kahler nilpotents, Ann. Scient. Ec. Norm. Sup. 28,307-316, (1995). 6. L. A. Cordero, M. Fernbndez, A. Gray, Symplectic manifolds with n o Kahler structure, Topology 25, 375-380, (1986). 7. J. M. DardiB, A. Mkdina, Alg6bres de Lie IcahlLriennes et double extension, J. Algebra 185,774-795, (1986). 8. P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kahler manifolds, Invent. Math. 29, 245-274, (1975). 9. S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Diff. Geom. 44, 666-705, (1996). 10. M. Fernbndez, V. Muiioz, On the formality and the hard Lefschetz property for Donaldson symplectic manifolds, preprint 2002, arxiv: math. SG/0211017. 11. M. Fernbndez, V. Muiioz, J. Santisteban, Cohomologically Kahler Manifolds with no Kahler Metrics, Intern. J. Math. and Math. Sciences (to appear). 12. A. Floer, Symplectic fixed points and holomorphic spheres, Commun. Math. Phys. 120,575-611, (1989).

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13. R. Gornpf, A new construction of symplectic manifolds, Ann. of Math. 142, 527-597, (1995). 14. R. Gompf, On symplectically aspherical symplectic manijolds with nontrivial q ,Math. Res. Letters 5, 599-603, (1999). 15. P. Griffiths, J. W. Morgan, Rational hornotopy theory and differential forms, Progress in Math. 16,Birkhauser, 1981. 16. D. Guan, Examples of compact holomorphic symplectic manijolds which are not Kuhlerian 11,Invent. Math. 121,135-145, (1995). 17. S. Halperin, Lectures on minimal models, MQm. SOC.Math. France 230, 1983. 18. A. Hattori, Spectral sequence i n the de Rham cohomology offibre bundles, J. Fac. Sci. Univ. Tokyo 8, 298-331, (1960). 19. R. Ib&iez, J. Kedra, Y. Rudyak, A. Tralle, Symplectically spherical rnanifolds, preprint 2003, arxiv: math. SG/0103098. 20. D. McDuff, Examples of symplectic simply connected manifolds with no KGhler structure, J. Diff. Georn. 20,267-277, (1984). 21. W. P. Thurston, Some simple examples of symplectic manifolds, Pmc. Amer. Math. SOC.55, 467-468, (1976). 22. A. Tralle, J. Oprea, Symplectic manifolds with no Kahler structure Lecture Notes in Math. 1661,Springer-Verlag, 1997. 23. We A. Weinstein, Lectures on symplectic manifolds, Conference Board of the Math. Sciences, Regional Conference Series in Math. 29, Univ. North Carolina 1976, Arner. Math. SOC.Providence, Rhode Island, 1977.

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SOME NEW RESULTS IN TOPOLOGICAL CLASSIFICATION OF INTEGRABLE SYSTEMS IN RIGID BODY DYNAMICS *

A. T. FOMENKO AND P. V. MOROZOV Moscow State University, Faculty of Mechanics and Mathematics, Department of Differential Geometry and Applications, Vorobiovy Gori d. 1, 119992 Moscow, RUSSIA E-mails: [email protected], [email protected]

The current paper is a brief review of some recent result in the Liouville classification of integrable systems from rigid body dynamics. Namely, the results of the analysis of the Clebsch and Sokolov integrability cases are presented.

1. Integrable Hamiltonian systems on symplectic manifolds

1.1. Completely integrable Hamiltonian systems Consider a 2n-dimensional symplectic manifold ( M 2 n , w ) . Let H be a smooth function on M 2 n . The dynamic system v = sgradH is called a Hamiltonian dynamic system with n degrees of freedom with Hamiltonian H . The vector field sgrad H is defined by the equation w(1, sgrad H) = 1(H), where 1 is an arbitrary vector of the tangent bundle and 1(H)is the derivative of function H along 1. Such a system can be written in local coordinates (d, . . . , x 2 n ) as follows:

i = 1,.. . ,2n.

22 = {zz,H},

Here { , } is the corresponding Poisson bracket on M2 n defined by the equation:

{ f , 9) = w(sgrad f , w a d 9 ) . Definition 1.1. A Hamiltonian system is called completely integrable if there exists a set of smooth functions f 1 , . . . ,f n such that: * MSC.2000: 37515. Keywords ; bifurcations, Liouville equivalence, rigid body dynamics.

201

202 f 1 , . . . , f n are first integrals of v (which is equal to { f i , H } = 0 fulfilled for every i on all M ) , 2) f l , . . . ,f n are functionally independent, which means that their gradients are linear independent everywhere on 111, 3) { f i , f j } = 0, for all i and j , 4) the vector fields sgrad f i are complete.

1)

The completely integrable systems are often simply called integrable. The subdivision of the manifold M2" into the connected components of the common level surfaces of f l , . . . , fn is called the Liouville foliation of the dynamic system. 1.2. Liouville's theorem. In a neighborhood of the a regular common level surface of its first integrals, the topology of a completely integrable Hamiltonian system is fully described by the Liouville's theorem. Let Tc be a common level surface:

Tc

={x E

M I f z ( x )= & , i

=

1,.. . n } .

We call a common level surface Tc regular if the differentials dfi are linear independent everywhere on it. Theorem 1.1. (Liouville) Let v = sgradH be a completely integrable Hamiltonian system o n M 2 n , and let Tc be a regular common level surface of the first integrals f 1 , . . . , f " . Then: 1) Tc is a smooth Lagrangian submanifold in M2", which is invariant in respect to any of the vector fields v = sgradH and sgrad f 1 , . . . sgrad f,. 2) If the manifold Tc is compact, than each connected component of Tc is diffeomorphic to a n-dimensional torus T n . Such tori are called Liouville tori. 3) The Liouville foliation in a neighborhood U of a Liouville torus is trivial, i.e. fiberwise diffeomorphic to T" x Dn, where Dn is a n-dimensional disk. 4 ) I n the neighborhood U = T n x D" there exist regular symplectic coordinates s1,. . . , ,s, (PI,. . . , (P, (called the action-angle coordinates) with the following properties: a) s1,. . . ,s, are the coordinates o n the disk D", while (PI,. . . , cpn are the standard angular coordinates on the torus T", cp E R/2 7r

z.

203

b) w = C d p i A dsi. c) The action coordinates si are some function of the first integrals f i , . , . , f n . d ) I n the neighborhood U the Hamiltonian system can be expressed in the form: ii = 0,

,

(c?i = qi(s1,. . . s n ) ,

i = 1,.. .i = n.

Which means that the integral trajectories of w define a conditionally-periodic motion o n each torus. The proof of this well-known theorem can be found in [ 11.

1.3. Equivalence relations of integrable Hamiltonian systems In the theory of topological classification of integrable Hamiltonian systems there are two main equivalence relations under concern: the topological trajectory equivalence and the Liouville equivalence.

Definition 1.2. Integrable Hamiltonian systems ( M ,w) and (M’,w’) are called topologically trajectory equivalent if there exists a homeomorphism CP : M M’, that takes the oriented trajectories of the first system t o the oriented trajectories of the second systems.

-

Note that the time parameter along the trajectories need not be necessarily preserved.

-

Definition 1.3. Integrable Hamiltonian systems ( M ,w) and (M’, w’) are called Liouville equivalent if there exists a homeomorphism : M M’, that takes the leaves of the Liouville foliation of the first system t o the leaves of the Liouville foliation of the second system. In the current paper we state some new results of the Liouville classification of integrable systems from rigid body dynamics. Some results of the trajectory classification are given in [ 11, where one can also find numerous references to papers on the topic. 2. Fomenko-Zieschang invariants of integrable Hamiltonian systems with two degrees of freedom Further on in this paper we deal with systems with two degrees of freedom, that is when n = 2. Note, that in this case the integrability of a

204

system on M4 is guaranteed if there exists just one additional integral f , functionally independent with H. In the current section we give a brief introduction of the famous Fomenko-Zieschang invariant (also knows as the marked molecule), which completely describes the topology of the Liouville foliation of an integrable system on its regular isoenergy surface. The full version of this theory can be found in [ 11. 2.1. Isoenergy surfaces

An isoenergy surface is a %dimension surface in M4 given by the equation: = {x E M 4 ( H ( x )= h}. We consider only those h, for which Qiis compact and d H # 0 everywhere on Qi.Thus we guarantee that Qi is a compact smooth regular submanifold in M4 and that the Hamiltonian vector field w = sgrad H has no singularities. Along with the trajectory and Liouville equivalences on the whole symplectic manifold M4 we consider trajectory and Liouville equivalences on selected isoenergy surfaces. To define these notions properly one simply has to replace M4 with Q3 in Definitions 1.2 and 1.3. The point x E Q3 is called critical if the vectors sgrad H and sgrad f are linearly dependent in x. Note that the singular common level surfaces Tt in Q3 are precisely those that have critical points on themselves. It is the case when the Liouville theorem cannot be applied. Another important assumption concerns the properties of the critical points on Q 3 . Since sgrad H # 0 on Q3 the critical points are not isolated. That is why there is no sense in supposing the critical points are Morse. But there exists a natural analogue of Morse points for the case of dynamic systems.

Qi

Definition 2.1. An additional integral f is called Bott for the current isoenergy surface Q3 if the critical points make up non-degenerate critical submanifolds in Q3. In this case the critical points are called Bott critical points. By non-degenerate submanifolds we mean such submanifolds that the restriction of the additional integral f to its transversal is a Morse function. Obviously, if the system is integrable one can construct many additional integrals. But the fact that an integral is Bott or non-Bott also depends on the choice of this integral. For example, if we replace a Bott additional integral fi with the additional integral f,”the minimax circles will become non-Bott. Typically, in real mechanical and physical integrable systems the

205

Qi

additional integral obtained is Bott on all except but some finite set of singular values h = { h l ,. . . hm}. The cases of integrability, in which all the values of h are Bott, do exist but they are very few.

2.2. Critical sets on isoenergy surfaces Let us sum up the assumptions we made about Q3 to list the possibilities left for the structure of the critical set in Q 3 . The Bott condition implies that a connected component of the critical set is a non-degenerate submanifold of dimension 1 or 2. Moreover, it is a closed submanifold since Q3 is compact. Also on this manifold there exists a smooth non-singular vector field given by w = sgradH # 0. Hence the manifold’s Euler characteristic is zero. Consequently, it is either a circle, a torus or a Klein bottle. Later on we suppose that the critical sets are represented only by critical circles. The main reason for this lies in that the Liouville foliation in the neighborhood of a critical torus is trivial. Moreover, this torus can be made regular by a certain substitution of the additional integral that doesn’t affect the foliation. As to the Klein bottle, one simply needs to pass to a two-sheeted covering Q f 3 of Q3, to turn it into a critical torus (for details see [ 11). It is also worth mentioning, that in real physical and mechanical integrable systems, the later two types of critical submanifolds are rare exceptions, whereas the critical circles are standard. Also note that usually a critical circle is only a part of a critical common level set Tc, which topological structure may be much more complicated.

2.3. Neighborhoods of singular leaves on the isoenergy

surfaces The isoenergy surface Q3 is made up of a one-parameter family of the common level sets Tc of the integrals H and f , where the parameter is simply the value of f . After collapsing each connected component of Tc to a point we obtain a graph, which is the base of the Liouville foliation (see fig. 1). Every edge of this graph is the product of a fiberwise collapsing of a T 2 x (0,l) manifold. Whereas the nodes of the graph correspond to connected components of the critical common level sets. Typically, the number of regular Liouville tori changes after passing through the critical levels. Consider a 3-dimensional neighborhood of a singular leaf in Q3.

206

.............

It appears that in Bott case there is only a finite number of possible bifurcations in respect to the Liouville equivalence, if the numbers of tori before and after the singular level are fixed.

1

............

............

Definition 2.2. A 3-atom is a Liouville equivalence class of the 3-dimensional neighborhood of a connected singular leaf.

............

............ fig. 1.

The 3-atom can be regarded as a 3-dimensional manifold equipped with a one-para-

I-’

I I T

A’

2.

rameter LiouviIle foliation with just one connected singular leaf.

The

207

boundary of this manifold is comprised of a certain number of Liouville tori. Note that these tori are naturally subdivided into two categories: the tori that precede and follow the bifurcation (the negative and positive parts of the boundary). There exists an algorithm (see [ 11) that allows to produce directly all the possible 3-atoms for a fixed number of positive and negative boundary tori. The 3-atoms are usually expressed in capital Latin letter, possibly with indexes and asterisks. The 4 most common and simple 3-atoms ( A ,A * ,B , and C2) are presented in fig. 2. Now if we place the letter expressions of the corresponding 3-atoms in the nodes of the graph on fig. 1 we will obtain the the rough molecule of the system on Q3. The rough molecule contains the information on the structure of the base of the Liouville foliations and on the types of bifurcations at its singular leaves, thus allowing to recover the local structure of the Liouville foliation in the neighborhood of every leaf. For more detailed information and proofs of the facts given in this section consult [ 11. Here we only state the main topological theorem. Theorem 2.1. (A. T. Fomenko [ l ] )Consider a Liouville foliation of an integrable Hamiltonian system ( Q 3 ,v) with a Bott additional integral. Then: 1 ) The 3-dimensional neighborhood of a singular leaf of the Liouville foliation can be equipped with a Seifert fibration, which singular fibers, if any, are of (2) 1) type. 2) If the Seifert fibration has no singular fibers, the neighborhood has a topological structure of a P2 x S1manifold, where P2 is an orientable 2-dimensional surface with a boundary comprised by a finite number of disconnected circles. 3) Every fiber of the Seifert fibration (a circle) is a part of some leaf of the Liouville foliation. 2.4. Gluing matrices

3-atoms contain the information about the Liouville foliation in the neighborhood of singular leaves. To reconstruct the global structure of the Liouville foliation on Q3, one simply has to add the information about the homeomorphisms of their boundaries by which the 3-atoms are "glued" with each other through one-parameter families of regular tori. If we fix a pair of basis cycles in the fundamental group of each boundary torus the gluing homeomorphism can be expressed by an integer 2 x 2 matrix with a

208

f l determinant. The only problem is that the basis pairs can be chosen in many different ways. Yet it turns out that; a 3-atom determines a unique cycle on each of its boundary tori, called the uniquely defined cycle of this 3-atom. In case of the minimax atom A it is the cycle that collapses to a point when approaching the critical circle. In case of all the rest (saddle) atoms (A*, B , C2,. . . ) the uniquely defined cycle is the cycle that is isotopic to the fibers of the Seifert fibration. The orientation of the uniquely defined cycle is also fixed in a way according to the topology of the 3-atom (see [ 11). It appears that it is impossible to fix the second cycle of the basis pair in a similar way. Moreover, the representatives of a whole infinite set comprised of possible second cycles, which is determined by certain rules depending on the atom’s type, are in some natural sense equally good. Together with the uniquely defined cycle they make up the class of the admissible coordinate systems of the 3-atom (see [ 11 for an accurate definition). Inside this class the group of admissible change of coordinates is of a very simple structure. The numerical marks r, E and n are the invariants of the action of this group on the set of the gluing matrices. They can be calculated by simple formulas given in [ 11, if a set of gluing matrices for some choice of admissible coordinates is known. A pair of numerical marks r and E emerges on every edge of the molecule, whereas the mark n may emerge on a saddle atom or on a special group of saddle atoms, called the family. The numerical marks have natural topological interpretations. The rational mark r E {Qn [0,l),co} on an edge defines the unsigned index of intersection of the uniquely defined cycles of the 3-atoms that this edge links. The mark E E {l,-1} shows if their orientations are the same or different, when such a comparison is possible. Finally, the n E Z mark is the Euler invariant of the corresponding Seifert fibration formed by the family of saddle atoms. In the end of this section we give the definition of the Fomenko-Zieschang invariant and state the main theorem. Definition 2.3. The Fomenko-Zieschang invariant, or the marked molecule, is a rough molecule W , equipped with numerical marks ri,Ei and nk. Theorem 2.2. ( A . T. Fomenko, H. Zieschang [ l ] )Two integrable Hamiltonian systems ( v , Q 3 ) and ( v ’ , Q ’ ~ )are Liouville equivalent i f and only i j their marked molecules coincide.

209

3. Some new results of Liouville classification in rigid body dynamics 3.1. The phase space Let us define the pair (M4,w ) that emerges in rigid body dynamics. Consider the Lie algebra e(3), corresponding to the Lie group E(3), the 6-dimensional group of motions of R3.The linear space e(3)* is equipped with a Lie-Poisson bracket of two arbitrary smooth functions f and g:

{f,g}(z) = d [ d z f , d z g I ) , IC

E e(3)*,

I,] is the commutator in Lie algebra e(3).

In standard coordinates s1,s2,s3,R1, R2, R3,

on e(3)* this Lie-Poisson bracket can be expressed by the following formulas:

{si,sj} = E i j k S k ,

{Ri,S j } = EijkRk,

{Ri,R j } = 0,

where

Let H ( S ,R ) be a smooth Hamiltonian on e(3)* and consider a system of Kirghoff differential equations:

{SZ,H}, Rg RZ and

Ri = {Ri,H } . (3.1) Functions f1 = R: f i = SIR1 S2R2 S3R3 lie in the kernel of the Lie-Poisson bracket, hence being the first integrals of (3.1). It’s easy to prove that the restriction of system (3.1) to every common level surface of functions f 1 and f 2 : si =

+ +

M;

= { f i= Rf

+

+

+ Rg + Ri = 1,f 2 = SIR1 + S2Rz +S3R3 =g},

is a Hamiltonian system with two degrees of freedom. Ad: surfaces are smooth submanifolds in e(3)*, diffeomorphic to T * S 2 . The symplectic structure on M: is given by the restriction of the Lie-Poisson bracket of e(3)’. Each restricted system on M i will be integrable if there exists a function F ( S ,R ) on e(3)*, functionally independent with H ( S , R ) ,f l and f 2 such that { H , F } = 0. Thus an arbitrary smooth function H ( S , R ) on e(3)* yields a one-parameter family of Hamilton systems (either integrable or non-integrable). The parameter g has the physical meaning of the axis of gravity from rigid body dynamics.

210

3.2. Classical cases of integrability: Euler, Lagrange, Kovalevskaya

The specific Hamiltonians H and additional integrals F in the classical integrable cases of Euler, Lagrange and Kovalevskaya can be given as follows: Euler case:

s,2

s;

2A1

2A2

H=-+-+Lagrange case:

H

s3” ,

2A3

F=S;+SZ+Si

1 (S2 1 + S Z + sx3” )+R3,

F=S3.

=-

2

Kovalevskaya case:

F

=

(Q S2 +

a2R2

-

alR1)

2

+ (a- alR2 - a2R1 A

Ai, A , aj are real parameters of the corresponding mechanic systems. The calculation of the Fomenko-Zieschang invariants for Euler and Lagrange tops is given in [ 11. Three different Liouville foliations of the isoenergy surfaces appear in each case. The corresponding marked molecules are presented in fig. 3. Euler

E = l

1) A-

r=o

A

Lagrange t = l

1) A-

r=o

A fig. 3

211

The topology of the foliation in Kovalevskaya case is considerably more complicated. The Liouville invariants were first completely settled in [ 21 with the help of the so called method of loop molecules, introduced in this paper. It appeared that ten different types of Liouville foliation emerge in the Kovalevskaya case. The method of loop molecules was also successfully applied in the larger part of discovered cases of integrability in rigid body dynamics. As a result the Fomenko-Zieschang invariants have been calculated in Zhukovski, Gorjatchev-Chaplygin [ 31, Sretenski, Clebsch [ 41 and other cases of integrability.

3.3. Liouville classification of the Clebsch case In this section we settle two important corollaries of Liouville classification of the well-known Clebsch case of integrability. The integrable Clebsch case corresponds to a mechanical system of a rigid body, fixed in its center of gravity, in a parallel vertical force field, where the module of the forces changes linearly with the height. The Hamiltonian and the additional integral can be expressed as follows:

H

s,2 + s; + s3” + t

=-

2A1

2Az

2A3

-

2

+

( A I R : A2R;

+ A3Ri),

Here E E R\{O} is a real parameter, proportional to the linear force change coefficient. In case e = 0 we have the Euler top. Due to an obvious symmetry, we get isomorphic Liouville foliations on M i and Hence onwards we consider g 2 0. Below we state some results achieved by Pogosyan [5, 6, 71 and Oshemkov [ 8 , 9 ] , that are essential to the Liouville classification of the Clebsch case. It is easy to prove that all variants of H and K can be presented as linear combinations of two commuting functions Ho and Fo of the following type:

+ + + + +

+

+

Ho = ( S ; S; S,”) ( c ~ R : c2R; C ~ R ; ) , Fo = ( C I S ? C~SZ ~35’32)- (cIR: c;R; cZRZ),

+ +

+

+

such that c1 cz q = 0. Thus we practically reduce the number of parameters from four to two. Henceforth we consider the integrable system

212

induced by HO and Fo, but the "isoenergy" surfaces we are interested in are now given by the equation Q: = {x E M;lH(z) = a H o ( x ) p F o ( z ) = c}. Not all combinations of a and ,B should be regarded. A. A. Oshemkov proved that in case E => 0, the lines f = -(a/,B)h cover zone I on coordinate space R2(h,f) in fig. 4. In case E < 0 zone I 1 is covered.

+

fig. 4.

Essential information on the topological structure of the Liouville foliation on M: can be collected from the momentum mapping:

Ho x Fo : M;

-

W h , f)

>

II:

-

( H o ( z ) F, o ( I I : ) ) .

The image of critical points of the momentum mapping is called the bifurcation diagram. The bifurcation diagrams for the Clebsch case were first obtained by Pogosyan [5, 6, 71. It appears that depending on the value of g one must distinguish 4 different cases:

(I) g = O (see fig. 5 (a));

(2)

0 < g2 < P l (fig. 5

< g2 < P2

(4)

g2

(3) Pl

(fig. 5 (c));

> P z (fig. 5 ( 4 ) .

Here PI, p:! are some definite constants depending only on c1, c2, c3. The curves of the bifurcation diagrams are images of critical common level sets of the integrals. Each smooth curve corresponds to a corank 1 Bott bifurcation of one type. Note that in general a specific common level set may be disconnected, and hence be comprised of several disconnected bi-

213

f

f

fig. 5 .

furcations, possibly of different types. The smooth curves on fig. 5 are denoted by small Greek letters. The bifurcation types of these curves, obtained by Pogosyan [ 5, 6, 71 and Oshemkov [ 8, 91, are as follows: curve

61,63

bifurcation type 2A 2B

Y1,YZ

CZ

al,a 2 , a3, P, 62

-

+

The curves az(as), Y Z ( ( Y ~and ) /? have linear asymptotes f = c3 h c1 c2, f = c2 h c1 c3 and f = c1 h c2 c3 respectively when h +co . By M , N and P we denote the images of corank 2 critical points in M:,

+

+

214

namely the points where dHo = dFo = 0. They make up the so-called non-degenerate equilibrium points of the system (see [ 11). Their types are as follows:

M N P P P

center-center saddle-center saddle-saddle focus-focus center-center

2 ( A x A) 2 ( A x Cz) (C2

x

C2)/&

2 ( Ax A)

g>o all g 0 < g2 < pl P1 < g2 < PZ g2 > P2

+

The coordinates of M , N and P are (g2 ci, ci(g2 - ci)),i = 1 , 2 , 3 respectively. The return points z1,.t2 are the images of corank 1 non-Bott singularities, called saddle-node ( g > 0) and pitchfork (g = 0) (see [ 1, 21). The areas between the curves of bifurcation diagram, called cells, are filled with regular points of momentum mapping. The preimages of such points are a union of a certain number of disconnected Liouville tori. Thus every point of cell I (see fig. 5) is covered by 2 symmetric tori. Similarly, cell I I is also covered by a pair of symmetric tori. The situation with cell I I I is slightly different: each point is covered by 2 pairs of symmetric tori. One pair collapses on S2 A-curve, whereas the other sweeps through 62-curve into cell I I without bifurcations. As we can see, the preimage of &-curve is actually comprised of 2 regular Liouville tori and 2 singular common level sets, namely 2 critical minimax circles. It is worth bearing in mind that the image of momentum mapping is a projection of the base of Liouville foliation of the system on R2. In general, the base of Liouville foliation is a 2-dimensional complex with natural stratification, induced by the rank of singularities. As we can see it is not necessarily embedable in EX2. We are interested in the structure of Liouville foliation of the isoenergy surfaces, that is the Liouville foliation in the preimage of lines Q! h P f = c. The information stated above gives us the rough molecules of these foliations. Hence to obtain the Fomenko-Zieschang invariants one simply has to calculate the numerical marks T , E , n. This was done by the loop molecules method [ 21. The calculation is fully states in [ 41. We won’t give technical details of the calculation in this paper, but the idea of loop molecules method can be briefly stated as follows. Consider a small circle around a singular point of the bifurcation diagram. The preimage is a 3-dimensional manifold equipped with a Liouville foliation. The FomenkoZieschang invariant of this foliation is called the loop molecule of the singular point. Since it is a local invariant we need to know only the local structure of

+

215

the system to obtain it. Moreover, the singularities presented by such points as M , N , P, z1 and 2 2 of the bifurcation diagram are typical for integrable Hamiltonian systems. The same types of singularities in respect to the local Liouville foliation can be found in many other cases of integrability. Their loop molecules are well known. Thus to calculate the loop molecules we actually have to prove that the singularities are of a certain type. On the other hand, loop molecules can be very helpful when calculating the global isoenergy Fomenko-Zieschang invariants.

6)

21, 2 2

0

(SZO)

(x2)

r b A fig. 6.

All six loop molecules of the Clebsch top are presented in fig. 6. In case of the focus-focus singularity (point P ( p 1 < g2 < p 2 ) ) , the monodromy

216

matrix plays the role of the loop Liouville invariant. Finally, in the Clebsch case we obtain ten different types of isoenergy Liouville foliation for different values of g and H . The corresponding marked molecules are given in fig. 7. c = l r=O

1) A-

A

(X2)

€=l

€=l

€=l

€=-1

T=Ao\ €=l

r=o e = l

1

€ =I E = -1 r = m

6)

A-

A

(X2)

€=l 7)

hA

r= €=l

fig 7.

217

Now consider the molecule of big energies H (see fig. 8) for the Clebsch top. Note that it is the same for all g . But the marked molecule of big energies of the Euler top is also identical to it. Thus we obtain

t = l

E = l

E = l

t = l

fig. 8.

Theorem 3.1. (P. V. Morozov [ 4 ] ) For suficiently big values of energy the Clebsch top is Liouville equivalent to the Euler top. In other words, for all values of parameter F , principal moments of inertia Ai and axis of gravity g of the Clebsch case and for all values of principal moments of inertia A’j and axis of gravity g’ of the Euler case there exists a level of energy ho, such that for all h, h’ > ho the Liouville foliation on Qiof the Clebsch case is isomorphic to that on Qi,of the Euler case. The established Liouville equivalence has a natural physical interpretation. We have already mentioned that we can obtain the Euler top out of the Clebsch top if we replace the linearly changing forces with constant forces. The constant force field is equal to the zero force field, since the rigid body is fixed in its gravity center. Evidently, the top in the Clebsch case can be supplied with such an amount of energy that it will not feel the relatively small external force field and will be moving almost purely according to its inertia like in the Euler top. Theorem 3.1 states the mathematical essence of this fact.

t = l

t = l

E = l

t = l

fig. 9.

Now consider the Clebsch and Euler cases under big values of axis of gravity g. The sets of isoenergy marked molecules are the same. They are given in fig. 9. The bases of the Liouville foliations are also isomorphic. The natural intention is to compare the Liouville foliations of the two systems on the whole M i . By applying simple topological technics one can prove:

218

Theorem 3.2. (P. V. Morozov [ 4 ] ) Under suficiently big values of axis of gravity the Clebsch and Euler tops are Liouville equivalent as integrable systems o n 4-dimensional symplectic manifolds. In other words, for all values of parameter E , principal moments of inertia Ai of the Clebsch top and for all values of principal moments of inertia A’j of the Euler top there exists a value of the axis of gravity go, such that for all g, g’ > go the Liouville foliation on M: of the Clebsch case is isomorphic to that on M , of the Euler case. This equivalence also has a natural physical interpretation. On a fixed level of energy H = h, the increase of axis of gravity results in the turning of the angular momentum toward the vertical position, which means that almost all motion happens in the horizontal plain. When the value of g is sufficiently big, the Clebsch top simply doesn’t feel the vertical inhomogeneity of the force field.

3.4. Marked molecules of the Sokolov case The Sokolov case of integrability, recently discovered with the help of computer methods (see [lo]),is defined by the Hamiltonian:

The additional integral can be expressed as:

F

+ 5’;+ Sz + 2 (R2S3

R3S2)+ Ri + Ri) + 2 S3(5’2- R3)(R1S1+ R2S2 + R3S3).

= 5’;(Sl

-

(34

The bifurcation diagrams of the momentum mapping, the types of Bott bi furcations and the topological types of the isoenergy surfaces were obtained by P. E. Ryabov in [ 111. Depending on the value of g one must distinguish 4 different cases: 1 (fig. 10 (b)); (2) 0 < g < 2

(1) 9 = 0 (fig. 10 (a));

(3)

1

2 < g < 1 (fig. 10 (c));

(4) g

> 1 (fig. 10 (d)).

The bifurcations in the preimages of the smooth curves are of the following types:

:2A

( ~ 1

( ~ 2 : 2 A a3:2A

( ~ 4 : 4 A a5:2A

,B1 : 2B

,&:

71 : C2

2B

,&:2B

72:2C2

a6:2A

61 : 2A

6 2 : 2A”

219

& 0 1

z (712

(714

y2z2 P 3N

(715

71

M

M

fig. 10.

The types of the non-degenerate equilibrium points are classified as follows:

M N P

center-center saddle-center saddle-saddle

2(A x A) 2 (A x B ) B x C2

g >0 0 < g < l/2 0
220

Q L R

saddle-center center-center saddle-center

2 ( A x Cz) 2 ( Ax A) A x Cz

1/2 < g < 1 g > 1/2 g>l

Points zi belong to the corank 1 non-Bott singularities. molecules of the Sokolov top see fig. 11.

For the loop

r=o

1)

A-A

( X

2)

2) N

A-

7)

fig. 11.

In the Sokolov case the loop molecules method doesn't give all the numerical

221

marks. But the Topalov formula [ 121, that links the numerical marks with the topological characteristics of the isoenergy surface, allowed to obtain the final answer. All twelve Fomenko-Zieschang invariants of the Sokolov top are presented in fig. 12.

e = l

r=o

,.--

r=m

A ,4.!bB-

€=lA

t = l

€=l

r=o r = m

t = l

r=o

A

n=l'.,

t = l

r=o

t = l

'A

= -1 r=m E

e = l

Ar = m G E = l

B

m

A

A

t = l

fig. 12.

A-A ( ~ 2 ) ,-n=-2 ~-A?-A- E = l

= r=o

r=m

t = l

r=o

222

As corollary we obtain Theorem 3.3. (P. V. Morozov) In case of s u f i c i e n t l y big values of energies, the Sokolov t o p with a non-zero axis of gravity is Liouville equal to t h e Kovalevskaya t o p with a zero axis of gravity. See fig. 6 for the corresponding marked molecule. The Sokolov top also gives a rare example of a real physical system with critical tori in its phase space (see 2.2). Namely, the preimage of the left vertical line on the bifurcation diagram in case g = 0 consists of a oneparameter family of critical tori

References 1. A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems. Geometry. Topology. Classification., Izd-vo UdGU, 1999. 2. A. V. Bolsinov, P. Richter, A. T . Fomenko, The loop molecules method and the topology of the Kovalevskaya top, Matematicheskiy sbornik, Vol. 191, No 2, 3-42, (2000). 3. 0. E. Orel, The rotation vector of integrable cases that are reduced to Abel equations. Trajectory classification of the Gorjatchev-Chaplygin system, Matematicheskiy sbornik, Vol. 186, No 2, 105-120, (1995). 4. P. V. Morozov, Liouville classification of the integrable Clebsch case systems, Matematicheskiy sbornik, Vol. 193, No 10, 113-138, (2002). 5. T. I. Pogosyan, Construction of bifurcation sets in one rigid body dynamics system, Mekh. tverd. tela, vip. 12, Kiev: Naukova dumka, 3-23, (1980). 6. T. I. Pogosyan, Areas of possible motion of the Clebsch case. Critical case, Mekh. tverd. tela, vip. 16, Kiev: Naukova dumka, 1983, 19-24, (1983). 7. T. I. Pogosyan, M. P. Kharlamov, Bzfurcation set and integral manifolds of a rigid body dynamics system in a linear force field, PMM, Vol. 43, 419-428, (1979). 8. A. A. Oshemkov, Description of isoenergy surfaces of integrable Hamiltonian systems with two degrees of freedom, Trudi seminara PO vectornomu i tenzornomu analizu, Vol. 23, Moskva, MGU, 122-132, (1988). 9. A. A. Oshemkov, Calculation of Fomenko invariants f o r the m a i n integrability cases of rigid body dynamics, Trudi seminara PO vectornomu i tenzornomu analizu, Vol. 25, No 2, Moskva, MGU, 23-109, (1993). 10. V. V. Sokolov, A new integrable case of the Kirghoff equations, Teoreticheskay i matematicheskay phisika, Vol. 129, No 1, 31-37, (2001). 11. P. E. Ryabov, Bifurcations of first integrals in the Sokolov case, Teoreticheskay i matematicheskay phisika, Vol. 134, No 2, 207-226, (2003). 12. P. Topalov, Calculation of Fomenko-Zieschang invariant for the for the m a i n integrability cases of rigid body dynamics, Matematicheskiy sbornik, Vol. 187, NO 3, 143-160, (1996).

THE CRYSTAL DUALITY PRINCIPLE: FROM GENERAL SYMMETRIES TO GEOMETRICAL SYMMETRIES*

FABIO GAVARINI Universitci degli Studi d i Roma “Tor Vergata” Dipartimento d i Matematica Via della Ricerca Scientifica 1, I-00133Roma, ITALY E-mail: gavariniamat. uniroma2.it

We give functorial recipes to get, out of any Hopf algebra over a field, two pairs of Hopf algebras bearing some geometrical content. If the ground field has zero characteristic, the first pair is made of a function algebra F[G+]over a connected Poisson group and a universal enveloping algebra V ( g - ) over a Lie bialgebra g- : in addition, the Poisson group as a variety is an affine space, and the Lie bialgebra as a Lie algebra is graded; apart for these last details, the second pair is of the same type, namely (F(G-1, U ( g + ) ) for some Poisson group G- and some Lie bialgebra g+ . When the Hopf algebra H we start from is already of geometric type the result involves Poisson duality: the first Lie bialgebra associated to H = F[G] is g* (with g := L i e ( G ) ) , and the first Poisson group associated t o H = U ( g ) is of type G*, i.e. it has g as cotangent Lie bialgebra. If the ground field has positive characteristic, then the same recipes give similar results, but for the fact that the Poisson groups obtained have dimension 0 and height 1, and restricted universal enveloping algebras are obtained. We show how all these “geometrical” Hopf algebras are linked t o the original one via 1-parameter deformations, and explain how these results follow from quantum group theory.

“Yet these crystals are to Hopf algebras but as is the body to the Children of Rees: the house of its inner jire, that is within it and y e t in all parts of it, and is its life”

N . Barbecue, “Scholia”

* MSCZOOO: 16W30, 14L17, 16S30, 17B62-63, 81R50. Keywords: Hopf algebras, algebraic groups, Lie algebras. 223

224

1. Introduction Among all Hopf algebras over a field k, there are two special families which are of relevant interest for their geometrical meaning. The function algebras F[G]of algebraic groups G and the universal enveloping algebras U ( g ) of Lie algebras g, if Char (k) = 0 , or the restricted universal enveloping algebras u(g) of restricted Lie algebras g, if Char(k) > 0 ; to be short, we call both the latters “enveloping algebras” and denote them by U ( g ) , and similarly by “restricted Lie algebra” when Char@) = 0 we shall simply mean “Lie algebra”. Function algebras are exactly those Hopf algebras which are commutative, and enveloping algebras those which are connected, cocommutative and generated by their primitives. In this paper we give functorial recipes to get, out of any Hopf algebra, two pairs of Hopf algebras of geometrical type, namely one pair (F[G+],U(g-)) and a second pair ( F [ K + ] , U ( L ) )In . addition, the algebraic groups thus obtained are Poisson groups, and the (restricted) Lie algebras are (restricted) Lie bialgebras. Therefore, to each Hopf algebra, encoding a general notion of “symmetry”, we can associate in a functorial way some symmetries - “global” ones when taking an algebraic group, “infinitesimal” when considering a Lie algebra - of geometrical type, where the geometry involved is in fact Poisson geometry. Moreover, the groups concerned are always connected, and if Char(k) > 0 they have dimension 0 and height 1, which makes them pretty interesting from the point of view of arithmetic geometry (hence in number theory). The construction of the pair (G+,g-) uses pretty classical (as opposite to “quantum”) methods: in fact, it might be part of the content of any basic textbook on Hopf algebras (and, surprisingly enough, it is not!). Instead, to make out the pair (IT+,€‘-)one relies on the construction of the first pair, and make use of the theory of quantum groups. Let’s describe our results in some detail. Let J := K e r ( c H )be the augmentation ideal of H (where c H is the counit of H ) , and let J := { Jn},EN

G-J ( H ) the associated graded J” . One proves that J is a Hopf alge-

be the associated J-adic filtration, vector space and H” := H /

nnEN

:=

bra filtration, hence E? is a graded Hopf algebra: the latter happens to be connected, cocommutative and generated by its primitives, so H F U ( g - ) for some (restricted) Lie algebra g- ; in addition, since 2 is graded also g- itself is graded (as a restricted Lie algebra). The fact that i? be cocommutative allows to define on it a Poisson cobracket (from the natural A

225

Poisson cobracket V := A - AoP on H ) which makes fi into a graded co-Poisson Hopf algebra, and eventually this implies that g- is a Lie bialgebra. So the right-hand side half of the first pair of “Poisson geometrical” Hopf algebras is just g . On the other hand, one consider a second filtration - increasing, whereas J is decreasing - namely Q which is defined in a dual manner to J : for each n E N,let 6, the composition of the n-fold iterated coproduct followed by the projection onto J @ , (note that H = k.1, @ J ) ; then D := {D, := Ker(6,+l)}nEn. Let now 3 := GD ( H ) be the associated graded vector space and H‘ := UnEND, . Again, one shows that Q is a Hopf algebra filtration, hence 2 is a graded Hopf algebra: moreover, the latter is commutative, so 3 = F[G+] for some algebraic group G+. One proves also that fi = F[G+] has no non-trivial idempotents, thus G+ is connected; a deeper analysis shows that in the positive characteristic case G+ has dimension 0 and height 1; in addition, since 3 is graded, G, as a variety is just an affine space. The fact that 3 be commutative allows to define on it a Poisson bracket (from the natural Poisson bracket on H given by the commutator) which makes into a graded Poisson Hopf algebra: this means G+ is an algebraic Poisson group. So the left-hand side half of the first pair of “Poisson geometrical” Hopf algebras is just g . The relationship among H and the “geometrical” Hopf algebras i)and g can be expressed in terms of “reduction steps” and regular 1-parameter deformations, namely

where the “one-way” arrows are Hopf algebra morphisms and the “twoways” arrows are 1-parameter regular deformations of Hopf algebras, realized through the Rees Hopf algebras R b ( H ) and R > ( H V )associated to the filtration D of H and to the filtration-J of H V . The construction of the pair (K+,L ) uses quantum group theory, the basic ingredients being R b ( H ) and R > ( H V ) .In the present context, by quantum group we mean, loo&y speaking, a Hopf k[t]-algebra (t an indeterminate)

Ht such that either

(u)

I

Ht t H t

E

F[G] for some connected Poisson

group G - then we say Ht is a QFA - or (b) H t l t Ht U ( g ), for some restricted Lie bialgebra g - then we say Ht is a QrUEA. Formula (*)

I

says that Hl := R b ( H ) is a QFA, with Hl t Hi -

S

=

F[G+], and

226 A

also that HY := R > - ( H ) is a QrUEA, with H,V/t HY % H % U ( g - ) . Now, a general result - the “Global Quantum Duality Principle”, in short GQDP - teaches us how to construct from the QFA H; a QrUEA, call it (Hi)”, and how to build out of the QrUEA H y a QFA, say (H:)’; then ( H ; ) ” / t (Hi)”

(HY)’/t(HF)’

2

EU

( L ) for some (restricted) Lie bialgebra t-

, and

F[K+] for some connected Poisson group K + . This

provides the pair (K+,t-). The very construction implies that (Hi)” and (H: ) ’ yield another frame of regular 1-parameter deformations for H’ and H”, namely

which is the analogue of (*). In addition, when Char(k) = 0 the GQDP also claims that the two pairs (G+, g-) and (K+,t-) are related by Poisson duality: namely, t- is the cotangent Lie bialgebra to G+ , and g- is the cotangent Lie bialgebraof K+ (in short, we write t- = gx and K+ = G*_). Therefore the four ‘‘Poisson symmetries” G+, g- , K+ and t-, attached to H are actually encoded simply by the pair (G+, K+). In particular, when H‘ = H = H” from (*) and (a) together we find F[G+]

-

7 H’ 0-t-+l

1 t t - + O

U(L)

(HI)”

( =U ( g c ) if Char(k) = 0)

I1 H

U(g-)

O t t - 1

H,V

I1

H”

l-t-+O c-------)

(HY)’

( = F[G?]

F[K+]

if Char(k) = 0)

which gives four different regular 1-parameter deformations from H to Hopf algebras encoding geometrical objects of Poisson type (i.e. Lie bialgebras or Poisson algebraic groups). When the Hopf algebra H we start from is already of geometric type, the result involves Poisson duality. Namely, if Char(k) = 0 and H = F[G] then g- = g* (where g := Lie(G)), and if H = U ( g ) = U ( g ) then Lie(G+) = g*, i.e. G+ has g as cotangent Lie bialgebra. If instead Char (k) > 0 we have only a slight variation on this result. The construction of and needs only “half the notion” of a Hopf bialgebra: in fact, we construct A for any “augmented algebra” A (i.e.,

fi

A

227

roughly, an algebra with an augmentation, or counit, that is a character), and we construct for any %oaugmented coalgebra” C (i.e. a coalgebra with a coaugmentation, or “unit”, that is a coalgebra morphism from k to C). In particular this applies to bialgebras, for which both 3 and are (graded) Hopf algebras; we can also perform a second construction as above using (B;)’ and (By)’ (thanks to a stronger version of the GQDP), and get from these by specialization at t = 0 a second pair of bialgebras ((Bl)VI , (Bl)’l then again (B;)’I % U ( L ) for some restricted t=O

t=O

):

t=O

,

Lie bialgebra t- , but on the other hand (BY)’( is commutative and t=O with no non-trivial idempotents, but it’s not, in general, a Hopf algebra! Thus the spectrum of is an algebraic Poisson monoid, irreducible t=O as an algebraic variety, but it is not necessarily a Poisson group. It is worth stressing that everything in fact follows from the GQDP, which -in the stronger formulation - deals with augmented algebras and coaugmented coalgebras over 1-dimensional domain. All the content of this paper can in fact be obtained as a corollary of the GQDP as follows: pick any augmented algebra or coaugmented coalgebra over &, and take its scalar extension from k to k[t];the Iatter ring is a I-dimensional domain, hence we can apply the GQDP, and every result in the present paper will follow. This note is the written version of the author’s talk at the international workshop “Contemporary Geometry and Related Topics”, held in Belgrade in May 15-21, 2002. We dwell somewhat in detail upon the very constructions under study, but we skip proofs and other technicalities, which are postponed t o a forthcoming article (namely [ 51). Finally, a few words about the organization of the paper. In Section 2 we collect a bunch of definitions, and some standard, technical results. In Section 3 we introduce the “connecting functors” A H AV (on augmented algebras) and C w C’ (on coaugmented coalgebras), and the (associated) “crystal functors” A H A^ and C H ; we also explain the relationship between these two pairs of functors with respect t o Hopf duality. Section 4 cope with the effect of connecting and crystal functors on bialgebras and Hopf algebras. Section 5 considers the deformations provided by Rees modules, while Section 6 treats deformations arising from the previous ones via quantum group theory, introducing the “Drinfeld-like functors”. In Section 7 we look at function algebras and enveloping algebras, we collect all our results in the “Crystal Duality Principle”, and explain how this result can be also proved via quantum group theory.

(@‘)’I

228

2. Notation and terminology 2.1. Algebras, coalgebras, and the whole zoo. Let k be a field, which will stand fixed throughout. In this paper we deal with (unital associative) k-algebras and (counital coassociative) k-coalgebras in the standard sense, cf. [ 81 or [ 11; in particular we shall use notations as in [loc. cit.]. For any given (counital coassociative) k-coalgebra C we denote by coRad(C) its coradical, and by

1

G(C) := { c E C A(c) = c @ c } its set of group-like elements; we say C is monic if lG(C)I = 1; we say C is connected if coRad (C) is one-dimensional: of course “connected” implies “monic”. Any (unital associative) k-algebra A is said idempotent-free (in short i. p. -free) iff it has no non-trivial idempotents. We call augmented algebra the datum of a unital associative J!-algebra A together with a distinguished unital algebra morphism g : A k (so the unit u : J! A is a section of g): these form a category in the obvious way. We call indecomposable elements of an augmented algebra A the elements of the set Q ( A ) := J,/J: with J , := Ker ( g : . A k). We denote A+ the category of all augmented k-algebras. We call coaugmented coalgebra the datum of a counital coassociative k-coalgebra C together with a distinguished counital coalgebra morphism u : J! C (so E : C k is a section of g ) , and let 1 := ~ ( 1, ) a group-like element in C : these form a category in the obvious way. For such a C we call primitive the elements of the set

-

-

-

-

P ( C ) : ={ c E C ( A ( C ) = C @ ~ + ~ @ C ) We denote C+ the category of all coaugmented k-coalgebras. We denote B the category of all k-bialgebras; clearly each bialgebra B can be seen both as an augmented algebra, w.r.t. g = E = E* (the counit of B ) and as a coaugmented coalgebra, w.r.t. g = u = U, (the unit map of B ) , so that 1 = 1 = 1,: then Q ( B ) is naturally a Lie coalgebra and P ( B ) a Lie algebra over R . In the following we’ll do such an interpretation throughout, looking at objects of B as objects of A+ and of C+. We call ‘FIA the category of all Hopf k-algebras; this naturally identifies with a subcategory of B. We call Poisson algebra any (unitaL) commutative algebra A endowed with a Lie bracket { , } : A 8 A A (i.e., (A,{ , }) is a Lie algebra) such

-

229

that the Leibnitz identities {ab,c} = {a,c}b+a{b,c},

{a,bc} = {a,b}c+b{a,c}

hold (for all a, b, c E A ) . We call Poisson bialgebra, or Poisson Hopf algebra, any bialgebra, or Hopf algebra, say H , which is also a Poisson algebra (w.r.t. the same product) enjoying

A({.,b)) = @(a>, A(b)}

1

€({a1

b}) = 0 , S ( { a ,b } ) = { S ( b ) ,S b > }

for all a, b, c E H , the condition on the antipode S being required in the Hopf algebra case, where the (Poisson) bracket on H @ H is defined by

{ a @ b,c@ d} := {a,b} @ c d

+ a b @ {c,d}

(for all a, b, c, d E H ) . We call co-Poisson coalgebra any (counital) cocommutative coalgebra C with a Lie cobracket 5 : C C €9 C (i.e., (C, 5) is a Lie coalgebra) such that the co-leibnitz identity

-

(id €9 A)

O

c

(W) =

(a)

(+(l))

€9 a ( 2 )

-

+ 0 1 , 2 ( 9 l ) €9 5(.(2))))

holds for all a E C , where 0 1 , 2 :Cm3 Cm3is given by x1 €9 5 2 €9 x 3 H We call co-Poisson bialgebra, or co-Poisson Hopf algebra, any bialgebra, or Hopf algebra, say H , which is also a co-Poisson coalgebra (w.r.t. the same coproduct) enjoying

x2 @ x1 @ 23.

+

6(ab) = 6 ( a )A(b) A ( a )6(b), ( E @ € ) ( S ( a ) )= 0, b ( S ( a ) )= ( S @ S ) ( 6 ( a ) ) for all a, b E H , the condition on the antipode S being required in the Hopf algebra case. Finally, we call bi-Poisson bialgebra, or bi-Poisson Hopf algebra, any bialgebra, or Hopf algebra, say H , which is simultaneously a Poisson and co-Poisson bialgebra, or Hopf algebra, for some Poisson bracket and cobracket enjoying

( W a ,b } ) =

{w, 4) +( 4 4 7 W)}

for all a, b E H (see [ 21, [ 61 and references therein for further details on the above notions). A graded algebra is an algebra A which is Z-graded as a vector space and whose structure maps m and u are morphisms of degree zero in the category of graded vector spaces, where A @ A has the standard grading inherited from A and k has the trivial grading. Similarly we define the graded versions of coalgebras, bialgebras and Hopf algebras, and also the graded versions

230

of Poisson algebras, cePoisson coalgebras, Poisson/cePoisson/bi-Poisson bialgebras, and Poisson/co-Poisson/bi-Poisson Hopf algebras, but for the fact that the Poisson bracket, resp. cobracket, must be a morphism (of graded spaces) of degree -1, resp. +l. We write V = eZEzKfor the degree splitting of any graded vector space V.

2.2. Function algebras. According to standard theory, the category of commutative Hopf algebras is antiequivalent to the category of algebraic groups (over k): then we call Spec ( H ) (spectrum of H ) the image of a Hopf algebra H in this antiequivalence, and conversely we call function algebra or algebra of regular functions the preimage F[G]of an algebraic group G. Note that we do not require algebraic groups to be reduced (i.e. F[G]to have trivial nilradical) and we do not make any restrictions on dimensions: in particular we deal with pro-affine as well as affine algebraic groups. We say that G is connected if F[G]is i.p.-free; this is equivalent to the classical topological notion when dim(G) is finite. Given an algebraic group G, let JG := Ker(EF(Cl) be the augmentation ideal of F[G];the cotangent space of G (at its unity) is g x :=

JG/J2

= Q(F[G])

endowed with its weak topology; the tangent space of G (at its unity) is the topological dual g := (gx)* of g x : this is a Lie algebra, the tangent Lie algebra of G. If Char(k) = p > 0 , then g is a restricted Lie algebra (also called ‘‘FLie algebra”). We say that G is an algebraic Poisson group if F[G] is a Poisson Hopf algebra. Then the tangent Lie algebra g of G is a Lie bialgebra, and the same holds for gx . If Char (k) = p > 0 , then g and gx are restricted Lie bialgebras, the poperation on g x being trivial.

2.3. Enveloping algebras and symmetric algebras. Given a Lie algebra g, we denote by U ( g ) its universal enveloping algebra. If Char(k)= p > 0 and g is a restricted Lie algebra, we denote by u(g) = U ( g ) / ( { x p - &I x E g }) its restricted universal enveloping al-

I

gebra. If Char@) = 0 , then P ( U ( g ) ) = g . If instead Char@)

=p

> 0,

I n E N})

, the latter carrying a natural structure of restricted Lie algebra with X I P I := XP . Note then that

then P ( U ( g ) ) = g” := Span

({xp”

U ( g ) = u(goo) for any Lie algebra g , so any universal enveloping algebra can be thought of as a restricted universal enveloping algebra. Both U ( g ) and u(g) are cocommutative connected Hopf algebras, generated by

231

g itself. Conversely, if Char(k) = 0 then each cocommutative connected Hopf algebra is the universal enveloping algebra of some Lie algebra, and if Char@) = p > 0 then each cocommutative connected Hopf algebra H which is generated by P ( H ) is the restricted universal enveloping algebra of some restricted Lie algebra (cf. [7], Theorem 5.6.5, and references therein). Therefore, in order to unify the terminology and notation we call both universal enveloping algebras (when Char($) = 0 ) and restricted universal enveloping algebras “enveloping algebras”, and denote them by U ( g ) ;similarly, with a slight abuse of terminology we shall talk of “restricted Lie algebra” even when Char($) = 0 simply meaning “Lie algebra”. Thus enveloping algebras are simply the objects of the category of cocommutative connected Hopf algebras generated by their primitive elements, regardless of the characteristic of the ground field. If a cocommutative connected Hopf algebra generated by its primitive elements is also co-Poisson, then the restricted Lie algebra g such that H = U ( g ) is indeed a (restricted) Lie bialgebra. Conversely, if a (restricted) Lie algebra g is also a Lie bialgebra then U ( g )is a cocommutative connected co-Poisson Hopf algebra (cf. [ 21). Let V be a vector space: then the symmetric algebra S ( V ) has a natural structure of Hopf algebra, given by

A ( z ) = z @ 1 1 + 1 @ 1 z ,E

( Z ) = ~ ,

and

S(z)=-z

for all 2 E V . If g is a Lie algebra, then S(g) is also a Poisson Hopf algebra w.r.t. the Poisson bracket given by {z, y}s(E) = [z, y], for all z, y E g . If g is a Lie coalgebra, then S(g) is also a co-Poisson Hopf algebra w.r.t. the Poisson cobracket determined by 6S(#)(z) = 6,(z) for all z E g . Finally, if g is a Lie bialgebra, then S(g) is a bi-Poisson Hopf algebra with respect to the previous Poisson bracket and cobracket (cf. [ 61 and references therein). 2.4. Filtrations. Let

{Fz}zEZ =: F : ( ( 0 ) C ) ... C F-1 & Fo C FI & ... ( C V )

ezEZ

be a filtration of a vector space V. We denote by GF ( V ):= F,/F,-l the associated graded vector space. We say that is exhaustive if V F := UrEZ F, = V ; we say it is separating if V, := F, = ( 0 ) . We say that a filtered vector space is exhausted if the filtration is exhaustive; we say that it is separated if the filtration is separating. A filtration = { F Z } z E Zin an algebra A is said to be an algebra filtration iff

nzEZ

m(Fe @ Fm)

F[+,

for all l , m, TI E

Z.

232

Similarly, a filtration gebra filtration iff

A(F,)

F

=

{F,},Ea in a coalgebra C is said to be a coal-

G C,.+,,,Fr

@ F,

for all z E Z.

Finally, a filtration F = { F t } z E Zin a bialgebra, or in a Hopf algebra, H is said to be a bialgebra filtration, or a Hopf (algebra) filtration, iff it is both an algebra and a coalgebra filtration and - in the Hopf case - in addition S(F,) C F, for all z E Z.The notions of exhausted and separated for filtered algebras, coalgebras, bialgebras and Hopf algebras are defined like for vector spaces with respect t o the proper type of filtrations.

Lemma 2.5. Let F be a n algebra filtration of a n algebra A . T h e n GF -( A ) i s a graded algebra; if, in addition, it i s commutative, then it i s a commutative graded Poisson algebra. If E i s another algebra with algebra filtration 9 and q5 : A E is a morphism of algebras such that q5(F,) C @, f o r all z E Z , then the morphism G(q5): GF ( A ) Gg, ( E ) associated to 4 i s a morphism of graded algebras. In addition, if GF( A ) and GG - ( E ) are Commutative, then G ( $ ) is a morphism of graded commutative Poisson algebras. T h e analogous statement holds replacing “algebra” with “coalgebra”, “commutative” with “cocommutative” and “Poisson” with “co-Poisson”. I n addition, i f we start f r o m bialgebras, or Hopf algebras, with bialgebra filtrations, or Hopf filtrations, then we end up with graded commutative Poisson bialgebras, or Poisson Hopf algebras, and graded cocommutatiue co-Poisson bialgebras, o r co-Poisson Hopf algebras, respectively.

-

-

Sketch of proof. The only non-trivial part is about the Poisson structure on C;c(A) and the co-Poisson structure on GF ( C ) (for a coalgebra C). Indeed, let F := { F Z } z E Zbe an algebra filtration of A . For any F E F , / F , - ~ , jj~Fc-Fc-1 ( z , [ ~ Z ) , l e tX E F , , resp. Y E F ~ b,e a lift of ?, resp. of g : then [z, y] := (zy - yz) E F,+c-l because GF ( A ) is commutative; therefore we define

-

{SIT-/}:=

[Xc,Y1

= [%Yl

modFz+c-2 E

Fz+c-1/Fz+c-2.

This gives a Poisson bracket on GF - ( A ) making it into a graded commutative Poisson algebra. Similarly] if F := { F Z } z E Zis a coalgebra filtration of C , for any T E F,/F,-1 ( z E Z ) , let z E F, be a lift of 3 : then (C) is cocommuV(5) := ( A ( z )- Aop(z)) E C r + s = z - l Fr @ F, for GF tative, so the formula S(3)

:=

V(z)

V(z) mod

C

r+s=z-1

FT@Fs E

C T+S=Z

(Fr/Fr-l) @ (Fs/Fs-l)

233

defines a structure of graded cocommutative cc-Poisson coalgebra onto GF 0

(c).

Lemma 2.6. Let G be a coalgebra. If := { F z } z E Z is a coalgebra filtration, then CE := U F, i s a coalgebra, which injects into C , and ZEZ

is a coalgebra, which C surjects onto. The same holds for algebras,- bialgebras, Hopf algebras, with algebra, bialgebra, Hopf algebra filtrations respectively. 0

3. Connecting functors on (co)augmented (co)algebras 3.1. The -filtration J on augmented algebras. Let A be an augmented algebra (cf. Subsection 2.1). Let J := K e r ( g ) : then J := { J P n := Jn}nEN is clearly an algebra filtration of A, called the g-filtration of A ; hereafter we shall consider it as a Z-indexed filtration, by trivially completing it ( J , := A V z E -N+). We say that A is €-separated if J is separating, i.e. J" := J" = (0). Next (trivial) lemma points out some properties of J :

nnEA

Lemma 3.2. (a) J is a n algebra filtration of A, which contains the radical filtration of A, that is J" 2 Rad(A)" for all n E N where R a d ( A ) is the (Jacobson) radical of A . (b) If A is g-separated, then it is i.p.-free.

(c) A'

:= A/(-),,,

J" is a quotient augmented algebra of A, which

0

is 5-separated. Now we come to the first, somewhat relevant result:

~ln,,,

Proposition 3.3. Mapping A H AV := J" gives a welldefined functor f r o m the category of augmented algebras to the subcategory of g-separated augmented algebras. Also, the augmented algebras A of the latter subcategory are characterized by A' = A . 0 Remark 3.4. It is worth mentioning a special example of g-separated augmented algebras, namely the graded ones (i.e. the augmented algebras

234

A with an algebra grading such that g, is a morphism of graded algebras w.r.t. the trivial grading on the ground field k), which are also connected (i.e. their zero-degree subspace is the k-span of 1, ). Then one easily proves Every graded connected augmented algebra A is €-separated, or equivalently A = A" . 3.5. Drinfeld's S.-maps. Subsection 2.1). For every n E

Let C be a coaugmented coalgebra (cf. -+ H@" by

N, define An: H

Ao := E , A' := id, , and An := ( A @

o A"-'

-

For any ordered subset @ = (21,. . . ,z k } G (1,. . . , n } with il define the linear map j , : H B k H@'" by

if n 2 2 .

< ... < i k ,

j,(al @ . . . @ a k ) := bl @ . . . @ b, with bi := 1 if i $ @ and bi,,, := a, for 1 5 m .< k ; then set

A, := j , o Ak , A0 := Ao and 6a :=

xQca ( - l ) n - ' Q ' A ~,

60 := E .

By the inclusion-exclusion principle, this definition admits the inverse formula A, = 6 9 . We shall also make use of the shorthand notation 60 := 6 0 , 6, := 6{1,2,...,,I . Next lemma is a trivial, technical one:

x*=,

Lemma 3.6. Let a , b E C . Then (u) 6, = ( i d , - g o c ) B n o A n (b) The maps 6, (and similarly the coassociative, i. e.

forall n E N + ; 's, for all finite @ C

) 0 6n = 6n+e-1 for all n,e,s E N, 0 5 s 5 n - 1 .

(id:"

@ Se @

N) are

7

0

3.7. The d.-filtration 0 on coaugmented coalgebras. Let C be as above, and take notations of Subsection 3.5. For all n E N , let D, := Ker(6,+1) : then D := { Dn}ncN is clearly an ascending filtration of C , the b.-filtration of C ; hereafter we shall consider it as a %-indexed filtration, by trivially completing it ( D , := (0) V z E -N+). We say that C is &.-exhausted if D is exhaustive, i.e. UnENDn= C . Next lemma highlights some properties of 0, in particular it shows that it is a refinement of the coradical filtration. We use the notion of "wedge" product, namely X A Y := A-l (C @ Y X @ C) for all subspaces X, Y of C , with AIX:= X and X := (A" X) A X for all n E N+ .

An+'

+

235

Lemma 3.8. (a) D o = k . I ,

D,

and

= A-l(C@Dn-l+Do@C) = /\,+'Do

for all n E N.

(b) Q is a coalgebra filtration of C, which is contained in the coradis the ical filtration of C , that is 0,& C, i f C := {Cn}nGN coradical filtration of C . C is connected D = C. ( c ) C is 6.-exhausted (d) C' := UnEND, is a subcoalgebra of C : more precisely, it is the irreducible (hence connected) component of C containing I. 0 Here is the second relevant result, the (dual) analogue of Proposition 3.3:

Proposition 3.9. Mapping C H C' := UnEND, gives a well-defined functor from the category of coaugmented coalgebras to the subcategory of 6. -exhausted (=connected) coaugmented coalgebras. Moreover the coaugmented coalgebras C of the latter subcategory are characterized by C'= C.0 Remark 3.10. the characterization of connected coalgebras in terms of the S.-filtration given in Lemma 3.8(c) yields easy, alternative proofs of two well-known facts. (a) Every graded coaugmented coalgebra C i s connected; (b) Every connected coaugmented coalgebra is monic. 3.11. Connecting functors and Hopf duality. Let's start from A E ,A+, with J := K e r ( g ) : the n-th piece of its g-filtration is

J"

Im(JBn-HBn-

Pn

H)

where the left hand side arrow is the natural embedding induced by J H and p n is the n-fold iterated multiplication of H . Similarly, let C E C+: the s-th piece :=

-

A'+l

1

of its 6.-filtration is D, := Ker HH@("+l) J@("+') where the right hand side arrow is the natural projection induced by (id, - 2 o E ) : H J and As+1 is the ( s 1)-fold iterated comultiplication of C. In categorial terms, Ker is dual to Im, the iterated comultiplication is dual to the iterated multiplication, and the above embedding JBT L_) H@'" is dual to the projection H B T --+JBT (for r E N ) . Therefore

-

(

+

(a) The notions of g-filtration and of 6.-filtration other;

are dual to each

236

nnEN

(b) The notions of AV := A / J” and of C’ := UnEW D, are dual to each other; (c) The notions of g-separated (for an augmented algebra) and 6.exhausted (for a coaugmented coalgebra) are dual to each other.

Remark 3.12. (u) ( )” and ( )’ as ‘%onnecting”functors. We now explain in what sense both AV and C’ are ‘‘connected” objects. Indeed, C‘ is truly connected, in the sense of coalgebra theory (cf. Lemma 3.8(d)). On the other hand, we might expect that AV be (or correspond to) a “connected” object by duality. In fact, when A is commutative then it is the algebra of regular functions F [ V ]of an algebraic variety Y ; the augmentation on A is a character, hence corresponds to the choice of a point POE V : thus the augmented algebra A does correspond to the pointed variety ( V ,PO). Then AV = F[Vo] where Yo is the connected component of V containing PO! This follows at once because AV is i.p.-free (cf. Lemma 3.2). More in general, for any augmented algebra A , if AV is commutative then its spectrum is a connected algebraic variety. For these reasons, we shall use the name “Connecting functors” for both functors A H AV and C H C’ . (b) Asymmetry of connecting functors on bialgebras. Let B be a bialgebra. As the notion of BV is dual to that of B’, and since B = B‘ implies that B is a Hopf algebra (cf. Corollary 4.4(b)), one might dually conjecture that B = BV imply that B is a Hopf algebra. Actually, this is false, the bialgebra B := F[Mut(n,L)] yielding a counterexample: F[Mut(n,k)] = F[Mut(n,L)IV,and yet F [ M u t ( n , k ) ]is not a Hopf algebra. (c) Hopf duality and augmented pairings. The most precise description of the relationship between connecting functors of the two types uses the notion of “augmented pairing” :

-

Definition 3.13. Let A E A+, C E C+ . We call augmented pairing between A and C any bilinear mapping ( , ) : A x C Jk such that, for all I C , Z ~ , Q E A and y E C , (El

. 2 2 , Y) = (21 c3 2 2 , A(Y)) := C(&l,Y(I)). ( 2 2 , Y(2))

(LY) =

1

dz).

€(?I), (Ell) =

For any B , P E I3 we call bialgebra pairing between B and P any augmented pairing between the latters (thought of as augmented algebras and coaugmented coalgebras as explained in Subsection 2.1) such that, for all

237

z E B , y1,y2 E

P , we have also, symmetrically,

For any H , K E 'HA we call Hopf algebra pairing (or Hopf pairing) between H and K any bialgebra pairing such that, in addition, (S(z),y) = (z,S(y)) for all z EH, y E K . We say that a pairing as above is perfect on the left (right) if its left (right) kernel is trivial; we say it is perfect if it is both left and right perfect.

-

Theorem 3.14. Let A E A+, C E C+ and let IT: A x C k be an augmented pairing. Then IT induce a filtered augmented pairing 7rf : A" x C' k and a graded augmented pairing 7rG : G-j ( A ) x GQ(C) k (notation of Subsection 2 4 ) , both perfect on the right. If in addition 7r is perfect then 7rf and 7rG are perfect as well. 0

-

-

4. Connecting and crystal functors on bialgebras and Hopf algebras

4.1. The program. Our purpose in this section is to see the effect of connecting functors on the categories of bialgebras and of Hopf algebras. Then we shall move one step further, and look at the graded objects ass* ciated to the filtrations J and D in a bialgebra: these will eventually lead to the "crystal functors" , the main achievement of this section. From now on, every bialgebra B will be considered as a coaugmented coalgebra w.r.t. its unit map, hence w.r.t the group-like element 1 (the unit of B ), and the corresponding maps 6, ( n E N) and &-filtration D will be taken into account. Similarly, B will be considered as an augmented algebra w.r.t. the distinguished algebra morphism f = E (the counit of B ) , and the corresponding -filtration (also called €-filtration) J will be considered. We begin with a technical result about the "multiplicative" properties of the maps 6,. Lemma 4.2. ([6], Lemma 3.2) Let B E t3, a , b E B , and finite. Then (a) b+(ab) = C A " Y = + ~ A ( ~ ) ~ ;Y ( ~ ) (b) if Q, # 8 , then

@GN, with Q,

238

Lemma 4.3. Let B be a bialgebra. T h e n J and trations. If B i s also a Hopf algebra, then J and filtrations. Corollary 4.4. (a)

D D

are bialgebra filare Hopf algebra

0

Let B be a bialgebra. T h e n

I

B" := B n n E N J ni s a n €-separated (i.p.-free) bialgebra, which B surjects onto.

(b) B' := UnENOn i s a 6,-exhausted (connected) Hopf algebra, which injects into B : more precisely, it i s the irreducible (actually, connected) component of B containing 1. (c) If in addition B = H i s a Hopf algebra, then HV i s a Hopf algebra quotient of H and HI i s a Hopf subalgebra of H . 0

Theorem 4.5. respectively. (a)

Let B be a bialgebra, J , D its €-filtration and 6, -filtration

B^

:= G J ( B ) i s a graded cocommutatzve co-Poisson Hopf algebra generated by P(GJ - (B)) , the set of its primitive elements. Therefore B^ U ( 0 ) as graded co-Poisson Hopf algebras, f o r some restricted Lie bialgebra 0 which i s graded as a Lie algebra. In particular, i f Char@) = 0 and dim(B) E N then = k.1 and

s

0 = (01.

(b)

g

:= Go ( B ) i s a graded commutative Poisson Hopf algebra. Therefore, 2 F[G] for some connected algebraic Poisson group G which, as a variety, i s a (pro)afine space. If Char(k) = 0 then g E F [ G ] i s a polynomial algebra, i.e. F [ G ] = L[{x~}~,,] (for some set 2);in particular, i f Char (k) = 0 and dim(B) E N then B = k.1 and G = (1). If Char (k)= p > 0 then G has dimension 0 and height 1, and i f k is perfect then g 2 F[G] i s a truncated

-

polynomial algebra, i.e. F[G]= k [ { ~ i }/ (~{ ~z /~} ]~ ~ (for ~ ) some

0

set 1).

4.6. The crystal functors. It is clear from the very construction that mapping A H A := G-j ( A ) (for all A E A + ) defines a functor from A+ to the category of graded, augmented %-algebras; similarly, mapping A

239

C H 2; := GD(C) (for all C E C+) defines a functor from Cf to the category of graded, coaugmented k-coalgebras. The first functor factors through the functor A H A’, and the second through the functor

CHC‘. The analysis in the present section shows that when restricted to kbialgebras the output of the previous functors are objects of Poissongeometric type (Lie bialgebras and Poisson groups): therefore, the functors BH and B H (for B E B ) on k-bialgebras are “geometrification functors”, in that they sort out of the generalized symmetry encoded by B some geometrical symmetries; we’ll show in Section 5 that each of them can be seen as a “crystallization process” (in the sense, loosely speaking, of Kashiwara’s motivation for the term “crystal basis” in quantum group theory: we move from one fiber to another, very peculiar one, within a 1-para.meter family of algebraic objects), so we call them ‘krystal functors”. It is worth stressing that, by their very construction, applying either crystal functor one looses some information about the initial object, while at the same time still saving something. So B^ tells nothing about the coalgebra structure of B” (for all enveloping algebras - like B^ - roughly look the same from the coalgebra point of view), yet it grasps some information on its algebra structure; on the other hand, conversely, 5 gives no information about the algebra structure of B’ (in that the latter is simply a polynomial algebra), but instead it tells something non-trivial about its coalgebra structure. We finish this section with the natural improvement of Theorem 3.14 :

- -

Theorem 4.7. Let B , P E B and let 7r : B x P k be a bialgebm pairing. Then 7r induces filtered bialgebm pairings 7rf : BV x P’ k, 7rf : B‘ x Pv k, and graded bialgebra pairings 7rG : B^ x k, x k; 7 r f and 7rG are perfect o n the right, 7rf and 7rG o n the 7rG : left. If in addition 7r is perfect then all these induced pairings are perfect as well. If in particular B , P E ‘HA are Hopf algebras and 7r is a Hopf algebra pairing, then all the induced pairings are (filtered or graded) Hopf algebra pairings. 0

--

240

5. Deformations I - Rees algebras, Rees coalgebras, etc.

5.1. Filtrations and “Rees objects”. Let V be a vector space over k, and let {F,},,, := : ((0)

) ... c F-, & . * -

F-1 C Fo C_ F1

c ... C F,

C_

... ( 5 V )

be a filtration of V by vector subspaces F, ( z E Z).First we define the associated blowing space to be the k-subspace BF(V) of V [ t , t - l ](where t is any indeterminate) given by BF(V) := CzEZ tZF, ; this is isomorphic to the first graded module” associated to ( V , F ) ,i.e. F, . Second, we let the associated Rees module be the k[t]-submodule R&(V) of V [t,t-l] ; easy computations give k-vector space isomorphisms generated by BF(V) -

ezEZ

R> ( V ) / ( t- l)R&(V)

E

u

F,

=:

V F , R> ( V ) / t R k ( V ) ”= GF(V)

ZEZ

ezEZ

where GF(V):= F,/F,-1 is the second graded module associated to ( V , F ) . In other words, R & ( V )is a k[t]-module which specializes to UzEZF, for t = 1 and specializes to G-F ( V )for t = 0 ; therefore the k-vector spaces UzEZF, and GF(V) can be seen as 1-parameter (polynomial) deformations of each other via the 1-parameter family of k-vector spaces given by R& -( V ), in short

We can repeat this construction within the category of algebras, coalgebras, bialgebras or Hopf algebras over k with a filtration in the proper sense (by subalgebras, subcoalgebras, etc.): then we’ll end up with corresponding objects B,(V), R&(V),etc. of the like type (algebras, coalgebras, etc.). In particular we’ll cope with Rees bialgebras.

5.2. Connecting functors and Rees modules. Let A E A+ be an augmented algebra. By Lemma 3.2 the €-filtration J of A is an algebra filtration: therefore we can build out of it the associated Rees algebra %?>(A). By the previous analysis, this yields a 1-parameter deformation -

I pick the terminology about (associated) graded modules from Serge Lang’s textbook “Algebra”.

a

241

LC I

where hereafter we use notation M := M (t - c)M for any k[t]-module M and any c E k . Note that all fibers in this deformation are isomorphic as vector spaces but perhaps for the special fiber at t = 0 , i.e. exactly A , for at that fiber the subspace Jmis "shrunk t o zero". This is settled passing from A to A', for which we do have a regular 1-parameter deformation, i.e. one in which all fibers are pairwise isomorphic (as vector spaces), namely A

where we implicitly used the identities h

h

AV = R",(AV)l t=O = R>(A)I t=O

=

A.

The situation is (dually!) similar for the connecting functor on coaugmented coalgebras. Indeed, let C E Ct be a coaugmented coalgebra. Then by Lemma 3.8 the d,-filtration Q of C is a coalgebra filtration, thus we can build out of it the associated Rees coalgebra Rb(C). By the previous analysis, the latter provides a 1-parameter deformation

C':= u D n =R&(C)I nEN

t=l

l c t - 0

Rb(C) -

Rh(C)I -

t=O

E

-

G D ( C )= : C . -

(5.2)

Note that all fibers in this deformation are pairwise isomorphic vector spaces, so this is a regular 1-parameter deformation.

5.3. The bialgebra and Hopf algebra case. We now consider a bialgebra B E 0. In this case, the results of Section 5 ensure that BV is a bialgebra, B^ is a (graded, etc.) Hopf algebra, and also that R;(BV) is a k[t]-bialgebra (because J is a bialgebra filtration). Therefore,-using Theorem 4.5,formula (5.1) becomes

for some restricted Lie bialgebra g- as was g in Theorem 4.5 (a) (for later purposes we need to change symbol). Similarly, by Corollary 4.4 we know that B' is a Hopf algebra, g is a (graded, etc.) Hopf algebra and that Rb(B')is a Hopf k[t]-algebra (because D(B') is a Hopf algebra filtration of%'): thus, again by Theorem 4.5, formula (5.2) becomes (5.4)

242

(noting that Rb(B’)= Rb(B) because D(B’) = D ( B ) )for some Poisson algebraic group-G+ as was G in Theorem 4.5 (b). “Splicing together” these two pictures one gets the following scheme:

This drawing shows how the bialgebra B gives rise t o two Hopf algebras of Poisson geometrical type, namely F[G+]on the left-hand side and U ( g - ) on the right-hand side, through bialgebra morphisms and regular bialgebra deformations. Namely, in both cases one has first a “reduction step”, i.e. B H B’ or B H B V ,(yielding “connected” objects, cf. Remark 3.12 (a)), then a regular 1-parameter deformation via Rees bialgebras. Finally, if H E W d is a Hopf algebra then all objects in (5.5) are Hopf algebras too, i.e. also HV - over k - and R > ( B V) over k [ t ]. Therefore (5.5) reads

with the one-way arrows being morphisms of Hopf algebras and the twoways arrows being 1-parameter regular deformations of Hopf algebras. In the special case when H is connected, i.e. H = H I , and “coconnected”, that is H = H V , the scheme (5.6) looks simply

which means we can (regularly) deform H itself to “Poisson geometrical” Hopf algebras.

Remark 5.4. (a) There is no simple relationship, a priori, between the Poisson group G+ and the Lie bialgebra g- in (5.5) or (5.6), or even (5.7): examples do show that; in particular, either G+ or g- may be trivial while the other is not. (b) The Hopf duality relationship between connecting functors of the two types explained in Subsections 3.11 - 3.14, extend to the deformations built upon them by means of Rees modules. Indeed, by the very constructions one sees that there is a neat category-theoretical duality between the definition of R>(A)and of R & ( C )(for A E A+ and C E C + ) . Even more, Theorem 3.14 “extends” ( i n a sense) to the following result:

-

k be Theorem 5.5. Let A E A+, C E C+ and let 7r: A x C a n augmented pairing. T h e n the pairing 7r induces a n augmented pairing

:~R > ( A )x

-{

Rb(C) -

k[t] which is perfect o n the right. If in addition

7

r

T

i s perfect then R;(A) =

Rb(C)=

243

q E A(t)

1

7rt

}

(q,K ) E k[t], V K E Rb(C) and

{ K E C ( t )l ~ t ( q , E~ k) [ t ] ,V q E R L ( A ) } ,

-

where we denote S @ := k ( t )@k S f o r S E { A , C } and where 7rt : A(t) x C ( t ) k ( t ) is the obvious k(t)-linear pairing induced by 7 r , and r r is ~ perfect as well. If A and C are bialgebras, resp. Hopf algebras, then everything holds with bialgebra, resp. Hopf algebra, pairings instead of augmented pairings. 0

6. Deformations I1 - from Rees bialgebras t o quantum groups 6.1. From Rees bialgebras to quantum groups via the GQDP. In this section we show how, for any k-bialgebra B , we can get another deformation scheme like (5.5). In fact, this will be built upon the latter, applying (part of) the "Global Quantum Duality Principle" explained in [ 3, 41, in its stronger version about bialgebras. Indeed, the deformations in (5.5) were realized through Rees bialgebras, namely R>( B ) and RL ( B ) : these are torsion-free (actually, free) as k[t]-modui&, hence o n e c a n apply the construction made in [loc. cit.] via the secalled Drinfeld's functors to get some new torsion-free k[t]-bialgebras. The latters (just like the Rees bialgebras we start from) again specialize to special bialgebras at t = 0 ; in particular, if B is a Hopf algebra the new bialgebras are Hopf algebras too, and precisely "quantum groups" in the sense of [loc. cit.]. The construction goes as follows. To begin with, set BY := R i ( B ): this is a free, hence torsion-free, k[t]-bialgebra. We define

(BY)'

:=

{ b E BY [ 6,(b)

E t " ( B y ) @ nV, n E N}

.

-

On the other hand, let B6 := R L ( B ): this is again a free, hence torsionfree, k[t]-bialgebra. Using notation J' := Ker ( E : BL k[t]) and also B'(t) := k ( t ) B; = k ( t )@k B' , we define

(BI)"

:=

C t-"(J')"

"20

=

C (t-'J')n n20

( C: B ' ( t ) ).

The first important results about (BY)' and ( B l ) vare the following:

244

Proposition 6.2. B o t h (BY)’ and (B;)’ are free (hence torsion-free) k [ t ] bialgebras; moreover, the mappings B H (BY)’ and B H (B;)’ are functorial. The analogous results hold for Hopf k-algebras, replacing “bialgebra(s)” with “Hopf algebra(s)” throughout. 0 Proposition 6.3. ( a ) (BY)’lt=l := (B,V)y(t- l)(BY)’ e BV as k-bialgebras.

0

(b) (BI)VI := (B;)’/(t - l)(B:)V2 B’ as k-bialgebras. t=l The key fact then is next result:

Theorem 6.4. ( a ) (BY)’lt=o := (BY)’/t (BY)’ i s a commutative, i.p.-free k-bi-

-

algebra. Furthermore, if p := Chax(k) > 0 then each non-zero element of Ker ( 6 : (BY)’l k) has nilpotency order p . t=O i s the function algebra F [ M ] of some conTherefore (BY)’I nected Poisson a & % t z i c monoid M , and if Char@) > 0 then M has dimension 0 and height 1. If in addition B = H E lid then (H:)’l t = O i s a Hopf k-algebra, and K+ := Spec((HY)’l t = O = M is a (connected) algebraic Poisson group.

)

(b) (B$)vlt=o := (B;)’/t (B$V i s a connected cocommutative Hopf

’)“Lo>.

k-algebra generated by P Bt

((

Therefore (B;)’I (c)

t=O

= U(t-) f o r some Lie bialgebra t-

.

If Char@) =

0 and B = H E lid is a Hopf k-algebra, let H = U ( g - ) and fi = F[G+]as in Theorem 4.5: then (notation of ( a ) and (b)) K+ = G*_ and t- = g; , that i s coLie ( K + ) = gA

0

and t- = coLie(G+) as Lie bialgebras.

6.5. Deformations through Drinfeld’s functors. The outcome of the previous analysis is that for each k-bialgebra B E B a second scheme - besides (5.5) - is available, yielding regular 1-parameter deformations, namely (letting p := Char (k)) O c t + l (Bi)V

B’-B--+BV<

1-t-0

(BY )’

F[M].

(6.1)

245

This provides another recipe, besides (5.5), to make two other bialgebras of Poisson geometrical type, namely F[M] and U(t-), out of the bialgebra B , through bialgebra morphisms and regular bialgebra deformations. Like for (5.5), in both cases there is first the "reduction step" B H B' or B H B" and then a regular 1-parameter deformation via k[t]-bialgebras. However, this time on the right-hand side we have in general only a bialgebra, not a Hopf algebra. When B = H E Ed is a Hopf k-algebra, then (6.1) improves, in that all objects therein are Hopf algebras too, and morphisms and deformations are ones of Hopf algebras. In particular M = K , is a (connected Poisson algebraic) group, not only a monoid: at a glance, letting p := Char(k) 2 0 , we have

This yields another recipe, besides (5.7), to make two new Hopf algebras of Poisson geometrical type, i.e. F[K+]and U(t->, out of the Hopf algebra H , through Hopf algebra morphisms and regular Hopf algebra deformations. Again we have first the "reduction step" H H H' or H t+ H", then a regular 1-parameter deformation via Hopf k[t]-algebras. In the special case when H is connected, that is H = H I , and %oconnected", that is H = H", the scheme (6.2) takes the simpler form, the analogue of (5.7),

which means we can (regularly) deform H itself to Poisson geometrical Hopf algebras. In particular, when H' = H = HV patching together (5.7) and (6.3) we find

U(g-)

O t t + l

HY

I1

HV

l e t + O t------f

(HY)'

F[K+]

( =F[G*_]if

Char(k) = 0 )

246

which gives four different regular 1-parameter deformations from H to Hopf algebras encoding geometrical objects of Poisson type (i.e. Lie bialgebras or Poisson algebraic groups).

6.6. Drinfeld-like functors. The constructions in the present secV tion show that mapping B H (Bi) and mapping B H (for all B E B ) define two endofunctors of B. The output of these endofunctors describe objects of Poisson-geometric type, namely Lie bialgebras V and connected Poisson algebraic monoids: therefore, both B H (BI) and B H (BY)’lt=o (for B E B ) are geometrification functors on k-bialgebras, just like the ones in Subsection 4.6, which we call “Drinfeldlike functors”, because they are defined through the use of Drinfeld functors (cf. [ 3 , 4 ] )for quantum groups. Thus we have four functorial recipes - our four geometrification functors -to sort out of the generalized symmetry B some geometrical symmetries. Hereafter we explain the duality relationship between Drinfeld-like functors and the associated deformations:

It=o

It=O

-

It=O -

Theorem 6.7. Let B , P E 8 , and let 7r : B x P k be a k-bialgebra pairing. Then 7r induces a k[t]-bialgebra pairing 7rL : (BY)’ x (P,’)” V k[t] and a k-bialgebra pairing 7rLIt=o : (BY)’(,=,x (P,’) k. If 7r is perfect and Char@) = 0 , then the induced pairings are perfect as well, and

and

-

where S ( t ) := k ( t ) @k S for S E { B ,P } and 7rt : B ( t ) x P ( t ) k(t) is the obvious k(t)-linear pairing induced b y 7r . If B , P E ‘HA are Hopf k-algebras then everything changes accordingly. 0

7. Poisson duality and the Crystal Duality Principle 7.1. Crystal functors and Poisson duality. In this section we show that when connecting functors are applied to Hopf algebras encoding a classical symmetry - an algebraic (maybe Poisson) group, or a universal enveloping algebra of a Lie (bi)algebra (maybe restricted, if Char@) > 0 )

247

- we know in advance the result of applying some connecting or crystal

functors. Namely, in case of function algebras of algebraic Poisson groups or (restricted) universal enveloping algebras of Lie bialgebras the outcome is explicitly expressed in terms of the dual Lie bialgebra or Poisson group. These cases are general, as we can always give an algebraic group the trivial Poisson group structure, and any (restricted) Lie algebra the trivial Lie cobracket to make it into a Lie bialgebra.

-

Theorem 7.2. Let H = F[G] be the function algebra of a n algebraic Poisson group. T h e n F[G] i s isomorphic t o a bi-Poisson Hopf algebra, namely (with p := Char@)) ifp=O

then F[G] 2 S(gx) and

if p > 0 then F[G] 2 S(gx)/({

?fpn(s)

lz EN(F[G])}),

(notation of Section 2) where N ( F [ G ] )i s the nilradical of F [ G] ,p"(") is the order of nilpotency of x E N ( F [ G ] )and the bi-Poisson Hopf structure of S(gx)/(

{

:pn(s)

1

z E N ( F [ G ] ) } ) i s the quotient one f r o m S(gx) .

In

particular, i f G is reduced then F[G]=" S(gx) . Sketch of proof. This follows essentially from definitions of F[G]and of gx (cf. Section 2). 0

Theorem 7.3. ( a ) Let Char@) = 0 . Let g be a Lie bialgebra. T h e n U ( g ) i s a biPoisson Hopf algebra, namely U ( g ) 2 S(g) = F [ g X ] (notation of Section 2), where the bi-Poisson Hopf structure o n S(g) i s the canonical one. (b) Let Char@) = p > 0 . Let g be a restricted Lie bialgebra. T h e n u(g) is a bi-Poisson Hopf algebra, namely

-

h_

-

ux)

S(g)/({xP

I z E g})

=

F[G*] (notation of Section 2)

1

where the bi-Poisson Hopf structure o n S ( g ) / ( { z P z E g}) is induced by the canonical one o n S(g) and G* denotes a connected algebraic Poisson group of dimension 0 and height 1 whose cotangent Lie bialgebra is g . Sketch of proof. By its very definition, the filtration D of U ( g ) or u(g) is just the natural filtration given by the order of differential operators. From

248

0

this the claim follows,

7.4. The Crystal Duality Principle. To sum up, we can finally tide up - once more - all results presented above in a single formulation, the “Crystal Duality Principle”. In short, we provided functorial recipes to get, out of any Hopf algebra H , four Hopf algebras of Poisson-geometrical type (arranged in two couples), hence four associated Poisson-geometrical symmetries: this is the “Principle”, say. The word “Crystal” reminds the fact that the first couple - out of which the second one is sorted too - of special Hopf algebras, namely ( H ,H ) , is obt,ained via a crystallization process (cf. Section 4.6). Finally, the word “Duality” witnesses that if Char(%) = 0 then Poisson duality is the link between the two couples of special Hopf algebras (thus only two are the relevant Poisson geometries associated to H ) and that if H is of Poisson-geometrical type then the crystal functor yielding a Hopf algebra of Hopf-dual type is ruled by Poisson duality (in any characteristic).

--

7.5. The CDP as corollary of the GQDP. The construction of Drinfeld-like functors passes through the application of the Global Quantum Duality Principle (=GQDP in the sequel): thus part of the Crystal Duality Principle (=CDP in the sequel) is a direct consequence of the GQDP. In this section we briefly outline how the whole CDP can be obtained as a corollary of the GQDP (but for some minor details); see also [ 3, 41, 56. For any H E Ed, let Ht := H [ t ]= %[t] @k H . Then Ht is a torsionless Hopf algebra over k[t], hence one of those to which the constructions in [ 3, 41 can be applied: in particular, we can act on it with Drinfeld’s functors considered therein, which provide quantum groups, namely a quantized (restricted) universal enveloping algebra (=QrUEA) and a quantized function algebra (=QFA). Now, straightforward computation shows that the QrUEA - ( H ) , with is nothing but HY := R > - ( H ) , and the QFA is just Hl := R & A

H

E

H,“I

and

E

t=O

Hi1

t=O

. It follows that all properties of

and

fi spring out as special cases of the results proved in [ 3, 41 for Drinfeld’s functors, but for their being graded. Similarly, the fact that H’ be a Hopf subalgebra of H follows from the fact that Hi itself is a Hopf algebra (over %[t]) and H’

= HI1

t=l

; instead, H“ is a quotient Hopf algebra of H be-

, cause H,” is a Hopf algebra (over k [ t ] )hence a Hopf algebra, and finally H” =

t=l

:= Ht//n,,,t”HV

is

. The fact that Hl and Hl be

regular 1-parameter deformations respectively of H’ and H” is then clear

249

by construction. Finally, the parts of the CDP dealing with Poisson duality also are direct consequences of the like items in the GQDP applied to H; and to H z (but for Theorem 7.3 (b)). The cases of (co)augmented (co)algebras or bialgebras can be easily treated the same, up to minor changes.

Acknowledgements These notes are the author’s contribution for the Proceedings of the workshop (Contemporary Geometry and Related Topics” (Belgrade, May 15-21 , 2002). The author thanks the organizers (especially Neda and Zoran) for inviting him to this very interesting conference.

References 1. N. Abe, Hopf algebras, Cambridge Tracts in Math. 74, Cambridge Univ. Press, Cambridge, 1980. 2. V. Chari, A. Pressley, A guide to Quantum Groups, Cambridge Univ. Press, Cambridge, 1994. 3. F. Gavarini, The global quantum duality principle: theory, examples, and applications, preprint, (2003), arxiv: math. qA/0303019. 4. F.Gavarini, T h e Global Quantum Duality Principle, to appear (2003). 5. F. Gavarini, T h e Crystal Duality Principle, preprint, (2003), arXiv: math.QA/0304163.

6. C. Kassel, V. Turaev, Biquantization of Lie bialgebras, Pac. Jour. Math. 195, 297-369, (2000). 7. S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series in Mathematics 82,American Mathematical Society, Providence, RI, 1993. 8. M. E. Sweedler, Hopf algebras, Mathematics Lecture Notes Series, W. A. Benjamin Inc., New York, 1969.

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WILLMORE SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD*

Z. H u t Department of Mathematics, Zhengzhou University, 450052, Zhengzhou, People's Republic of CHINA E-mail: [email protected]. cn

H. LIS Department of Mathematical Sciences, Tsinghua University 100084, Beijing, People's Republic of CHINA E-mail: [email protected]. edu. cn

Let N"+P be an (n + p)-dimensional Riemannian manifold and I : A4 N"+P an isometric immersion of an n-dimensional Riemannian manifold M . A mapging I : M + Nn+p is called Willmore if it is an extremal submanifold of the following Willmore functional

where S = x m , i , j ( h $ ) zand H are respectively the norm square of the second fundamental form and the mean curvature of the immersion I , dv is the volume element of M . In this survey paper, we calculate the Euler-Lagrangian equation of W(z) for an n-dimensional submanifold in an (n + p)-dimensional Riemannian manifold Nn+p and give many applications as well as many examples of Willmore submanifolds.

1. Introduction

-

Let (Nn+p,h) be an oriented smooth Riemannian manifold of dimension n + p . Let M" be a smooth manifold of dimension n and x : M" Nn+p be a differentiable immersion. By imposing on M" the induced metric we * MSC2000: 53C42, 53A10. Keywords ;Willmore functional, Willmore submanifolds, minimal submanifolds. t Partially supported by grants of CSC, NSFC and Outstanding Youth Foundation of Henan, China. Partially supported by the Alexander von Humboldt Stiftung and Zhongdian grant of NSFC. 25 1

252

can suppose M to be Riemannian and x to be an isometric immersion. We will agree on the following range of indices:

1< i , j , k , . . - I n ; n + l
1I A , B , C , . . . < n + p .

Let { e A } be a local orthonormal basis for TN"+p with dual basis { W A } such that when restricts to M", {ei} is a local orthonormal basis for T M and {e,} is a local orthonormal basis for the normal bundle of x : Mn Nn+p. We write the second fundamental form I1 and the mean curvature vector H of x by 1 I1 = H=h: e, := H a e, . (1.1) h: wi @ w j e,, n.

-

c i,,

i>jP

c a

2

Let S = 1111 = xi,j,,(hG)2 and H = [HI be the norm square of I1 and the mean curvature of x : M -+ Nn+plrespectively. We define the following non-negative function on M p2 = S - n H 2 ,

(1.2)

which vanishes exactly at the umbilic points of M . Willmore functional is the following non-negative functional (see [ 91, [ 341 or [431)

w ( ~:=)J,

pn dw = J,(s

- n H 2 ) q dv.

It is well known (cf. [ 9 ] , [ 341 and [43]) that this functional is invariant under conformal transformations of N"+P. We use the term Wallmore submanifolds to call critical points of W ( x ) ,because when n = 2 and Nn+p = Sn+p, the functional essentially coincides with the well-known Willmore functional W ( x )and its critical points are the Willmore surfaces. The famous Willmore conjecture says that W ( x )2 4n2 holds for all immersed tori z : M S 3 . The conjecture was approached by Willmore [46], Li-Yau [ 251, Montiel-Ros 1281, Ros [ 381, Langer-Singer [ 171, HertrichJeromin-Pinkall [15], Cai M. [6], Kusner R. [ l S ] , Topping P. [42] and many others (see [ 451, [ 471 and references therein).

-

Compared with the fact that the Willmore functional has been extensively studied when n = 2 and N2fp is the Euclidean space R2+P (or the sphere S2+p or the hyperbolic space W2+pl because W is invariant under conformal transformations of ambient space), in higher dimensional case, n 2 3, the research for W ( z )received little attention during the same period. We note that in [ 341, the authors stated the formula of Euler-Lagrangian equation of W ( x ) for an n-dimensional submanifold in an (n p)-dimensional

+

253

Riemannian manifold, but they did not write down the proof. In [43] C. P. Wang got the Euler-Lagrangian equation of W ( z )for submanifolds z : M" S n + p without umbilic points in terms of Mobius geometry. In [ 131, the authors calculated the second variation formula of W ( z )for Willmore submanifolds z : M" -+ S n + p without umbilic points in terms of Mobius geometry and gave many examples of Willmore subrnanifolds. In [ 181, [ 201, the second author established the integral inequality of Simons' type for compact Willmore submanifolds in 3: : M" S n + P and gave the characterizations of Willmore tori and Veronese surface (see Theorem 4.1). --f

-

We organize the paper as follows. In Section 2, we give a review of ndimensional submanifolds in an (n+ p)-dimensional Riemannian manifold. In Section 3, we calculate the Euler-Lagrangian equation for the critical points of W ( z )for the most general case. For the purpose of later's application, we also write out the equations for some important special cases. In Section 4, we present some canonical examples of Willmore submanifolds in the ( n p)-dimensional unit sphere S n + P . In Section 5, we study Willmore complex submanifolds in a complex space form. In Theorem 5.1, we prove that every complex submanifold with constant scalar curvature in a complex space form is Willmore. In Theorem 5.2, we prove that every complex curve in complex space form is Willmore. These two theorems provide abundance of examples of Willmore submanifolds, at least locally. In Section 6, we study Lagrangian (or totally real in the more general setting) Willmore submanifolds in complex space forms. After writing out the Euler-Lagrangian equation for such cases (see Theorem 6.1 and Theorem 6.2), we prove in Proposition 6.2 that every minimal and Einstein totally real submanifold in a complex space form is Willmore.

+

2. Submanifolds in Riemannian manifold In this section, we give a review of submanifolds in the most general Riemannian manifold, using the method of moving frames which we refer to [ 101 for more details. We will follow the notations in the first section. Let { W A B } be the connection forms of the structure equations

( N n + p ,h ) , they

dwA = ~ ~ A B A w B ,W B

are characterized by

A B + ~ B A = ~ ,

(2.1)

254

where RABCDare the components of the Riemannian curvature tensor of (Nn+p, h). Now we restrict t o a neighborhood of x : M -+ Nn+p. Let O A , OAB be the restriction of W A , W A B t o M . Then we have

oa = o .

(2.3)

Taking its exterior derivative and making use of (2.1), we get

C oai A ei

=0.

(2.4)

i

By Cartan’s lemma we have

- -

from which we can define the second fundamental form I1 and the mean curvature vector H of x : M Nn+p as stated in (1.1). We note that H is a normal vector field over M . x : M Nn+p is called a minimal submanifold if H = 0 . If we denote by Rijkl the Riemannian curvature tensor of M ,then from (2.2) and (2.5), we get the Gauss equation

R ajkl . . - R-ajkl ..

+ C(hFk hyl - h$ hyk),

(2.6)

a

Rijij+ n2 H 2 - S,

R= i,j

where Rik and R are the Ricci curvature and scalar curvature of M , respectively. If we denote by Rapij the curvature tensor of the normal connection in the normal bundle of x : M N n f p , i.e.,

-

then from (2.2), (2.5) and (2.9), we get the Ricci equation Rapij = & p i j

+ c ( h y kh f j

-

-

hjgc hfi).

(2.10)

k

Taking exterior derivative of (2.5) and making use of (2.1), (2.2), we get the Codazzi equation of x : M Nn+p

h?. ajk

-

- h? a k j. = & k j ,

(2.11)

255

where the covariant derivative h3k is defined by

k

k

P

k

Define the second covariant derivative h;,,

by

1

1

Taking exterior derivative of (2.12), we have the following Ricci identities

m

P

m

We define the first, second covariant derivatives and Laplacian of the mean curvature vector field €3 = C , H a e , in the normal bundle of x :M Nn+p as follows

-

P

i

H,'$Oj = dH: ALHa=CH,yi,

+

H:

Oji

+

(2.16)

kk.

(2.17)

H " = -1C h a 72

i

H,{ Op,,

k

Let f be a smooth function on M , we define the first, second covariant derivatives fi, fi,j and Laplacian o f f as follows i

i

3. Euler-Lagrange equation of the Willmore functional

-

To calculate the first variation of the Willmore functional W ( X Ofor ) xo : M Nn+p, we assume, without loss of generality, that M is compact with (possibly empty) boundary a M . If otherwise, we will consider the variation with compact support. Let x : M x R + Nn+p be a smooth variation of xo such that z(., t ) = 20 and d x t ( T M ) = d z o ( T M ) on d M for each small t , where q ( p ) = ~ ( p , t ) . We call such variation an admissible variation of XO. We note that the two boundary conditions for an admissible variation disappear if d M = 4.

256

Along x : M x R -+ N n f p , we choose a local orthonormal basis { eA} for TNn+P with dual basis { w A } , such that { e i ( . , t ) } forms a local orthonormal basis for xt : M x { t } -+ Nn+P. Since T * ( M x R ) = T*M @ T*R,the pullback of { W A } and { W A B } on Nn+p through x : M x R Nn+p have the decomposition

-

x*wa = V, d t , x*wij = 8 ,

+ V ,d t , x * w ~ ,= Bi, + Mi, dt,

X * W ~= Bi

+ Lij d t ,

(3.1)

x*w,p = 8,p

+ Nap d t , (3.2)

where { V , , V, , Lij , Mi,, N a p } are local functions on M x R with Lij = -Lji, Nap = -No, and

-1

V= d xt=CVidx,(ei)+CV,e,, dt t=o i

-

(3.3)

(2

is the variation vector field of xt : M Nn+p. We note that the one forms {Oi, 8 i j , Oi,, a,@} are defined on M x { t } , for t = 0 , they reduce to the forms with the same notation on M . We denote by dM the differential operator on T * M , then d on T * ( M x R).

Lemma 3.1. Under the above notations, we have

257

Proof. These are direct calculations. In fact, substituting (3.1) and (3.2) into the following equations, respectively d(x*wi) = x*(dwi) = x*

(c

wij A wj

+

c

wia A w a ) ,

a

j

and comparing the terms in T*M A d t for each equation on the both sides, we can get (3.4), (3.5) and (3.6), respectively. 0

Lemma 3.2.

Proof. Differentiating (3.7) with respect to t and using (3.4), (3.6) we get

k

-

k,P

Covariant differentiating (3.5) over M x { t } and using the Codazzi equation (2.11) for xt : M Nn+p, we get

258

Set

a = J' in (3.11) and sum over a with

using

Likh& = 0 , we get

From (3.11) and the fact that

c

5' = C ( h ; ) 2 ,

c

Nap h$ h; = 0 ,

Ljk hri hg

= 0,

i,j,k,a

i,j>ff,P

we obtain

(3.13)

For xt : M

-

+

C

H a hg h$ VP+

Nn+p,

C H a RaipiVp.

(3.14)

i,a,P

i,j,cdJ

we consider the Willmore functional

W ( z t )=

,/

'

( S - n H 2 ) dv .

(3.15)

Noting p2 = S - n H 2 , we can rewrite W ( x t )as (3.16) From (3.13) and (3.14), we get (3.17)

259

&om (3.4), we have (3.18) n

i

=

( C K , i - n C H a V a ) 8,A...A8,. a

i

From (3.16)-(3.18),we get

We note that Y

k

k

k

and that our variation is admissible, thus a t each point on d M we have V, = Va = 0 and

-)at

0 = -a( d x t ) = d ( axt

at

=

c

dKei

i

+CdVaea, a

from which we can deduce Va,i = 0 on a M . It follows from (3.19), (3.20) and Green's formula that

From (3.3) and (3.21) with restriction t o t = 0, we have proved

260

- [c

Theorem 3.1. T h e variation of the Willmore functional depends only o n the normal component of the variation vector field. A submanifold x :M Nn+p i s a Willmore submanifold af and only if

0 = p"-2

h$ hf, h&

+

c

Rpiaj

c

+ pn-2h;ij}

(3.22)

H Ph$ h;

h$ -

iAP

i,j,k,P

Gj7P

- H " A ( P " - ~ )- p n P 2 A L H a - 2

c

(p"-"),H:

7

i

n -k 1 5 a 5 n + p .

Remark 3.1. As we have pointed out in the introduction, Pedit and Willmore stated an equivalent version of (3.22) in [ 341, where they did not write down the proof and they used very different notations. We also note that for some special cases, Theorem 3.1 was obtained in [ 181, \ 20 1, 29 1, [37], [43], [ 4 4 ] and [ 2 ] (also see [ l ] [41]). , When the ambient space is of constant curvature, i.e., N n f p = R"+P (4 is a space form of constant sectional curvature c, the equation (3.22) of Willmore submanifold x : M R"+p(c) can be simplified.

-

Theorem 3.2. (1201, 1341) A submanifold x : M" Willmore submanifold if and only if

0 = p"-2

+

[

,x

hij P hi, P hEj -

-

c

HPh$ h; - p H

wA P i,%P ( n - 1)H " A ( P " - ~ )+ 2 ( n - 1)

c

R"+p(c) is a

-1

(p"-')>,

(3.23)

H:

i

+ (n-l)p"-2A'Ha-~(pn-2)i n

+ 15 a 5 n+p.

.(nH"Sij - h $ ) ,

Ll

ij

Proof. In the present case, we have

,

R A B C D = c (SAC ~ B D SAD ~ B C )

(3.24)

which gives Rpiaj

= c ~ a bij p ,

C

Ripiai = n c &pa,

(3.25)

i

and therefore it follows that (3.26)

26 1

Furthermore, the Codazzi equation (2.11) now reads h;k = h g j , applying it we obtain that

C h G j = n H ,zs

1

Ehgij=nAiH".

(3.27)

43

j

Putting (3.26) and (3.27) into (3.22), we obtain (3.23). For the special case n = 2, we have

-

Corollary 3.1. (1191,1441) A 2-dimensional submanifold x :M 2 N2+P is Willmore if and only i f

A'H"

- 2H2Ha

+ C H P h i h;

3 5 CY 5 2 + p .

= 0,

(3.29)

Proof. Now (3.22) takes the form hij P hik P hEj -

0= AkP -

H P h z h; - p2H"

+

4AP

A'H"

+ c R p i u jh i -CHPRpiai, i,j,P

h2ij

(3.30)

i ,j

35Q52+p.

i,P

From the Gauss equations ( 2 . 7 ) and ( 2 . 8 ) ,we have

P

i

p2 = 2 H 2 -k

C

(3.31)

R i j i j - R.

i,j

Note that for 2-dimensional case, we have (3.32) Inserting (3.31) and (3.32) into (3.30) we get (3.28).

0

262

If N2+P= R2+P(c), we have

From (3.28),(3.25)and (3.33),we get (3.29). This proves Corollary 3.1. Another important application of (3.28)is the case when p = n = 2 and N4 = CP2(4) is the complex projective plane with its canonical complex structure J and canonical Fubini-Study metric g of constant holomorphic sectional curvature 4. Recall that for any isometric immersion 2 : M CP2(4) of an oriented surface M , we have a well-defined Kahler function C on M defined by C = g(Jel,e2),where (el,e2].is any orthonormal basis of T M . Surfaces with C = f l and C = 0 are called respectively complex and Lagrangian (cf. Section 5 and Section 6). For completeness, we conclude the EulerLagrangian equation of W ( x )for this special case, which essentially is due to Montiel and Urbano, cf. Proposition 5 of [29]where the Willmore functional was decomposed into two modified Willmore functional. To state the result by using method of moving frames, we choose an oriented local orthonormal basis {eA}1
-

Jel = C e 2

+ d m e4,

Je3 = C e 4 -

d

z e2,

+d

z e3,

Je4 = -Ce3 - d

el.

Je2 = -Gel

Then we have

Corollary 3.2. An isometric immersion x : M and only if

A*Ha

+ (3+ 3 C 2 - 2 H 2 )H a

-

(3.34)

CP2(4) is Willmore if

263

Using (3.34) and (3.36), we have

c

Rvv . . . .- 2 + 6 C 2 ;

kikjk

h$ = (2 t 6 C2)H a ;

iAk

i,j

(3.37)

CRgi,iHP=5H"-3C2H", i,P

C

+ 3 (1 - C2)(h;, + hg2), 2 H a + 3(1 - C2)(h?,+ h!l)7 2 Ha

hz =

Rpiaj

Gj,P

if

cr = 3;

if

Q

(3.38)

= 4.

Differentiating C = g(J e l ,e2), we get

c

c,i wi

= g(Jdel, .2)

+ g ( J e l ,d e 2 )

(3.39)

i

=g(JxwlAeA7e2)f g ( J e 1 , x w 2 A e A ) A

=

c{

A

W 2 A g(Jel 7 e A )

}

-W1A d J e 2 , e A )

A

=

dTZF (w24 - w13) = J1-cZ C ( h i , - h:i) wi, i

which gives C,i =

JTZF ( h i , - h:*) ,

(3.40)

i = 1,2.

Putting (3.40) into (3.38), we get

C Rpiaj hz = iAP

(8 - 6 C 2 ) H"

+3 d m C , 1 ,

if cr = 3;

(8 - 6 C 2 )Ha - 3 d m C , 2 ,

if

Q

(3.41)

= 4.

hGij. By (2.11), we have

Finally, we come to calculate

xiR,i, 3 xig ( Je,, ei)g(Jei, can be calculated: (3.43) R;, -e2 3 c J C 3 , R;, Ri1 0.

where R:j =

=

=

ej)

=

=

=

By using (3.43) and the following definition of covariant derivatives

C k

Rj,kwk

dR:j t

C Rk k

w k j $-

CP

&j

UP,,

(3.44)

264

we have

k

k

k

k

On the other hand, by exterior differentiating both side of the first equation of (3.34) and then making inner product with e3, we get

i

Now (3.45)-(3.47) and (3.40) give

cR;i,i

+ hf2) = 6C2H3+ 3

= 3 (Cd-),1+3C2(hg2

d z C , 1 .

i xR&= ,j-3

(Cdz),2+3C2(h;2

+

=6 C 2 H 4 - 3 d Z C , 2 .

j

It then follows from (3.42) that 2 ALH a + 6 C ' H a + 3 d p.. 2A'-H" + 6 C 2 H " 3 J-C,,, 23 2 3

i,i

=

i

m C,1,

if a = 3; if a = 4.

-

(3.48)

Putting (3.37), (3.41) and (3.48) into (3.28), we obtain (3.35). This proves Corollary 3.2. 0

-

Remark 3.2. Let x : it4 @P2(4)be a compact Lagrangian surface, in [ 29 ] Montiel-Urbano defined the following modified Willmore functional

W-(x) = /M( /HI2+ 2 ) dv.

(3.49)

We can check by use of (3.14), (3.18) and (3.37) that the Euler-Lagrangian equation of W-is A'H"

+ (1

-

3 C 2 - 2 N 2 ) H"

+

c

HPh$hG

ki,P

= 0,

3<

5 4. (3.50)

265

4. Examples of Willmore submanifolds in Sn+P(l) In this section, we will present some examples of Willmore submanifolds in the sphere Sn+P(l). Before that we first recall the following well known fact

Example 4.1. ( [ 441 ) It follows from (3.29) that every minimal surface in N2+p(c) is automatically Willmore. From 131, we know that there exist many canonical examples of minimal surfaces in S"(1). In particular, the following Veronese surface is a minimal surface in S4(l),thus it is a Willmore surface. Let (x,y,z) denote the canonical coordinate system in R3 and let u = (u1, u2, u3, u4, us)the canonical coordinates in R5. Veronese minimal surface u : S2 S4(1) is defined by 1 1 1 u 1 = - y z , u2 = - 2 2 , u3 = - xy,

-

(a)

&

u4 = 1 ( x 2 - y2), 2&

&

u5

&

=1 (x2

6

+ y2 - 2 2 2 ) ,

+ +

where x 2 y2 z2 = 3. We note that there are much more abundance of non-minimal Willmore surfaces in N2+P(c), see, e.g., [ 2, 8, 11, 12, 22, 27, 30, 351, among many others. In [ 5 ] , [24], (231, [33] and [36], readers can find some related results about Willmore surfaces.

-

N"+p(c) may be not When n 2 3, a minimal submanifold z : M" Willmore, even it has constant scalar curvature. Take for example, the Clifford minimal tori

cm,n-.m := Sm(

r

%) x S n - m ( / F )

-

Sn+'(l), 1 5 m 5 n - 1,

are not Willmore, except for the case n = 2 m (see [ 181 or [ 131). From (3.23), we can check the following fact

Proposition 4.1. ([13],[ 3 4 ] ) Every minimal and Einstein submanifold in N"+p(c) i s Willmow. Remark 4.1. The examples ( n # 2 m ) above show that the Einstein condition in Proposition 4.1 can not be weaken to the condition of constant scalar curvature. Applying Proposition 4.1, Guo-Li-Wang [ 131 constructed many minimal Willmore submanifolds in S"+P( 1) (see example 1 and example 2 in [ 131).

266

However, the following example shows that a Willmore submanifold in N”+p(c) can be neither minimal nor Einstein.

Example 4.2.([13], [20]) Wnl,...,n,+l := Snl(al)x...xS”P+l(ap+l) isan n-dimensional Willmore submanifold in Sn+P(l), where n = nl . +np+l and ai are defined by

+.

Furthermore, Wnl,... rnp+l is a minimal submanifold in Sn+P(l) if and only if it is Einstein with

When p = 1, we have the following important example

Example 4.3. ([ 181,[ 131) Willmore tori.

are Willmore hypersurfaces in Snf’(l). According to [ 181, [ 131, we call them Willmom tori. In [ 181 and [ 201, the second author proved the following integral inequality of Simons’ type for n-dimensional compact Willmore submanifolds in S n + p (cf. [ 39 I)

Theorem 4.1. ([18],[ Z O ] ) Let M be a n-dimensional (n 2 2 ) compact Willmore submanifold in a (n+p)-dimensional unit sphere Sn+p. Then we have

I n particular, if

then either p2 = 0 and M is totally umbilic, or p2 = case, either p = 1 and M is a Willmore torus Wm,,-,; and M is the Veronese surface.

f i .I n the latter or n

=

2, p = 2

267

For n = 2, the following result was proved by Li [ 191 (also see Li-Simon

[all) Theorem 4.2. (1191, [21]) Let M be a compact Willmore surface in a n (2 + p)-dimensional unit sphere S 2 + P . T h e n we have /,p2(2

- 53 p 2 ) dv 5 0.

(4.3)

I n particular, if O I p 2 < -4 3'

(4.4)

then either p2 = 0 and M is totally umbilic, or p2 = 413. I n the latter case, p = 2 and M is the Veronese surface. In the following, we give some examples of Willmore hypersurface in Sn+l( I ) , which are also isoparametric but in general are neither minimal nor Einstein. For an isoparametric hypersurface in S"+'(l), p2 and the mean curvature H are both constant, by dropping the upper index a = n 1 the Willmore condition (3.23) becomes

+

hij

h j k hki

-2SH

+ n H 3 = 0.

(4.5)

iAk

-

Example 4.4. Cartan's isoparametric hypersurfaces. For n = 3, 6, 12, 24, let z, : M" S"+'(l) be the n-dimensional compact minimal isoparametric hypersurface constructed by E. Cartan [ 71. Each of these hypersurface has three distinct principal curvature A1 = &, A2 = 0, X.3 = -8 with the same multiplicities. We can show by using (4.5) (see [ I S ] ) that all of these are Willmore hypersurfaces and they are not Einstein.

Example 4.5. Nornizu's isoparametric hypersurfaces. Let SY+'(I) = { ( z 1 , z z , . . ., ~ 2 ~ + 1 , 5 2 ~ + E2 ) R ~ + ~ ( c= ~I}, ~ for~ x ~ n = 2 r 2 4. We define a function on Sn+'(1) by

According to Nomizu [ 31 1,

Mt"

=

{ z E Sn+l(l)l F ( z ) = cos22t

7T

(4.7)

268

defines an isoparametric family of hypersurfaces in Sn+l ( 1 ) . The principal curvatures of the hypersurface Mp for a fixed t can be computed:

For this family of isoparametric hypersurfaces, a straightforward calculation from (4.5) and (4.8) shows that Mp is a Willmore hypersurface in S n + l ( l ) if and only if

t =t,

:= cot-l

n+2 We also note that from (4.8) we can see that Mp is minimal if and only if t = t m := cot-' A further computation shows that t , = t , := t o if and only if n = 4, then the corresponding four principal curvatures are given by A 1 = - 1, A 2 = 1 , X3 = 4 - 1 , X4 = 1. Therefore M t is a minimal, Willmore and non-Einstein hypersurface in S 5 ( l ) . We note that, when n = 4, example 4.5 is due to Cartan.

w.

-a

a+

-a+

5. Willmore complex submanifolds in a complex space form

Let ( M m ( 4 c ) ,g, J ) be a (complex) m-dimensional complex space form of constant holomorphic sectional curvature 4c with associated Kahler metric gandcomplexstructureJ. Let { e A } = { e l , . . . , e , ; e l * = J e l , . - . , e , . =Je,} be a local orthonormal basis for TM with dual basis { w A } . We make use the notation of e(A*)* = - e A , SA*B = - 6 ~ pand the indices range convention A , B , C , D = 1,.. . ,m, 1*, . . . ,m*. Then the Riemannian curvature tensor of Mm(4c) satisfies RABCD = c ( ~ A C SBD -

+

-

AD ~

B C

SA*C 6 B * D - SA*D SB*C

+ 2 6 A * B 6C.D)-

(5.1)

Assume z : M n M"(4c) is an isometric immersed submanifold of (real) dimension n. For p E M , we have a direct sum decomposition TpM = TpM @ Tp'M. Recall that M is called a complex (resp. totally real) submanifold of fi if J(T,M) c (T,M) (resp. J ( T p M ) c T t M ) €or each p E M . A totally real submanifold M of is called Lagrangian if m = n. We note that a complex submanifold is minimal and has even dimension.

a

269

For complex submanifolds in a complex space form, we can strengthen Proposition 4.1 by proving

-

Theorem 5.1. Let x : M Mm(4c) be a complex submanifold with constant scalar curvature. Then M is Willmore.

Proof. Since M" is a complex submanifold, n = 2 r is even. We can choose {eA} such that, restricted to M , { e l , . . . ,e,,el. = J e l , . . . ,e,* = Je,} is a local orthonormal basis for TIM. During the proof of Theorem 5.1, we further use the following convention on the range of indices: , r , l * , . . . ,r*;

i , j , k , l = l;..

a , P = r + I , . . . , m , ( r+ I ) * , - - .,m*;

1 , . . . ,T ;

a, b, C , d

=

A,p = r

+ I , . . . ,m.

Then we have ([48. p. 1811)

hz,,,

=

-haa b = -ha

hz* = hzbb'= -hzi.

(5.2)

Now it follows from the minimality of M" and Theorem 3.1. that M" is a Willmore submanifold in M"(4c) if and only if

{ ( P " - ~ ) , . h: + (p"-')),

+

273

+ pn-2

hGj

h&},

Va.

i,j

From the Codazzi equation (2.11), the minimality of a straightforward calculation gives

c

h z j = 0,

j

M n and (5.1), (5.2),

C Rpiajh$ = 0, C h$hfkh& = 0, V a .

(5.4)

i,j,k,P

i,j#

-

Combining (5.3) and (5.4), we have proved the following:

Proposition 5.1. A complex submanifold x : M" more if and only i f for each a, ( P " - ~ ) . . h; = 0.

M m ( 4 c ) is Will-

%>3

Note that in the present situation, (2.8) gives that p2 = S =

Rijij

+ 2) c - R.

- R = n (n

(5-5)

i,j

Theorem 5.1 then follows from (5.5) and Proposition 5.1. When n = 2, a complex submanifold is called a complex curve. Previous proposition implies

270

Theorem 5.2. Every complex curve in a complex space f o r m is Wallmore.

+

Example 5.1. ([ 481) Let @P+' be the ( n 1)-dimensional complex projective space with constant holomorphic sectional curvature 4. Then the (n + 1)-dimensional complex quadric ((20,z 1 , .

. . ,Z " + l )

E

@n+2

I + z; + . . . + zo"

2 Z"+1

= 0)

defines a compact complex hypersurface Q" in It is well known that Q" is Kahler-Einstein and thus has constant scalar curvature. From Theorem 5.1, Q" is a Willmore submanifold in @P"+l. 6. Willmore Lagrangian submanifolds in a complex space

-

form

In this section, we will consider the case that IC : M" &P(4c) is a totally real submanifold and also the Lagrangian submanifold ( see section 5 for their definition). We will use the following convention on the range of indices in this section:

A , B , C , D = 1 , . . - , m , l * , . . .,m*;

i , j , k , Z = 1 , 2 , . . . ,n.

From the assumption, we can choose { e A } such that, restricted to M", { e l , .. . , e n } is a local orthonormal basis for T M " . The Gauss and Codazzi equation now read

Rijkl = C (6ik djl - 6il 6jk)

+ C (h:k hyl - h$ hj",),

(6.1)

Q

h?. z j k = h? z k j.'

vi,j,k,a.

(6.2)

We also have the relations (cf. [ 481) .*

h? jk

- hj' i k = hk: zj 7

vi,j,k.

(6.3)

From (5.1) and (6.3), a direct calculation gives:

- for a E {I*,... , n * }

c

R+j

G%O

hf: = 4 n c H a ,

c H P R p i Q i= (n i,P

-for a E { n + 1 , . . . , m ; ( n + l ) * , . . . , m * }

+3)cH".

(6.4)

271

Applying (6.1), (6.4) and (6.5) in Theorem 3.1, we get

Theorem 6.1. A totally real submanifold x : M" if and only if (1) For

(Y

E

-

Mm(4c) is Willmore

{ l * , . - . ,n*}, there hold

(2) For a E { n

+ 1 , . . . ,m, (n + l)*,. . . ,m*},there hold

We will emphasize the important special case, n = m, of Lagrangian submanifolds. We note that in recent years, due to their backgrounds in mathematical physics, special Lagrangian submanifolds have been extensively studied (see [ 141, [ 32] and [40]).

Theorem 6.2. A Lagrangian submanifold x : M" if and only if

-

M"(4c) is Willmore

The following result is an immediately consequence of Theorem 6.1, which can be proved by a similar argument as in proving Corollary 3.1.

272

Proposition 6.1. A totally real surface x : M 2 if and only af

A'H"

- 2 H 2 Ha + 3 c H a

+

-

H P h c h; = 0,

Q

U m ( 4 c ) i s Willmore = 1*,2*;

(6,9)

%P

- 2 H 2 H"

A'H"

+ C H'hc

h; = 0 ,

QI

E

(3,

.a7

m, 3*, *., m*}. (6.10)

W>P

Corollary 6.1. ( [ 2 9 ] ) Every totally real minimal surface in a complex space form is Willmore. Remark 6.1. Proposition 6.1 can be compared with Proposition 5 in [ 29 1. Counterpart of Proposition 4.1 holds for totally real submanifolds in complex space forms.

-

Proposition 6.2. Ewery minimal and Einstein totally real submanifold x : M" U m ( 4 c )as Willmore.

Proof. From (6.6) and (6.7), we only need to prove that

c

r2

i,j,k,P

P

hij

P

hi, hZj

+

c

;

(P"-2)i,j h . . = O .

(6.11)

ki

Since M is minimal and Einstein, the Gauss equation (6.1) gives

Ch$hfk = ( n -

R

( n - 1 ) C S j k - -bjk, n

1 ) C b j k - Rjk =

i,P

p2 = S = n(n - 1)c - R = const,

(6.12) (6.13)

which imply that (6.11) holds. Thus we prove Proposition 6.2. Example 6.1. Let RP"(1) be the n-dimensional real projective space with constant sectional curvature 1. RP"(1) can be isometrically immersed into CP"(4) as a totally geodesically Lagrangian submanifold. From Proposition 6.2 we know that RP"(1) is a (compact) Willmore submanifold of CP"(4). Example 6.2. ([ 321) The Clifford torus

Consider the isometric embedding of ( n

T" c (CPn(4).

+ 1)-torus

this embedding is Lagrangian in Cnfl and it is minimal in S2"+l(1). Since the standard action by S1 on Cnfl restricts to both the above torus Tn+'

273

and S2"+l(l),we take the quotients of these. The induced quotient metric on GP" as the quotient S2"+'(l>/S1has holomorphic sectional curvature 4. The torus T" := T"+l/S1 in the CP"(4) is both Lagrangian and minimal. Since T" is flat, it follows from Proposition 6.2 that T" is a Willmore Lagrangian submanifold in CP"(4).

-

Example 6.3. Whitney sphere. Let @ : S2(1)

C2 be defined by

then @ is a Lagrangian Willmore surface and is called the Whitney sphere (see [Sl). Acknowledgments The authors have done this research work during their stay in the institute of mathematics of TU Berlin. They would like to express their thanks to Prof. Udo Simon and Prof. Franz Pedit for their interest and helpful discussions. They also would like to express their thanks to the referee for helpful comments. References 1. W. Blaschke, Vorlesungen uber Differential Geometrie-111, Springer, Berlin, 1929. 2. R. L. Bryant, A duality theorem f o r Willmore surfaces, J. Diff. Geom. 20, 23-53, (1984). 3. R. L. Bryant, Minimal surfaces of constant curvature in S", Trans. Amer. Math. SOC.209, 259-271, (1985). 4. F. Burstall, D. Ferus, K. Leschke, F. Pedit, U. Pinkall, Conformal Geometry of Surfaces in S4 and Quanternions, Lecture Notes in Mathematics 1772, Springer -Verlag, Berlin - Heidelberg, 2002. 5. F. Burstall, F. Pedit and U. Pinkall, Schwarzian derivatives and the flows of surfaces, (2001), a r x v : math.DG/0111169. 6. M. Cai, Lp Willmore functionals, Proc. Amer. Math. SOC. 127, 569-575, (1999). 7. E. Cartan, Sur des familles remarquables d'hypersurfaces isoparame'triques dans les espaces spherigues, Math. Z. 45, 335-367, (1939). 8. I. Castro and F. Urbano, Willmore surfaces of R4 and the Whitney sphere, Ann. Global Anal. Geom. 19, 153-175, (2001). 9. B. Y. Chen, Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital. 10, 380-385, (1974).

274 10. S. S. Chern, Minimal Submanifolds in a Riemannian Manifold (mimeographed), University of Kansas, Lawrence, 1968. 11. N. Ejiri, A counter example f o r Weiner's open question, Indiana University Math. J. 31, 209-211, (1982). 12. N. Ejiri, Willmore surfaces with a duality in S N ( l ) , Proc. London Math. SOC.,I11 Ser. 57, 383-416, (1988). 13. Z. Guo, H. Li and C. P. Wang, T h e second variation formula f o r Willmore submanifolds in Sn, Results in Math. 40, 205-225, (2001). 14. R. Harvey and H. B. Lawson, Calibmted geometries, Acta Math. 148,47-157, (1982). 15. U. Hertrich- Jeromin and U. Pinkall, Ein beweis der Willmoresechen vennutung fur kanaltori, J. Reine Angew. Math. 430, 21-34, (1992). 16. R. Kusner, Comparison surfaces f o r the Willmore problem, Pacific J. Math. 138, 317-345, (1989). 17. J. Langer and D. Singer, T h e total squared curvature of closed curves, J. Diff. Geom. 20, 1-22, (1984). 18. H. Li, Willmore hypersurfaces in a sphere, Asian J. of Math. 5, 365-378, (2001). 19. H. Li, Willmore surfaces in S n , Ann. Global Anal. Geom. 21, 203-213, (2002). 20. H. Li, Willmore submanifolds in a sphere, Math. Research Letters 9, 771790, (2002). 21. H. Li and U. Simon, Quantization of curvature for'compact surfaces in Sn, preprint No 732/2002, TU Berlin, to appear in Math. Z . 22. H. Li and L. Vrancken, N e w examples of Willmore surfaces in Sn, Ann. Global Anal. Geom. 23, 205-225, (2003). 23. H. Li, C. P. Wang and F. E. Wu, A Mobius characterization of Veronese surfaces in Sn, Math. Ann. 319, 707-714, (2001). 24. H. Li, C. P. Wang and G. S. Zhao, A new Mobius invariant function for surfaces in S3, preprint, (2003). 25. P. Li and S. T. Yau, A new conformal invariant and its applications t o the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69, 269-291, (1982). 26. W. P. Minicozzi, T h e Willmore functional o n Lagmngian tori: its relation t o area and existence of smooth minimizers, J. Amer. Math. SOC.8, 761-791, (1995). 27. S. Montiel, Willmore two-spheres in the four-sphere, Trans. Amer. Math. SOC. 352, 4469-4486, (2000). 28. S. Montiel and A. Ros, Minimal immersions of surfaces by the first eigenfunctions and conformal area, Invent. Math. 8 3 ( l ) , 153-166, (1986). 29. S. Montiel and F. Urbano, A Willmore functional for compact surfaces in the complex projective space, J. Reine Angew. Math. 546, 139-154, (2002), arxiv: math .DG/0002155. 30. E. Musso, Willmore surfaces an the four-sphere, Ann. Global Anal. Geom. 13, 21-41, (1990). 31. K. Nomizu, Some results in E. Cartan's theory of isoparametric families,

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Bull. Amer. Math. SOC.79, 1184-1188, (1973). 32. Y. G. Oh, Second variation and stabilities of minimal Lagmngian submanifolds in Kahler manifolds, Invent. Math. 101,501-519, (1990). 33. B. Palmer, T h e conformal Gauss m a p and the stability of Willmore surfaces, Ann. Global Anal. Geom. 9, 305-317, (1991). 34. F. 3. Pedit and T. J. Willmore, Conformal geometry, Atti Sem. Mat. Fis. Univ. Modena XXXVI, 237-245, (1988). 35. U. Pinkall, Hopf tori in S3,Invent. Math. 81,379-386, (1985). 36. M. Rigoli, T h e conformal Gauss m a p of submanifolds of the Mobius space, Ann. Global Anal. Geom. 5 , 203-213, (1987). 37. M. Rigoli and 1. M. C. Salavessa, Willmore submanifolds of the Mobius space and a Bernstein-type theorem, Manuscripta Math. 81,203-222, (1993). 38. A. Ros, T h e Willmore conjecture in the real projective space, Math. Research Letters 6, 487-493, (1999). 39. J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. 88, 62-105, (1968). 40. A. Strominger, S. T. Yau and E. Zaslow, Mirror symmetry is T-duality, Nuclear Physics B479, 243-259, (1996). 41. G. Thomsen, Uber Konforme Geometrie. -I: Grundlagen der K o n f o n n e n Plachentheorie, Abh. Math. Sem. Hamburg 3,31-56, (1923). 42. P. Topping, Towards the Willmore conjecture, Calc. Var. Partial Differential Equations, 11,361-393, (2000). 43. C. P. Wang, Mobius geometry of submanifolds in Bn, Manuscripta Math. 96, 517-534, (1998). 44. J. Weiner, O n a problem of Chen, Willmore, et., Indiana Univ. Math. J. 27, 19-35, (1978). 45. T. J. Willmore, Total curvature in Riemannian geometry, Ellis Horwood Ltd., 1982. 46. T. J. Willmore, Notes o n embedded surfaces, Ann. Stiint. Univ. Al. I. Cuza, Iasi, Sect. I a Mat. (N.S.) 11B,493-496, (1965). 47. T. J. Willmore, Riemannian Geometry, Oxford Science Publications, Clarendon Press, Oxford; 1993. 48. K. Yano and M. Kon, Structures on Manifolds, Series in Pure Mathematics 3,World Scientific Publishing Co. Singapore, 1984.

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SOME ASPECTS OF VISUALIZING GEOMETRIC KNOWLEDGE: POSSIBILITIES, FINDINGS, FURTHER RESEARCH *

DJORDJE KADIJEVICH Megatrend University of Applied Sciences, Belgrade, YUGOSLAVIA E-mail: [email protected]

Having underlined a rationale for visualizing the knowledge of geometry, the paper describes two tools whereby computer-assisted visualization can be realized. It then summarizes research findings regarding the use of these tools in mathematics education, suggesting promising directions for further research.

1. To visualize geometric knowledge or not? Whenever possible, teachers/learners should do it. Appropriate grounds for this approach can be found in philosophy, psychology and mathematics education. A summary of such grounds is given below. 0

0

Recalling a medieval distinction between ratio (producing an appropriate chain of reasoning) and intellagentia (comprehending a totality at once), it is useful to differentiate the knowledge of a mathematical truth from a holistic understanding of the same truth, (see [ 2 ] ) . It has been generally acknowledged that students advance their geometric thinking by climbing over the following levels: visualization, analysis, informal deduction, formal deduction and rigor, some of the latter ones may be unreachable "peaks" for many students (see

[ 11I). * MSC2UUU: 97-04, 97C80, 97U70 Keywords :dynamic geometry environments, Java applets, visualization, geometry.

t Work partially supported by the Serbian Ministry of Science, Technologies and Development under contract No MM1646. 277

278

”The National Assessment of Educational Progress found in 1982 that proof was the least liked mathematical topic of 17-year-olds, and less than 50% of them rated the topic as important” (see [ 1, 5. VII]). As it may appear at first sight, visualizations are not limited to argumentation based upon pictures (such ”proofs” can be found in [ 81, for example). Instead, they in general deal with drawings that may aid/advance our understanding of various geometric entities and processes. Such visualizations can be made alive by the use of computers - imagine that the square above the hypotenuse in Fig. 1. has been transformed into that pentagonally-shaped shaped object, by dragging point B along the drawn line perpendicular to the hypotenuse.

/

9

Fig. 1. IS a’ 4- b2 equal to c2 ?

(screenshot concerning a demo Geometer’s sketch)

2. Two tools for computer-based visualizations 2.1. Dynamic geometry environments These environments, such as Cabri-geometry ( http://education.ti.com/us/ product/software/cabri/features/features.html), the Geometer’s Sketchpad (h tt p :// www . keypress.com/s ketchpad /i ndex.htmI), and Euclid (htt p://www . dynageo.de/), simulate the Euclidean ruler and compass constructions, support such constructions by utilizing user-defined macros, and allow moving

2 79

certain parts of a constructed figure without changing its underlying geometric relations. By using such an environment, students can make and test their own conjectures. Two screenshots concerning a simple use of Geometer’s Sketchpad are presented in Fig. 2. Note that sketches from this software can be published on the Internet as Java applets by means of the JavaSketchpad program.

Fig. 2 . Would L ABC change if point B is dragged along the circumference of the circle?

2.2. Java applets These applets are programs written in Java http://java.sun.com, which are usually run inside a Web browser such as Microsoft Internet Explorer. Such programs support the creation of (dynamic) pictures, the elements of which can be: - displayed in a step-by-step fashion, which is applicable to describing

a construction task in geometry, to generating the sine curve by using the unit circle, or to providing a hint in problem solving; - transposed and rearranged, which is suitable for a visual ’,proof” of Pythagorean theorem; - dragged, which is beneficial to detecting geometric invariants, like the centroid of a triangle; - defined by some input data, which is applicable to graphing a function given by equation, to selecting a Platonic solid and an angle of its view, or to selecting the F distribution with the specified degrees of freedom;

280

-

matched against a specified rule, which is suitable for connecting algebraic and graphical representations of functions; and so on.

Skillfully developed collections of Java applets regarding mathematics education can be found at http://www.ies.co.jp/math/java/ (Manipula maths applets), http://www.saab.org/moe/start. html (Maths online), and http://illurninations.nctm.org/pages/912.html [Illuminations of NCTM). Fig. 3. presents a screenshot of an applet taken from the Manipula site.

Problem Two circles inters Point P is on one on the other. Line

B. What do you observe about triangleAPQ when point P moves on the circle?

Applet

Fig. 3. Task on triangle similarity

It is true that some visualizations can be produced with a pencil much easier than with the presented tools. But, the dynamic aspects of modifying a drawing can only be realized with such tools. Thus, the emphasis concerning visualizations should be on proofs based upon simple animations, which would, in a longer run, change the attitude of proof-denying students.

281

3. Research outcomes and further directions A summary of main research findings and suitable directions for further research is given below. 0

0

0

Although some features of appropriate geometric visualizations can be found in the literature (see, for example, [ 2]), a detailed list of these features, based upon cognitive, didactical or other issues, still awaits to be developed. Problem solving in school geometry requires moves between a theoretical (T) domain, comprising theoretical geometric entities and their properties, and a spatial-graphical (SG) domain comprising spatial-graphical geometric entities and their properties. A key point t o success in problem solving is thus linking these domains appropriately. An experience with Carbi geometry evidences that such a linkage is not fully established [ 71. Such an unfavorable outcome undoubtedly calls for developing DGE-based instructional units that can promote the links in question (T-SG links), which has not been so far explicitly treated in developed learning/teaching materials (see, for example, [I]). To develop T-SG appropriate DGE-supported lessons, the designer would not only utilize a problem-solving context where empiricism and deduction coexist and reinforce each other, but also help students realize the necessity of deductive arguments. Otherwise, empiricism will prevail, resulting in the following unfavorable outcome: Because it would be easy for students to check conjectures, they might develop a modus operandi that consisted of making a guess and trying it on a large number of cases. If the guess worked for all those, most of the students would feel no need for proof at all (see [ 10, p. 262

0

1)

The designer should also be aware that the drag mode is not heuristically neutral, since points that move and line segments that stretch and reduce differ from the traditional, paper-and-pencil entities, which promotes new styles of reasoning (see [3] ). Although mathematics educators have realized promising values of the use of Java applets-based courses (see, for example, http://www.geornetria.de/; see also Java View at http://www. javaview.de), recent electronic searches of the ERIC and MATHDI databases at http://www.askeric.org/Eric/adv- search.shtm and http://www .emis.de/ MATH/ DI /en /q u ick.htrnI, respectively, evidenced no studies concerning cognitive and affective features of

282

Java applets-based learning of mathematics. Such studies, like that of [ 91, may primarily be published in an on-line journal such as ON-Math (see http://my.nctm.org/eresources/journalhome.asp? iournal_id=6 ).

Having in mind that multimedia may be a powerful tool for knowledge construction (see [ 4 ]), future mathematics teachers should design Java applets-based multimedia lessons enabling/utilizing various learning paths, which would, inter alia, help them and their students establish links between procedural and conceptual mathematical knowledge (see [ 61). Although the explicit treatment of these links (or T-SG links in particular) is a very complex enterprise for multimedia designers, successful multimedia lessons can still be developed [ 51. 4. Coda

Kant once said: ”Perception without conception is blind; conception without perception is empty.” The message is clear: empiricism and theory are equated parts of any visualization, one influencing the other. Although computer-based visualizations still add a new dimension to learning activities, expectations regarding their utility should be realistic and not just technology-minded since, as Begle already underlined, ”mathematics education is much more complicated than you expected even though you expected it to be more complicated than you expected.”

References 1. D. Bannett, Exploring Geometry with the Geometer’s Sketchpad, Emeryville, CA: Key Curriculum Press, 1999. 2. B. Casselman, Pictures and Proofs, Notices of the AMS, 47, 1257-1266, (2000). 3. R. Holzl, How Does “Dragging” Affect the Learning of Geometry, International Journal of Computers for Mathematical Learning, l, No. 2, 169-187, (1996). 4. D. H. Jonassen, Computers as Mindtools for Schools, Upper Saddle River, NJ: Prentice Hall, 2000. 5. Dj. Kadijevich, Developing multimedia lessons in the pre-service developm e n t of mathematics teachers, Proceedings of the International Conference on Information and Communication Technologies in Education (ICTE2002),

283

6.

7.

8. 9.

10.

11.

Badajoz, Spain: Consejeria de Educacion, Ciencia y Tecnologia, Junta de Extremadura, I, 460-463, 2002. Dj. Kadijevich, Linking Procedural and Conceptual Knowledge, Proceedings on the XIX Symposium of the Finnish Mathematics and Science Education Research, University of Joensuu: Bulletins of the Faculty of Education 86, 21-28, (2003). C. Laborde, Dynamic geometry software as a window o n mathematical learning: empirical research o n the use of Cabri-Geometry, Proceedings of the Second Mediterranean Conference on Mathematics Education, Nicosia: Cyprus Mathematical Society & Cyprus Pedagogical Institute, I, 161-173, (2000). R. B. Nelsen, Proofs Without Words II: More Exercises in Visual Thinking, Washington, DC: The Mathematical Association of America, 2000. M. Pesonen, L. Haapasalo, & H. Lehtola, Looking at Function Concept through Interactive Animations, The Teaching of Mathematics, 5 , No 1, 3745, (2002). A. H. Schoenfeld, O n Having and Using Geometric Knowledge, Conceptual and Procedural Knowledge: the Case of Mathematics, Hilssdale, NJ: Lawrence Erlbaum, 225-264, (1986). P. M. van Hiele, Structure and Insight. A Theory of Mathematics Education, Orlando, FL: Academic Press, 1986.

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DUAL ALGEBRAIC PAIRS AND POLYNOMIAL LIE ALGEBRAS IN QUANTUM PHYSICS: FOUNDATIONS AND GEOMETRIC ASPECTS *

v. P. KARASSIOV~ P. N . Lebedev Physical Institute Moscow, RUSSIA E-mail: vkarast2sci.lebedev.m

We discuss some aspects and examples of applications of dual algebraic pairs (GI,&) in quantum many-body physics. They arise in models whose Hamiltonians H have invariance groups Gi. Then one can take G1 = Gi whereas another dual partner 8 2 = g D is generated by Gi invariants, possesses a Lie-algebraic structure and describes dynamic symmetry of models; herewith polynomial Lie algebras 4 = g D appear in models with essentially nonlinear Hamiltonians. Such an approach leads to a geometrization of model kinematics and dynamics.

1. Introduction

As is known, group-theoretical and Lie-algebraic methods yield powerful tools for both qualitative (adequate formulations of model kinematics and dynamics) and quantitative (dimension reduction of calculations) analysis of many physical problems [ l, 5, 101. In quantum many-body physics, where Hilbert spaces L of states and all physical observables 0 are given in terms of boson (ui,u:) and fermion (bj ,b:) operators with standard commutation relations (CR), Lie-algebraic structures arise in a natural way via using different bfm u r ,bj , b:) H boson-fermion mappings: (ui, Fa = Fa (ui,u? , bj ,b:) which introduce generators Fa of finite-dimensional Lie (super)algebras g = Span{F,} as (super)symmetry operators and simultaneously as basic dynamic variables (i.e. 0 = O({F,))) yielding a most adequate formu* SMS Numbers: 17B, SlQ, 81V. Keywords : polynomial Lie algebras, cluster dynamics, noncommutative algebraic manifolds, generalized coherent states, quasi-classical approximations. 'Work is financially supported from the INTAS programme, grant 01-2122 (Russia).

285

286

lation of problems under study [ 51. Such algebras g generate Lie groups G = exp g = {exp F : F E g} with the key for applications group property of their elements: exp FI exp F2 = exp F3, Fi E g [ 11, [ l o ] . Depending on the behavior of model Hamiltonians H with respect to symmetry transformations one discerns two (used, as a rule, separately) symmetry types [ 1] : a) invariance groups Gi of Hamiltonians H : [Gi,H] E G i H - H G i = O ; b) dynamic symmetry algebras gD : [ g D , H ] C_ gD # 0 (* H E gD). In the first case Hamiltonians are considered to be functions in only Gi -invariant (Casimir) operators Rj (Gi) whose eigenvalues A, label energy levels EA=[A,], and dimensions d G i (A) of Gi- irreducible representations (IR) DA(Gi) are equal to the Ex- degeneracy multiplicities p(X). At the same time algebras gD already generate total spectra {E,} of "elementary" quantum system within fixed IRs DA(gD)and yield spectral decompositions

-W)I,D

=

c4 x 1

L(N,

L(A) = Span{/X;4

A

= D,(g

D

)/A)}

(1.1)

x

of the Hilbert spaces L(H) of many-body systems in (p-multiple) gD-invariant subspaces L(X) generated by actions of the g D - operators D;(gD) on eigenvectors /A) E L(H) of gD-invariant operators Ai. Subspaces L(X) describe formation of "macroscopic coherent structures" (gD-domains) in L(H) which are stable under the temporal evolution:

I@@)) E L ( X ) ==+ I@(t))= U H ( t )IQ(0))

E L(A), UH(t) = exp(-itH),

for H E gD. But a physical sense of c-numbers p, A j in Eq. (1.1)still remains unclear. At the same time within many-body models with Gi-invariant Hamiltonians one can reveal deep interrelations between Gi and gD symmetries which enable not only to elucidate this sense but also to formulate an unified invariant-algebraic approach for an efficient analysis of physical problems in such models [ 51. A natural formal description of the latter is given in terms of novel mathematical concepts of dual algebraic pairs (DAP) [ 41 incorporating actions of both groups Gi and algebras gD and polynomial Lie algebras (PLA) [ 121 arising as gD in models with essentially nonlinear Hamiltonians [ 5 1. The DAP techniques enabled us to elucidate a few non-trivial questions of quantum physics; however, a number of problems concerning applications of PLA is still unsolved [ 51, [ 61. In this work we briefly discuss these problems

28 7

and ways of their solution focusing the main attention on geometric aspects. At first we recapitulate fundamentals of the DAP and PLA formalism in the context of quantum many-body physics, restricting ourselves for the sake of simplicity by the boson case and referring to [ 51 for a general discussion. Then we discuss some aspects of our applications of the DAP techniques in quantum optics [ 51, [ 61 and outline prospects of further studies.

2. Dual algebraic pairs and polynomial Lie algebras in multiboson physics: a general analysis The notion of DAP extracted from the vector invariant theory of classical groups [ 141 by Howe [ 41 is defined in the context of many-boson systems bY

Definition 2.1. Let ai = (ai,)~=,,u+= (ai)t,i= 1 , . . . ,n be n pairs of boson vector operators transforming according to two mutually contragredient fundamental IRs D1(G) and D1(G) of a certain group G:

p= 1

p=1

Consider the associative algebra A& of vector invariants of the group G generated by finite (according to the vector invariant theory [ 141) basis BGI = {Ij : [Ij,GI = O},dC; of homogeneous polynomials Ij = Ij(ui,u:). Endowing it by the commuting operation [Ii,Ij] = Ii Ij - Ij Ii one obtains a Lie algebra g ( d & )with the basis BGI and defining CR

[Ii, Ij] = fij({IZ))

( [ I a ,f b c ]

+ [ I b , f c a ] + [Ic,fab] = 0 )

(2.2)

where f i j ( { I l ) )are (consistent with the Jacobi identities) polynomials in IZ stemming from CR for a i , a; and the invariant theory. By the construction two algebraic structures GI = G and 6 2 = g(d&)commute: [GI, G2] = 0 and have a common center C(G1 = G, G2 = g(d&))= C : [C, Gii=1,2] = 0. Then they are said to form DAP (GI, &) induced by the G-actions (2.1) on 21 = 'ui = ~ p a n { a + }v, = span{ai). The Definition 2.1. entails a very important for physical applications

288

Corollary 2.1. (sometimes inserted in the DAP definition). Let

L(v@"") = Span{I{nip})

3

n ( a & ) " i " O ) : aipl0) = 0)

= L ~ ( n m )(2.3)

i,P

be the Foclc space generated by actions of creation operators a$ o n the vacuum vector 10) and carrying (due t o Eqs. (2.1) and the G2 definition) reducible representations of both structures GI, &. T h e n there holds the decomposition

where L ( [ c i ] )are 8 &-invariant subspaces labeled by eagenvalues ci of elements Ci = Ci(ai,a t ) = C i ( I j ) of the center C = {Ci} and generated by joint actions D [ " i ] ( 6 18 ) D['aI(&) of both DAP components o n some reference vectors I[ci])E L~(71.m). CR defined by (2.2) yield finite-dimensional Lie algebras gO(d&) = Span{l:} = h only if all basic invariants 1; E BGI are quadratic polynomials I; = Fj(a$,ai,) that holds, e.g., for groups G = O ( n ) , U ( n ) , SpL2n). But in the general case bases BGI contain polynomials .fj = Ij(a$,ai,) = Tj of higher orders which form tensor operators t = Span{Tj} with respect to h : [h,t]= t. Then CR in (2.2) do not close to linear combinations of invariants Ij E BGI, and repeated commutators lead to infinite-dimensional Lie algebras g(d&),generally, not belonging to well-examined classes of the Kac-Moody algebras [ 51. Therefore, for physical applications it is useful to consider (retaining Eq. (2.4)) DAP with & = & ( B G I ) where &(&I) are defined as enveloping algebras generated by the bases BcI = h U t and appropriate specifications of CR (2.2). Such objects, also appeared in other contexts [ 121, are called as polynomial deformations of Lie algebras or simply PLA (in view of the absence in the general case one-to-one correspondences between root systems of PLA and usual Lie algebras [ 51, [ 61). PLA &(aGI)being, by the definition above, specific (t-tensor) extensions of usual Lie algebras h are also G-invariant subalgebras of the universal enveloping algebra U ( w ( n m ) )of the Weyl-Heisenberg algebra w(nrn) = Span{aia, It enables one to specify completely CR (2.2) for them and to develop their representation theory (unlike the case of arbitrary PLA [ 121). These constructions are especially simple when h-tensors

289

t consist of two Hermitian conjugated irreducible tensors t = t A + tX ,

t A = { T ) : [ T ) , T j X ] = 0 } ,t X = ( t A ) t , Then CR (2.2) are specified as follows

( a ) [ h h l = h,

( b ) [h,t A ] = t X , [h,t i ] = tX, (el

(2.5)

[c', T~X] = Pij(h;r),r c c

where Pij(h;r ) are polynomials of a fixed degree s 2 2 in Fj E h, Ri E r which are found with the help of the Jacobi identities and (2.5 ( b ) ) from the only polynomial PxX(.. . ) = Px (. . . ) (corresponding to "extremal" components T2,T; of tensors t T ; , t A )the ; latters, in turn, are determined by explicit expressions T;, T; E 24 (w(nm)). so, bases &I = h U (t = t A+ t X ) )centers , r and CR (2.5) define a special (very vast) class of PLA &(&I) = & T ( h ; t Aas ) the second component of the DAP (Q1 = G, &) connected with G via the appearance of P,r c C in CR (2.5). In fact, PLA &T(h;tA) can be also examined as abstract PLA beyond the DAP context that is of interest for finding their representations not containing in Eq. (2.4) (as it is the case for usual Lie algebras [ 11). As an illustration we consider two examples taken from physics [ 51, [ 61.

Example 2.1. A simplest example is given by: PLA &gl(h = u(1) = {Vo};tX= ziy) = {V+})defined by the bases B = {Vo,V+,V- = V l } , r = (R1 : [R1,Va]= 0) and CR

where (extracted from concrete physical models) polynomials Q(V0;R1) (of the degree s 1) determine the Casimir operators CE of this PLA:

+

C" = V+V- - Q(V0;R l ) , [CE7 Val = 0, C"ILF(71rn)

=

0 ( -e==Eq. (2.4)

1.

&El

(2.7)

The PLA ( ~ ( 1u) y; ) ) can be also viewed as polynomial deformations slrd(2) of the Lie algebra

d ( 2 ) = Span{Yo,Y* : [Yo,Y&]= &Y&,[Y-,Y+] = f2 Y o }

290

due to their connection via the generalized Holstein-Primakoff transformation [ 5 ] , [ 61 :

Yo =

Y-

- & - J,

Y+ = V+[$ ( V O ) ] - ~ / ~ , (2.8)

[Ya,Ro]= 0 = [ye,J ] ,

= (Y,)+,

where &,-J are invariant ”lowest weight” operators and functions $(Vo) are determined via polynomials Q(V0;R1). Furthermore, PLA , :€ ( ~ ( 1vy)) ) ; admit two conjugate realizations by (pseud0)differential operators of one complex variable z E C

V+ = Z ,

Vo = z d / d z

+ Ro,

+

V- = z-l[CE Q ( z d / d z

+ Ro;R i ) ] ,

+

with Q ( z d / d z Ro;R1) = Cizi - y k z k ( d / d z ) kbeing determined from (2.6) - (2.7), [ 51 and [ 61.

Example-2.2. The extension of the first one is given by the following data: PLA ( 4 2 ) ;vy’) where 4 2 ) = {Eij : [Eij,Ekl] = b j k E i l - d i l E k j } is the two-dimensional unitary Lie algebra, and vy) = {yf}is its 2-nd rank symmetric tensor. All components y; and K j = (
€El

[Eij,V&] adE,,V,Z = d j k y t

+ djlV&,

[Eij,Vkl]= -[Eij, V$]t,

of CR (2.5 ( b ) ) ) by u(2) adjoint actions

2 5; = adE21 2v12

vL7

2 V& = ad&, VL,

= -adElzV1l,

2v22 =

(2.10)

-ad&2V11

on the ”extremal” components Ti2)= VA ( U ~ E , ~=V0A= adi2,V&)and

Ti2)= V11 (adE2,V11= 0 = adi12V11)which together with Vo = E11/2 generate PLA slrd(2) ~!?$,(u(l);vf)) c €~,(u(2);v~)) with CR (2.6). N

Then, using Eqs. (2.10) and the Jacobi identities we can calculate all polynomials ‘ P i j ; k ~ ( { E i jR; l } ) = [Kj, V;] in specifications of CR (2.5 ( c ) )by the u(2) adjoint actions on

P

= Pli;ii(.. .),

e.g.,

P11;12(.. . )

1

= - adE,,P(. . . )

2

etc.

291

Evidently, this procedure of ”lifting” PLA s2Fd(2) to PLA €:(h; t’) is easily extended on the case of any h = u ( N )+ u ( M ) and their irreducible tensors t’; however, generalizations of Eqs. (2.8), (2.9) are still open problems (see 151).

And now we outline general features of DAP applications in examining multiboson models with the Hilbert spaces L ( H ) = L ~ ( n mand ) Gi-invariant Hamiltonians

where If;$(. . . ) = H E t ( . . . ) are polynomials of higher (23) degrees describing essentially nonlinear interactions [ 51. Then HGI E €(&I = h u t ) where quadratic terms in (2.11) belong to h, H;$ E t , and the DAP (61= Gi, 6 2 = € ( B G I ) = g D ) naturally arise in such models. Their use reveals a ”synergetic” role of Gi-invariance and leads via the introduction of three types . _ of collective variables related to T c C ( integrals - of motions ), g D ( ”cluster” dynamic variables ) and Gi (”hidden” intrinsic parameters) to a geometrization of model kinematics and dynamics that opens possibilities to apply geometrical methods [ 3, 9, 11, 131 for their analysis. Indeed, the Hamiltonians (2.11) can be reformulated in the Gi- invariant form: , I

i E T , (with some of coefficients f l j , v k being where F’ E h, T k E t , c equal to zero), and the decompositions (2.4) for L ( H ) = L ~ ( n mcan ) be viewed as specifications of Eq. (1.1) because subspaces L([ci])have a fibre bundle structure with fibres LE(aGr)([ci; v ] ) ( wL(X) in (1.1))generated by actions D[ci](E(BG~)) on (labelling the fibre bundle bases) vectors I[ci;v]) = D1ci](GY E Gi)J[ci]). Herewith dimensions dG;([ci])of the DIciI(Gi) IRs are equal to multiplicities p(X) in Eq. (2.4) and describe degeneracies of all energy levels within a given subspace L ( [ c i ] ) .At the (quasi)classical level of analysis, implemented via generalized coherent

292

states (CS) [ 9, lo], the decomposition (2.4) induces the fibre bundle representation

M ( H ) = UMIcJ({t:;

<,”H,

. .1 I M[“*I({
(2.13)

of the model phase spaces M ( H ) C Cnm = Span{a,p Ji = 1 7 . . . 7 n ; p = 1 , . . . ,m} where fibres M[cal({ g , b >El >% g7 , bsh({C,”)> = exP[C~b({<~})Fb>l of groups G,, GD which define G, 8 GD-orbit-type generalized CS, [ 101 I

D

I{Ca;
I$o)

1;$0)

sg,({tLal)>

sh({<,”}>

I$)o>,

(2.14)

E L([ca]) = S~an{l[ca]; Y ;6))

on L([c2])and implement a re-parametrization J { a , p } ) = I { a a p ( { c i ;C,”})}) of the Glauber CS

via the factorization ~ n m ( { W )= )

sg% ccr:>>Sh({
Dll(a)SLC{<,”>> s!a({<:al)>

of Dnm({aip})[ 71. However, direct generalizations of Eqs. (2.14) are less efficient for gD = € ( & I ) because explicit expressions for matrix elements ( [ c i ] ; ~ ; ~ ~ ( e x p [ C ~ b l ~ ] lare [ c absent. i];~;~.) On the other hand, the introduction of three classes of collective variables (Ci E r , I j E € ( B G I ) , G ~E Gi) leads to a dimension reduction of dynamical problems governed by Hamiltonians (2.12) in both Schroedinger and Heisenberg (for dynamic variables Ij = Fj,T’) pictures. Indeed, the Schroedinger and cluster Heisenberg (for I j ) equations can be written in

293

terms of only variables Ci,Ij:

where U H ( t ) is the time-evolution operator induced by H = HGI from Eq. (2.12) and Eqs. (2.15 ( b ) ) , in a sense, determine a generalized dynamics on noncommutative algebraic manifolds Mc"({Ii})= {Ij : c a ( { I i } )= Cf} (see (2.7)). If Hamiltonians (2.12) do not contain operators Tk E t both Eqs. (2.15) are solved by group-theoretical methods even for time-dependent HGI (see 151, IW a

a

Ij(t>= U H ( t ) Ij Ufi(t> =

C Ba(t)

~

j

, 13

E BGI

a

where the second (factorized) form of U H ( t ) is more adequate for physical calculations in comparison with the first one. However, such simple expressions are not valid for general (even time-independent) Hamiltonians (2.12) due to the absence of the group property for elements of e x p [ & ( B ~ ~ and )] nonlinearity of L ( { I j ( t ) } in ) Eq. (2.15 ( b ) ) , [ 51. In this case for U H ( ~ I)j,( t ) one can get only " Ij-power series" representations

na

UH(t) = C [ k j ]A E j ] ( t ) I,".

=u H W j } ; % (2.16)

I j ( t > = C [ k j ]B[;ijl(t)naI? ~'j({IjI;t) where the coefficients AEjI( t ) Bfkjl , ( t ) are determined from differentialdifference equations obtained via the substitution of Eqs. (2.16) in (2.15) and the use of CR (2.5) (see [ 51). These equations define (non-classical) special functions related also with solutions of differential equations stemming from realizations of the type (2.9) for PLA &(&I). However, at present, simple analytical expressions for these functions are absent even in the case of simplest PLA sZrd(2), [ 61, that necessitates to separate "principal parts" (or asymptotics) U i ( { I j } ;t ) ,q ( { I j } ;t ) in

u ~ ( { I j ) ;=u&({Ij};t) t) (1 + f ( [ C a ] ) F ' (+ t ). . . I , zj({Ij};t)= q ( { I j ) ; t ) 7

which possess special (simplifying physical calculations) properties and determine quasiclassical factors in model dynamics [ 6 ] . So, e.g., one

294

can take solutions of classical dynamic equations, obtained via averaging Eqs. (2.15 ( b ) ) , as suitable approximations for q ( ( I 3 } ;t). At the same time asymptotics U g ( { 1 3 } ; t can ) be obtained from (determined by gD CS [[GI; v ;I )= s&(a,,,(t)I[~,]; v ; ) E L E ( ' ~ l ) ( [ cv])) Z ; quasiclassical representations of UH( t ):

UH(t) = ~

p

*

k

p) w K[ca](ttlto)

o

[GI

x

c 1k z l

tt

; v ; ) ([GI; v ;Eo

I

(2.17)

v

where d p [ " ~ ] ( Iis~ a) E(BGI)-invariant measure on M [ " ~ ] + 'c ( IM ) ( H ) and the v-independent (in view of Eq. (2.12)) kernel

K[,] (It1600) = ( [CZI;v ;I 0 I U H (t) I [GI; v ; It )

has the &(&I)-path integral form [ 9, 111. Its calculation in the stationary phase approximation [ 111 determines U g ( { I j } t; ) . However, the problem of finding adequate S&(BG1)(t)is not still solved completely.

So, within the DAP framework Gi-invariance of HGI classifies states I+) E L ( H ) yielding potential kinematic forms for, generally, degenerate (with dG.([ci]) # 1) gD-domains L( [ c i ] ) . Non-degenerate gD-domains with the identical IR D[ci=o](Gi)= I describe completely Gi-invariant (Giscalar) subsystems having unusual (extremal) physical features while degenerate gD-domains have "rest" Gi characteristics stipulating an appearance of critical phenomena in L([ci])(see [ 51). At the same time CS techniques and associated path integral schemes provide efficient tools to solve dynamical problems enabling to reveal new cooperative phenomena in Gi-invariant models [ 61. Furthermore, Gi-invariance of L ( [ s ] allows ) to examine on L( [ci])Gi-dynamics determined by gD-invariant "intrinsic" Hamiltonians H(g: E gi = InGi) with considering gD-variables as "dummy" ones [ 5, 71. 3. Dual algebraic pairs in action: applications in polarization and nonlinear quantum optics

In this Section we demonstrate an efficiency of the DAP concept and techniques on recent examples of their applications in quantum optics. The first example [ 5, 71, manifesting the kinematic significance of DAP, is due to the gauge SU(2) invariance of free light fields described by Hamil-

295

tonians

Hfl

of the form (2.11) with

m = 2,

sij sap,

g ~ z p 0,

wa? = w i 23

H$$

=o

and the Hilbert space L ~ ( 2 n = ) Span{l{ni*))} where i = 1 , . . . , n , p = f label, respectively, spatiotemporal (frequency) and polarization (in the helicity basis) modes of light. Then, taking Gi = SU(2),

we get DAP

(GI

= S U ( 2 ) = Gi,G2 = s 0 * ( 2 m ) ,

s o * ( 2 m ) = Span{Eij, X i j ,

E~~=

Xt = ( X i j ) +:

C a$ a j p , xij

= ai+ aj-

-

ai-

aj+) = g D

-

h)

p=*

) specified by acting on L ~ ( 2 n ) .The decomposition (2.4) for L ~ ( 2 n is determining the "polarization domains" L(Ci

= P ) = SPan{lP; v ;K )

IP) =

(~+)p+u~~~({~ij})(~~)"'~P),

(.:-IZp

(3.1)

10))

in L ~ ( 2 n= ) C L(c1) as eigenspaces of the S U ( 2 ) Casimir operator

p2= ~

1

, +2 -2 (P+P-

+ P- P+>= ~1

E

C ( G ~= S U ( ~ ) , = G s~o * ( 2 m ) ) :

p2I p ; v ;4= C l ( P ) Ip; v ;4

+

i,

whose eigenvalues c l ( p ) = p ( p l ) ,p = 0 , 1 , . . . determine values p of the polarization (P)-quasispin replacing the non-gauge-invariant usual spin for light fields. This decomposition of L ~ ( 2 n provides ) a new (symmetry) treatment of polarization structure of light [ 5 , 71 that enabled us to reveal an unusual (coherent) sort of unpolarized light (P- scalar light) given by states 10,)

E L ( p = 0 ) = Span{ ( p = 0;v = 0; K.

# 0 ) 0: n(x;)"*J 10))

(existing for L ~ ( 2 n )n, 2 2 ) with characteristic property

P,=o,*JOp) =0

(OplP;"1P,"2P~JOp) = 0 , V a l +a2

+a0

2 1 (3.2)

+

of the "polarization vacuum". For n = 2 (when P, = Pi, Pz,) in view of Eq. (3.2) states of P-scalar light generalize so-called Bell states widely used in quantum physics for examining both fundamental (EPRparadox, teleportation etc.) and applied (design of quantum computers,

296

optical communication) problems [ 2 1. Furthermore, they give positive solutions of the problem of existence of non-stochastic waves of unpolarized light [ 51 (A. Fresnel, 1821) having the negative solution in classical optics. According to general remarks of Section 2 polarization domains L ( p ) are dynamically stable under Hamiltonians HG;m p = H j l + Hso*( 2 m ) Hsv(2) with

+

a

where Hso.( 2 m ) and H S ~ (determine, ~ ) respectively, dynamics of biphoton clusters Xf (including their production) and a purely polarization dynamics. These dynamics are adequately described in terms of the SU(2), g~ so*(2m)-orbit-type CS of the form (2.14) with SSU(2)(0

= exP(tP+ - t*P-),

sso*(2m)({CbU;Cbl})=exP(~[5,U~ii+l - C*Ei+li

+ 5x;+1

-

C*qli1)

which, in particular, yield elegant solutions of many quantum problems (such, e.g., as calculations of geometric phases [ 5 ] , developments of quantum tomography schemes and analysis of quantum interference patterns [71). The second example [ 5, 61, leading to applications of PLA formalism, is n;l given by models with Hamiltonians Hmps(n;s) = wo a: a0 HGI from (2.11), where gf = 0 and for s 2 2,

+

H;:

c

= H1(n;s) =

[gil...isa; . . .a t a o

l < i l ,...,i s < n

+ g;l,..isail . ..ai.a:1,

(3.4)

+

acting on the Hilbert space L F ( ~1) = Span{J{ni})}&l (the "dummy" label ,9 = 1 is omitted) and describing processes of multiphoton scattering. In the case of arbitrary gil,,,i, Hamiltonians HmPs(n;s) have the invariance groups Gi = C,@ U R (1) ~ with both discrete

(cs -- {ei2.rrkN/s },&, s

n

Eii, Eii = a:ai)

N = i=l

( u R 1 ( l ) = {exp(i$Rl)},

Ri

=

[N+sEoo])

and continuous

297

factors. Then r = { R l } ,and L?cr={Eij = a+aj,

yT.,,is =: a

. . . a t a o E .+"I, ( K1,,,is= ail

. . .ai,ao+ E

- },

J5)

where uy)is the s-rank symmetric u(n)-tensor, and the DAP

(GI= Gi = Cs 8 uRI(1)762 = g D

=€El(U(n);Vy)))

acts on L ~ ( n + l ) .In view of the Gi Abelian nature the decomposition (2.4) for LF(n 1) contains only non-degenerate finite-dimensional gD-domains

+

L([S

~ 2 1 )=

span{ I [GI;K )

O:

~ 2({Eij '1)

I [GI),

(~;t..l)~1

I [GI> =

(3.5)

10))

where c1 = lc = 0 , 1 , . . . , s - 1,cg = 2 j = 0 , 1 , . . . are determined by eigenvalues of Gi-invariant operators. At the same time, in view of CR (2.10), the Gi-invariant form (2.12) of the Hamiltonians H"p"(n; s) can be given by the expressions Hmps

(n;s) =

(where S E ( < )= exp{&j[&j Eij - (:i E j i ] } ) which are most suitable for analyzing Eqs. (2.15). However, nowadays we can get only (quasi)classical solutions of these equations, and besides, solely in the case n = l when PLA (u(n); uy))is reduced to suS(2) defined by Eqs. (2.6) [ 5, 61. For example, in this case Eqs. (2.15 (b)) are nonlinear analogs

&El

ih-dV0 =jv+-g*v-, dt

ih 5 dt = -aV+

ih-

dV= aVdt

- G* P(V0),

+ GP(V0) (3.7)

(where VO= [ N - S E O OV+ ] , = V,+l,V- = Vl...l)of the well-known linear Bloch equations for 4 2 ) . In turn, solutions of Eqs. (3.7) are equivalent to those of the only equation

+

- C ) - a2 Vo(t) 2 1 ij 12 P(Vo(t)) dt2 which, in the cluster mean-field approximation ( ( f ( { V a } )=) f ( { ( V a ) } ) ) , have quasiclassical solutions in terms of (hyper)elliptic functions [ 61 naturally arising in soliton theories [ 11, 131. On other hand, using Eqs. (2.8)

-d2vo(t) - a ( H

298

in this case one can transform linear Hamiltonians (3.6) to an essentially nonlinear form Hrnp"(l;s) =h[AYo+Y+g(Yo)+gt(Yo)Y+b(R1)],

dY0) = 9[ 4(VO) 7 (3.9) depending on variables Y, E 4 2 ) that enabled us to obtain (via path integral representations (2.17) with using SU(2) CS of the form (2.14)) quasi-classical SU(2)-asymptotics

L&{Y,};~) = exp

[ C U ~ ( ~ H) Y = ~,~]p," ( i ; S ) ,

(3.10)

i

of the evolution operators U H ( ~where ) time-dependent coefficients ui(t) are determined through solutions of classical versions of Eqs. (3.8), [ 5, 61.

4. Conclusion

So, we demonstrated natural appearances and an efficiency of DAP and PLA formalism in examining multiboson models with Gi-invariant Hamiltonians. In conclusion we outline some directions of further studies concerning physical applications. They incIude:

1) specifications of quasiclassical representations (2.17) for VH( t ) based on determining adequate form of generalized CS related to the exponentials E x p ( i j P ( & ~ ) )and on generalizations of the transformations (2.8) ; 2) extractions of their "group-like'' asymptotics (extending (3.10)) and examinations (in view of Eqs. (2.9)) of connections of latters with the Maslov quasiclassical asymptotics for partial equations in quantum mechanics [ 81 ; 3) applications of geometric methods [ 3, 131 in analysis (cf. [ 111, [ 133) of nonlinear operator evolution equations of the type (3.8) stemming from the "cluster" Heisenberg equations (2.15 ( b ) ) , (3.7) and their quasiclassical approximations (taking into account that Eqs. (3.9) together with transformations M$l({
Sil

Acknowledgments The author is thankful to Professor Z. RakiC for his attention to this work.

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References 1. A. 0. Barut and R. Racka, Theory of Group Representations and Applications, PWN- Polish Sci. Publishers, Warszawa, 1977. 2. D. Bowmeester, A. Ekert, A. Zeilinger, T h e Physics of Quantum Information, Springer - Verlag, Berlin e. a., 2000. 3. A. T. Fomenko, Differential Geometry and Topology. Additional Chapters (in Russian), Moscow University Press, Moscow, 1983; A. S. Mishchenko, Vector Bundles and Their Applications (in Russian), Nauka, Moscow, 1984. 4. R. Howe, Remarks on Classical Invariant Theory, Yale University Preprint, 1976; S. Sternberg, Some recent results o n the metaplectic representation, Lect. Notes Phys. 79 (117), (1978). 5. V. P. Karassiov, Algebms of SU((n)-invariants: structure, representations and applications, J. Phys. A25 (393), (1992); G-invariant extensions of Lie algebras in quantum many-body physics, ibidem A27 (153), (1994); Polynomial Lie algebras and associated pseudogroup structures in composite quantum models, Rep. Math. Phys. 40:2(235), (1997); Symmetry as a source of hidden coherent structures in quantum physics: general outlook and examples, Yad. Fiz. 63 (714), (2000); Symmetry approach to reveal coherent structures in quantum optics: general outlook and examples, J. Rus. Laser Res. 21 (370), (2000). 6. V. P. Karassiov, sl(2) variational schemes f o r solving one class of nonlinear quantum models, Phys. Lett. A238 (19), (1998); Cluster quasiclassical dynamics in multiphoton scattering models. Analytical results., J. Rus. Laser Res. 20 (239), (1999); V. P. Karassiov, A. B. Klimov, An algebraic approach to solving evolution problems in some nonlinear quantum models, Phys. Lett. A189 (43), (1994). 7. V. P. Karassiov, Polarization coherent states in action: partttial tomography of rnultimode quantum radiation, Bull. Lebedev Phys. Inst., Allerton Press N9(34), (1999); V. P. Karassiov, A. V. Masalov, Quantum interference of light polarization states via polarization quasiprobability functions, J. Opt. B4 (S366), (2002). 8. V. P. Maslov and M. V. Fedoriuk, Semi-classical Approximation in Qumtum Mechanics, D. Reidel, Dordrecht, 1981. 9. A. Odzijewicz, O n reproducing kernels and quantization of states, Commun. Math. Phys. 114 (577), (1988); Coherent states and geometric quantization, ibidem 150 (385), (1992). 10. A. M. Perelomov, Generalized Coherent States and Their Applications, Springer-Verlag, Berlin e. a., 1986. 11. R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, 1982. 12. K. Schoutens, A. Sevrin, P. Nieuwenhuizen, Commun. Math. Phys. 1 2 4 (87), (1989); M. Rocek, Representation theory of the nonlinear SU(2) algebra, Phys. Lett., B255 (554), (1991).

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13. I. A. Taimanov, Sekushchie abekevykh mnogoobraziy, teta-funktsii i solitonnye uravneniya [Sections of Abelian varietes, theta-functions and soliton equations], Usp. Math. Nauk 52:1(150), (1997). 14. H.Weyl, The Classical Groups, Princeton University Press, Princeton, 1939.

VISUALISATION AND ANIMATION IN DIFFERENTIAL GEOMETRY*

EBERHARD MALKOWSKY t Department of Mathematics, University of Giessen, Arndtstrasse 2, 0-35392 Giessen, G E R M A N Y E-mail: [email protected] Department of Mathematics, Faculty of Science and Mathematics, University of NiS, ViiSegradska 33, 18000 NiS, YUGOSLAVIA E-mail: [email protected]

VESNA VELICKOVIC Department of Mathematics, Faculty of Science and Mathematics, University of Nag, ViSegradska 33 18000 NiS, YUGOSLAVIA E-mail: [email protected]

In this paper, we give a survey of our own software for differential geometry and its extensions [5, 1, 2, 3, 41. Furthermore we deal with a few applications to represent some interesting classical results in differential geometry. Originally the software was aimed to support teaching by visualizing the results from differential geometry. But it also has applications in research and the engineering sciences. The software is open, that is, its source files are available to its users. Hence it can be extended. It uses OOP, and its programming language is PASCAL.

1. Introduction and Notations Visualization and animation are of vital importance for the modern methods in mathematical education. They strongly support the understanding * MSC2000: 68N19, 53A04, 53A05. Kevwords :Computer Graphics, Visualisation, Animation, Differential Geometry. t Work supported by Serbian ,Ministry of Science, Technologies and Development under contract No MMf646 and by the German DAAD foundation, grant number 911 103 102. Work supported by Serbian Ministry of Science, Technologies and Development under contract No MM1646

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of mathematical concepts. Most of all, this applies to geometry and differential geometry. We think that the application of a commercial graphics software package is neither a satisfactory approach for the illustration of the theoretical concepts, nor can it be used as their substitute. It cannot be the aim of education to teach students t o use some software package by instructing them which keys to press, and how to move the mouse, regardless of how convenient this may seem. The emphasis should be put on teaching the fundament a1 theoretical facts. Students should also be encouraged to write their own programmes for the visualization of the solution of problems. A successful completion of this task will not only give prove of the students’ correct understanding of the matter, but it will also add to their motivation. Moreover the students will considerably improve their command of a programming language and their techniques. In view of this, we developed an open software in PASCAL on programme level which provides the basic tools for computer graphics, in order t o offer an alternative to existing graphics software packages. The main purpose of our software is t o visualize the classical results in differential geometry on PC screens, plotters, printers or any other postscript device, but it also has extensions to physics, chemistry, crystallography and the engineering sciences. To the best of our knowledge, no other comparable, comprehensive software of this kind is available. The software is open which means that its source files are accessible t o the users, thus enabling them to apply it in the solutions of their own problems. This makes it extendable and flexible, and applicable to both teaching and research in many fields. In contrast t o this, almost all other available graphics packages are closed in general, the area below the user interface is inaccessible and consequently the software cannot be extended beyond the scope of solutions it offers. The software uses OOP, object oriented programming, and its programming language is PASCAL. The software is self-contained in the sense that no graphics package is needed other than PASCAL. The advantages of PASCAL are the hierarchy of objects and the polymorphy which is not available in some OOP languages. In the hierarchy of objects, a successor inherits all the data, in particular the methods and procedures, of its predecessors. Polymorphy means that virtual methods can be declared, a

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virtual method can be rewritten with the same name in a successor, and one may have more methods than one with the same name. The development of our software could not have been achieved without OOP. 2. The General Concept and Main Principles of Our Software The concept of our software is the strict separation of geometry from the technique of drawing. Its main principles are the use of line graphics, central projection and independent visibility checks. 2.1. Separation of geometry and drawing The main principle of our software is a strict separation of geometry from the technique of drawing. There is a Unit UG which contains all the objects types, procedures, methods etc. for the geometry of the objects t o be represented, and there is a unit Draw which contains the technical tools needed in the actual drawing process. This makes our software very transparent. We need only two units and may concentrate on geometry. 2 . 2 . Line graphics

We use line graphics which means that we only draw curves. Line graphics are most suitable for the representation of results in differential geometry. Surfaces are represented by families of curves on them, usually the parameter lines. Thus we do not need a special strategy or technique for the graphical representation of surfaces and only need two general objects.

At the root of the hierarchy tree on the geometry side, there is the general object type SurfaceT. It contains the methods needed in the mathematical description of surfaces, in particular the virtual methods ParToSur f and Surf ToPar t o transform parameter points into points on the surface and vice versa. At the root of the hierarchy tree on the side of the technique of drawing, there is the object type Curve3DT which contains all the methods and procedures needed to draw three-dimensional curves. Its successor UiT is needed t o draw parameter lines on surfaces. On the geometry side, ST is a successor of SurfaceT. It contains the special methods and procedures needed for the geometry of the given surface S . Now the procedures ParToSurf and SurfToPar have to be declared according to the parametric representation of the surface S . The object types SCT, a successor of CurveSDT, and SUiT, a successor of UiT, are needed to draw

304

the contour line and the parameter lines of S . Again, the geometry of S has t o be implemented into them. Geometry SurfaceT

Technique of Drawing

x ‘ I

SCT

ST

Geometry of S

UiT

SUiT

Figure 1. The hierarchy tree of objects for the representation of a given surface S

Curves may be given by parametric representations or equations and are approximated by polygons. Lines graphics causes no problem in the graphical representation of lines on surfaces or lines of intersections of surfaces, which is a major difficulty in many commercial software packages. A disadvantage of the use of line graphics is that we are faced with a contour problem.

2.3. Independent visibility check

We use an independent procedure Check t o test the visibility of our geometric objects. Thus we are free to manipulate t o show desirable but unrealistic effects or not to use any test at all for a fast first sketch.

Example 1. Dandelin’s spheres. The uniquely defined spheres that are both tangent to a cone along a circle line and an intersecting plane are called Dandelin’s spheres. They are tangent to the plane of intersection at the foci of the resulting conic section. The visibility of every point is immediately tested analytically. Thus our graphics are generated in a geometrically natural way. Two consecutive points in the approximation of a curve are joined by a straight line segment if and only if both of them are visible. We use interpolation t o close occasional gaps. This is less time consuming than increasing the overall number of points for the approximation. There is an option to either dot or omit invisible parts.

305

Figure 2.

Dandelin's spheres, concept, reality and desirable representation

3. The Procedure Check The visibility of points of a surface S is tested by the procedure check. There are two methods in Check, namely S.Visibility(P;V i s ) , which checks whether a point P on S is invisible with respect t o S in which case V i s = F A L S E , and S.NotHidden(Q;NotHid), which checks if an arbitrary point Q in space is hidden by S in which case NotHid = FALSE. The method S.NotHidden is very useful in the simultaneous representation of more surfaces than one.

A point P of a given surface S in a class S of surfaces is visible, that is Check = T R U E if and only if Vis = T R U E in S.Visibility(P;Vis) and Not.Hid = T R U E in S*.NotHidden(P;NotHid) for all S* E S \ { S } . Let S be a surface with a parametric representation ."(ui) = (d(u1,u2),x2(u1,u2), z3(u1,u2)),((u1,u2) E D),

(3.1) --f

where D c R 2is a domain, P E S be a point with position vector O P and C be the center of projection. Then V i s = FALSE if and only if + -t .'(ui) = O P t PC , for some ((ul,u2)E D ) and t > 0.

+

306

Figure 3.

T h e natural value for V i s

The visibility problem has been solved for explicit, ruled, potential, tubular and general screw surfaces, for exponential cones and surfaces of revolution. 4. Contour

We already pointed out that the use of line graphics results in a contour problem. Roughly speaking the contour of a surface separates its visible points from its invisible points. The representation of a surface without its contour line appears to be unfinished.

Figure 4.

A sphere without contour and with contour

We say that a point P on a surface S with a parametric representation (3.1) is a contour point if and only if --f dZ 82 n ' o P C = Q where Z = - X - , dul dU2 the contour line of a surface is the set of all its contour points.

307

P

Figure 5. The concept of a contour point

Figure 6. plane

An explicit surface and the projection of its contour line in the parameter

Figure 7. Level lines on an explicit surface

Our definition of a contour point is slightly coarse, but serves its purpose well, since it only excludes rare cases which are much more easily handled by a slight change in the perspective than by a more precise definition that involves a very time consuming additional check. The determination of the

308

contour line of a surface in the general case involves a numerical method t o find the zeros of a real-valued function of two real variables in a domain. An algorithm and its implementation can be found in [ 21. The same algorithm can be used to find the line of intersection of a surface with a plane. The contour problem and the problem to find lines of intersection of surfaces and planes has been solved in the general case.

Figure 8. The intersection of an explicit surface and an exponential cone

5 . Curves

In this section, we apply our software to the graphical representation of some results from the theory of curves. Let y be a curve in three-dimensional Euclidean space lR3 with a parametric representation Z(s) = (xl(s), x2(x),x 3 ( s ) ) with s the arc length along y. Then the vectors i71(s) = k(s), i7z(s) = ~ ( s ) / ~ ~ ~and ( s ) ~ ~ i73(s) = V ; ( s ) x &(s) are called tangent, normal and binormal vectors of y at s. The vector 2(s) and its length n(s) = ~ ~ ~are( called s ) the ~ ~vector of curvature and the curvature of y at s; n is a function which measures the deviation of a curve from a straight line. The value ~ ( s =) $ ( s ) 0 i73(s) is called the torsion of y at s , T is a function which measures the deviation of a curve from a plane.

A plane at a point P of a curve, orthogonal to the binormal vector at P, is called the osculating plane at P . The osculating circle of a curve y at a point P E y is the uniquely defined circle which is a second order approximation

309

of y in the osculating plane of y at P . Furthermore, the osculating sphere of y at a point P given by the parameter s is a sphere that is a third order approximation of y at s;it is uniquely defined whenever ~ ( s ~) ,( s#) 0, and its center an radius are given by

and

The fundamental theorem of curves states that the shape of a curve is uniquely defined by its curvature and torsion. Planar curves with K ( S ) = c.s where c is a constant are klothoids. Planar curves with K ( S ) = c / f i where c is a constant are the lines of intersections of a helix with a plane orthogonal to the axis of the helix.

Figure 9. Vectors of curvature and osculating circles of an epicycloid Osculating plane and osculating sphere of a helix

\ Figure 10.

Curves with ~ ( s = ) c . s and ~ ( s = ) c/&

310

Curves with the property that their curvature and torsion are proportional are called lines of constant slope; they have a constant angle with a given direction in space.

Figure 11. Orthogonal projections of lines of constant slope on a sphere and a paraboloid of rotation on t o planes

6 . Surfaces

In this section, we apply or software t o the graphical representation of some results from the theory of surfaces. We always assume that a surface S is given by a parametric representation

where D c EL2 is a domain and the component functions x k have continuous partial derivatives of order T 2 1. Furthermore, we use the familiar notations Zk(ui) = d Z ( u i ) / d u k ,Z j b ( u i ) = d 2 ( u i ) / d u j d u k for j , k = 1 , 2 , and $, gik, L i k for i, Ic = 1 , 2 and for i , j ,Ic = 1 , 2 for the (normalized) surface normal vectors, the first and second fundamental coeficients, and the second Christoflel symbols.

(ik}

Let y be a curve on S with a parametric representation Z(s) = Z(u2(s)). We write .'(s) = K ~ ( s q) s )

Then the values ture of y at s.

K ~ ( S and )

+ K ~ ( S d(s) )

where

~ ~ ( are s )called the

i?=

8 x 2.

geodesic and normal curva-

311

6.1. The normal and principal curvature First we consider the normal and principal curvature. The normal curvature of a curve at a point only depends on its direction; the extreme values of the normal curvature at a point are called principal curvatures and the corresponding directions are called principal directions. At each point of a surface there are two orthogonal principal directions. Curves on a surface the tangents of which coincide with the principal directions are called lines of curvature; they are solutions of the differential equation

Figure 12.

The principal directions of a curve. Lines of curvature on a tangent plane

Three families of surfaces are said to be a triple orthogonal system if any two surfaces from different families are orthogonal. Triple orthogonal systems sometimes are useful to find lines of curvature.

Example 2. Dupin’s theorem. The surfaces of a triple orthogonal system intersect in their lines of curvature.

312

Figure 13. Ellipsoid, hyperboloids of one and two sheets, their intersections and lines of curvature on them

A curve on a surface with identically vanishing normal curvature is called an asymptotic line, the corresponding direction is called asymptotic direction. Asymptotic lines are solutions of the differential equation L , j k duj duk = 0.

Figure 14.

Asymptotic lines on a monkey saddle, a catenoid and a Moebius strip

313

6.2. The geodesic curvature

A curve on a surface with identically vanishing geodesic curvature is called geodesic line. Let (9'2) denote the inverse of the matrix (912) of the first fundamental coefficients. Then

[ i j k ] = 2ij 2k

and

{ ;k}

= g ' i [ j ~ ]for

i,j,k = 1,2;

denote the first and second Christoflel symbols of the surface. The geodesic lines of a surface are the solutions of the following system of differential equations (6.1)

Example 3. The geodesic lines on a sphere are principal circles, that is intersections of the sphere with planes through its center.

Figure 15.

Principal circles and geodesic lines on a sphere

In a special case, the differential equations (6.1) for geodesic lines can be solved explicitly.

Example 4. ( [ 5 ] ) Let the first fundamental coefficients of a surface satisfy the conditions gll(u2) = gll(ul),

912(ui) = 0 and

g22(ui) = g22(u1).

(6.2)

Then the ul-lines are geodesic lines; the u2-lines are geodesic lines if and only if

314

if 80 E [0,27r) \ {7r/2,37r/2} and c = sign(sin80) c o s 8 0 for~ given ~ u&,ug,then a curve with a parametric representation Icc(ul)= Zc(u1,u2) is a geodesic line in a neighborhood of (u:,ug)if and only if u1

u"u1)

2 -ug

=f c

I&

4

drn

i

m

d

du.~

(6.3)

It follows from (6.3) that if the first fundamental coefficients of a surface S satisfy the conditions in (6.2) a geodesic line on S can take one of three characteristic shapes. If g22(u) > c2 then the integral exists for all u ' > uh, and the geodesic line exists for all such u'. If g22(u) = c2 for some u > uy, let iil = inf{u > u& : g22(u) = c'}. Then the integral has a singularity a t iil. If the singularity is removable, then the improper integral in (6.3) converges as u1 46' and the geodesic line is tangent to the u2-line corresponding to iil which is not a geodesic line, since dg22/du1 (a') # 0. If the singularity is not removable, then the improper integral in (6.3) diverges as u1 -+ 6'. The geodesic line now asymptotically approaches the u2- line corresponding to Q1 which is a geodesic line, since dg22/du1(ii1). By the uniqueness of geodesic lines the geodesic line cannot be tangent to the u2-li ne corresponding to G1. Surfaces of revolution are given by a parametric representation .'(uZ) = (r(u1)cosu2,r(ul)sinu',

h(u')), ( ( u 1 , u 2E) D = 11 x 12).

where r(u') > 0 on 11. Their first fundamental coefficients satisfy the condition in (6.2).

Figure 16. The three characteristic shapes of geodesic lines on surfaces of revolution

Geodesic lines are useful in the generalization of Cartesian and polar coordinate systems.

315

Geodesic parallel coordinates are a generalization of Cartesian coordinates in the plane. Let .'(ui)be a parametric representation of a surface S and y be a curve on S given by a parametric representation Z(s) = Z(ui(s)) (s E I ) where s is the arc length along y. For each s E I , we denote the geodesic line through s orthogonal to y by geod,. Let $,(s*) = y',(wi(s*))be a parametric representation of geod, where s* is the arc length along geod, and wB(0) = ui(s)(i = 1 , 2 ) . The lines that have constant distance along the geodesic lines geod, from the original curve y are called parallel curves of y. The geodesic lines geod, and the parallel curves define geodesic parallel coordinates. We introduce new parameters tii (i = 1 , 2 ) for the surface S such that the fil-lines are the parallel curves and the ,Ii2-lines are the geodesic lines. The transformation formulae are ui = ? & 1 ( 2 ) ,

Figure 17.

(i = 1 , 2 ) .

Geodesic parallel and polar coordinates

Geodesic polar coordinates are a generalization of polar coordinates in the plane. Let S be a surface with a parametric representation .'(ui) and P be a point on S. By geode we denote the geodesic line through P at an angle 0 with the u2-line through P. Let & ( s ) = y"(v;(s) be a parametric representation of geodo where s is the arc length along geode. The lines C, that have constant distance s along the geodesic lines geode are called geodesic circles. The geodesic lines geode and the geodesic circles define

316

geodesic polar coordinates. We introduce new parameters G i (i = 1,2) for the surface S such that the ti-lines are the geodesic lines geode and the G2-lines are the geodesic circles C,. The transformation formulae are uz= vh.(u ' 1),

(i = 1,2).

Leva-Civita parallel movement is a generalization of a movement of a vector along a straight line in the plane. Let S be a surface with a parametric representation .'(ui),P be a point on S and T ( P )be the tangent plane to S at P . Let z' = (t1,t2) be a vector in the tangent plane T ( P )where t1 and E2 are its components with respect to .'I and i$ at P . If the vector z is moved along a geodesic line on S such that the angles between the vectors and the geodesic line have a constant value then the movement is said to be parallel. When S is a plane then the straight lines are geodesic lines and a movement of a vector along a straight line is parallel. More generally, a parallel movement along an arbitrary curve on a surface is called Levi-Civita parallel movement and given by

6ti 6s

dti ds

Figure 18. Levi-Civita parallel movement along a geodesic line on a torus

7. Some Animations In this section, we deal with three examples of animations. Animations are extremely suitable to illustrate the geometric principle of construction for many curves. Here we deal with the construction of a so-called double egg line. Let Sl(0)be the unit circle line and A and B be distinct points on SI(0). Furthermore, let F denote the intersection of the straight line O A through the origin 0 and A with the straight line through B orthogonal to OA, and P denote the intersection of the straight line through OB and the straight line through F orthogonal to OB. A

317

double egg line is the set of all points P constructed in this way when B moves along Sl(0). A parametric representation for the double egg line is Z(t) = ( C O St ,~cos2(t) sin t ) (t E (O,27r)).

Figure 19. Construction of a double egg line

Let S be a surface with a parametric representation .'(uZ) ((u', u2) E 0 ) and surface normal vectors $(tiZ).Then the map from S t o the unit sphere which assigns to every point P of S the surface normal vector fi at P is called the Gauss (spherical) map.

Figure 20.

Gauss maps of parts of paraboloid of rotation

Finally we represent the transformation of a torus of a double egg lane with a parametric representation

+ C O S ~u') cosu2, ( R + C O S ~u')sinu2, cos2u' sinu')

2(ui)= ( ( R

318

((u1,u2) E D = ( 0 , 2 ~ ) where ~) R parametric representation

< 1 is a constant, to a torus with a

g(ui)= ( ( ~ + c o s u lcosu', ) ( ~ + c o s u l sinu2, ) sinul), ((u1,u2)= ( 0 , 2 7 r ) ~ ) . We consider the class T of surfaces St (t E [0,1]) with parametric repre) tg(u2) (1 - t ) Z ( u i ) ,((u1,u2)E D ) . sentations Z ( t ) ( u i =

+

Figure 21.

Transformation of a surface of revolution to a torus

References 1. M. Failing, Entwicklung numerischer Algorithmen zur computergrafischen Darstell ung spezieller Probleme der Differen tialgeome trie und Kristallographie, Ph.D. Thesis, Giessen 1996, Shaker Verlag, Aachen, 1996. 2. M. Failing, E. Malkowsky, Ein efizienter Nullstellenalgorithmus zur computergrafischen Darstellung spezieller Kurven und Flachen, Mitt. Math. Sem. Giessen 229, 11-25, (1996). 3. E. Malkowsky, An open software in OOP for computer graphics and some applications in differential geometry, Proceedings of the 20 th South African Symposium on Numerical Mathematics, 51-80, (1994). 4. E. Malkowsky, An open software in OOP for computer graphics in differential geometry, the basic concepts, ZAMM 76,Suppl 1, 467-468, (1996). 5. E. Malkowsky, W . Nickel, Computergrafik in der Differentialgeometrie, Vieweg-Verlag Wiesbaden, Braunschweig, 1993. 6. E. Malkowsky, V. VeliEkoviC, Some geometric properties of screw surfaces and exponential cones, Proceedings of the 10 th Congress of Yugoslav Mathematicians, Belgrade, 395-399, (2001).

COMPUTER GLUING OF 2D PROJECTIVE IMAGES *

G . V. NOSOVSKIY Moscow State University, Faculty of Mechanics and Mathematics, Department of Differential Geometry and Applications, Vorob’evy gory d. 1, 119992 Moscow, RUSSIA E-mail: [email protected]

A method is suggested of computer gluing of 2D projective images of the same object obtained from different points in the space. This problem is well-known in computer geometry [5, 6, 11. The suggested method is based on a general approach to recognize similar fragments in perturbated sets of objects which was suggested by A. T . Fomenko and the author [ 3, 41. The corresponding algorithm has a linear complexity with respect to the total number of pixels in the images and t o the number of groups of values which appear in the pixels.

1. Introduction One of the important problems in modern computer geometry is a problem of creating efficient computer algorithms for gluing together 2D projective images of the same object obtained by central projection from different points in the space. Such a problem arises in multiple view geometry, in stereophotogrammetry, etc. See for example [ 1, 5, 61. The solution of this problem requires computer algorithms of the recognition of conjugate points. But algorithms of this kind are either not very efficient or require specific additional information which in many cases may not be available. From purely geometrical point of view this problem could be formulated as a problem of computation of a projective mapping F , which bounds two (in general case - unknown) domains D1 and D2, which belong to the same f i n e coordinate map of a projective plane RP2 (see Figure 1). In order to solve this problem one must have an ability to recognize some pairs of points * MSCZOOO: 65D18, 51N05, 68T10. Keywords :computer geometry, multiple view computer geometry, computer vision, comput er graphics , projective geometry, pat tern recognit ion, stereophot ogrammet ry, pro j ective mappings.

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which correspond to each other by means of an unknown projective mapping F. In applications such points are sometimes called conjugate points with respect to F or simply conjugate points. We will follow this terminology.

F=?

Figure 1.

In this paper we suggest an algorithmic method of computer recognition of conjugate points - mostly for the case when the given 2D images are color (which implies that there is a sufficiently large number of possible values which could appear in the pixels of the images). This method is based on a general approach to the recognition of similar fragments in perturbated sets of objects which was suggested by A. T. Fomenko and the author [ 3, 41. The corresponding algorithm has a linear complexity with respect to the total number of pixels in the images and to the number of groups of values which appear in the pixels. We assume that the rule of making such groups was fixed at the very beginning.

-

It is known from projective geometry that in order t o completely define a projective mapping F : RP2 RP2 it is enough to know (absolutely precisely) two sets, each consisting of four arbitrary points in RP2, such that one of these sets is mapped to another by mapping F. In practical situations, however, one can not determine such points with absolute precision. They are obtained with some error. So, it is necessary to make estimations of the stability of the computed value of F with respect to admissible perturbations of the initial data. Such stability could be characterized by the value of the determinant of the coefficient matrix for the system of linear

321

equations which determine the elements of F. In Section 3 this determinant is calculated which makes it possible to formulate a rule of optimal choice of the configuration of conjugate points.

2. A method of recognition of conjugate points, based on the distribution of shifts between bounded patterns Assume that we have two photo (or video) images El and E2, which were obtained by two different cameras. Assume that the same 3D object X appears on both images El and E2. It means that we have two images of the same object X , which were obtained by central projection from two different points in the space by means of two different, generally non-parallel planes of projection (see Figure 1). The configuration of the centers and planes of projection is unknown. It is also unknown whether the pictures of the object X cover the whole images E l , E2 or only some parts of them. We will call two points (in the discrete cilSe - pixels) of the images El and E2 respectively conjugate if the same point of X was projected to these two points. Let us assume that both images El and E2 are digital or were already digitized. Assume that each of them is represented by a square matrix of size n x n. The value of the matrix element (ij) is a vector e; which characterizes the balance of colors and the intensity at the pixel (zj). Assume that the set of possible values of ef consists of many sufficiently different values. In our case we consider two values as sufficiently different if they can not (except for the cases of influence of random perturbations) appear in conjugate points of images El and E2. We assume that the set of possible values of e: was divided into finite quantity m of groups of values close to each other (i.e. we consider grouped sample). We will call these groups patterns and denote them by a1 , . . . ,a,. We denote the set of all possible patterns by

I = {ul,...,um}. Each of the images El,. .(Ic = 1 , 2 ) is represented as an n x n matrix with pattern numbers 1 L e y 5 m as matrix elements. We will assume that m is large enough. It will be so, for example, in the case of color images. In the case of black and white or grayscale images it is possible to increase the value of m by considering the whole block d x d, (d > 1) of pixels as an elementary cell of matrix Ek (Ic = 1,2).

322

Our purpose is to build an efficient algorithmic procedure of the recognition of conjugate cells of the images El and E2 in the situation when there is no a priori information about the object X and its location on the images El and E2. All information concerning the conjugate cells and their vicinities should be taken from the original images El and E2 which are ordinary pictures and do not carry any additional facilities for recognition. In this article we consider the case when the vertical axes of the rectangular images El and E2 respectively approximately correspond to each other. It will be so, for example, if the columns and rows of both matrices El and E2 approximately represent horizontal and vertical planes in the space. This is the case for many applications when the cameras are positioned either horizontally or vertically. If not, the algorithm could be modified to take in account a possible rotation. We will follow the non-parametric approach to the recognition of similar fragments suggested in the works of A. T. Fomenko and the author [ 3, 41. Assume that the size n of the matrices E l , Ez has the form n = N p for some natural numbers N and p . Consider the division of the matrices E l , E2 into square blocks of size p x p containing p 2 matrix elements each. The number of such blocks in El or E2 is equal to N 2 . We will denote them by E r p , 15 k 5 2, 15 a,@5 N :

Ek"P= {f$ : ( a - l)p < 2 5 a p , ( p - l)p < 2 5 P p } . In our algorithm we will interpret the blocks EFp as fragments consisting of pixels sufficiently close to each other. The number p (block size) is a parameter which should be chosen according to the concrete application. We will assume that p 2 2.

Definition 2.1. The shift of two cells e? E El and ebi' E E2 is the vector p with two natural numbers as components: .. ., p(ei2,ei3 ) = (2'-2,j'-j). (2.1) .I

Definition 2.2. We call two patterns ni,n.j E I bounded (with respect to the pair of matrices El and Ez with fixed division to blocks) if they appeared in the same block of any of these two matrices: 3 k ~ { 1 , 2 ) , 3 a , @ E { 1 , ..., N ) :

If the patterns by ni

-

ni.

ni,nj

E I

ni,njE~kQP.

(1 5 i , j 5 rn) are bounded we will denote this

323

., If two cells ey E El and e i 3 E .I

contain a pair of bounded patterns res., pectively we will also call these cells bounded and denote this by e y ei’ . E2

.I

N

In particular, any pattern which appeared in any of the images E l , E2, is bounded with itself with respect to matrices El ,E2,. We consider the cells and patterns which appear in these cells as elements of the matrices E1,E2 (or their blocks) and use the same symbol ”E” in both cases. We will start with the following construction. Using given matrices E l , E2 consider a random choice of two cells in El and E2 respectively. The corresponding probability space R, C, P is defined as follows. Let R = El x E2, C = 2” and P be the uniform distribution on R given by P{w} = l / n 2 for any w E R. Denote the cell randomly chosen from El by e y ( w ) , and the cell randomly chosen from E2 by eg”(w). Indices i = i ( w ) , j = j ( w ) , i’ = i’(w),j‘ = j ‘ ( w ) are random here but for simplicity we omit the argument w . Denote by bl = b l ( w ) the pattern which appeared in the first chosen cell ey((w) E El,and by b2(w) the pattern which appeared in the second chosen cell egj’(w) E E2. Let us define the random variable ( = ( ( w ) on R , C , P which is the shift between the chosen cells:

(2.2) Then with probability one 5 takes values in the domain of the natural 2dimensional lattice: {(il, i 2 ) : 1 - n I i , j 5 n - 1). Assume that there exists at least one pair of bounded cells belonging to El and E2 respectively. This assumption is natural in our case, i.e. when El and E2 represent two different images of the same object X. In this case, even in the presence of some perturbations, there should exist such a pair of patterns close to each other (i.e. belonging to the same block) in one image that at least one of them appears in another image also. This is enough for existence of bounded pairs of patterns. This assumption implies that the probability P(A) of the event A is greater then zero: A = {W : b l ( w ) b 2 ( ~ ) } . (2.3) Event A means that the chosen pair of patterns is bounded (see Definition 2.2). The first stage of our algorithm of the recognition of conjugate points includes a procedure of elimination of comparison of most pairs of fragments t ( w ) = p ( e y ( w ) , e:j’(u)>.

-

324

belonging to El and Ez respectively. This elimination is based on a preliminary analysis of the following two distributions:

fo(i,j) = P ( f ( 4 = ( i d } , f i ( i , j ) = P{((w)= (i,j)I A }

(2.4)

(1- n I i , j

I n - 1)

(2.5)

Here fo denotes the unconditional distribution of the above defined random variable f on the natural lattice in R2, and f1 denotes its conditional distribution under the condition A defined in (2.3). We will need the explicit form of the function fo. It is given in the following two lemmas.

Lemma 2.1. For any 1 - n 5 i , j 5 n - 1 the following relations hold:

Proof. Denote by V the clockwise rotation of a square matrix to the angle of 90" :

The transformation V is an automorphism on the set of cells of a matrix Ek. Therefore V could be considered as an automorphism of R = El x Ez which in the discrete case always preserves uniform distribution: P{Vu} = P{w}. Furthermore, if both matrices El and E2 were rotated by the same angle, then the shift vector between any pair of cells belonging to El and Ez respectively rotates by the same angle. Therefore:

Similarly:

fo(i,j)

= P { f ( V 2 W ) = (i,j)}= P

{ f ( w ) = (-2, - j ) } = h

- 2 ,

-j),

fo(i,j) = P ( f ( V 3 W ) = ( i , j ) } = P { f ( w >= (2, - j ) } = fo(i, 3). This completes the proof of the lemma.

0

According to this lemma, in order to find the distribution fo it is enough to calculate fo(i,j) for i , j 2 0. This is done in the following lemma.

325

Lemma 2.2. Let 0 < i , j _< n - 1. Then f o ( i , j ) = P { ( ( w ) = ( i , j ) }=

n2

1 2 fo(i,O) = - - -, n2 n3

fO(0,O) =

1 2.

(2.10)

The distribution fo does not depend o n the entries of matrices El and Ez. It is determined b y their size n only.

Proof. Let us prove (2.7). The probability that the shift ( i , j ) appears between two chosen cells e y ( w ) and e i j ‘ ( w ) according to the uniform distribution P is equal to the ratio of the number of appropriate chances to the total quantity of chances. The total quantity of chances is equal to n4 (the number of pairs of cells belonging to El and Ez respectively). In our case the number of appropriate chances is equal to the number of such cells in E l , that there exists another cell in E2 such that its shift from the first one is equal to ( i , j ) . It is clear that if such a cell in E2 exists then it is defined uniquely by the .first one. Notice that the number of cells in a square matrix n x n such that there exists another cell shifted from it by ( i , j ) ,is equal to n2 - ni - n j ij. Consequently we have

+

n2-ni-nj+ij 1 i+j ij -. n4 n2 n3 n4 Relations (2.8)-(2.10) could be proved by similar considerations. This completes the proof of the lemma. 0

fo(6.i)

+

=

Comment. It is easy to see that the function fo does not change if the shift vector between the cells e y E El*’ and eij’ E EY’O’ is defined by the following formula instead of (2.1): p ( e y , e;j‘)

= ((y’ - ( y , ~ ’- PI).

(2.11)

The difference between f l and fo has a spike at the values of typical shifts between the conjugate cells of the matrices El and E2 [ 3, 41. This spike could be recognized by standard statistical procedures. In this way the typical shift between conjugate cells is determined. The suggested algorithm of determining the conjugate cells in the matrices

El and E2 includes two stages. At the first stage the functions fo and f l

326

are calculated and compared by statistical procedures. As a result of this comparison the typical shifts between bounded patterns (with respect to El and E2) are determined. They are interpreted as shifts between conjugate cells [ 3, 41. Algorithmically, this stage of calculations could be done during one path through the set of the cells of the matrices El and E2. For fixed N and m the complexity of the corresponding calculations is linear with respect to the total number of cells 2n2. At the second stage the conjugate pairs of pixels are determined finally. At this stage we do not need to compare all pairs of fragments of the matrices El and E2 respectively. It is enough to consider only pairs of blocks (EYp,E;’@’),which contain sufficiently many such bounded pairs of cells that they are shifted to the typical values, determined at the first stage. This approach significantly decreases the amount of pairs of fragments in El and E2 which should be analyzed for the existence of conjugate cells. For the final determination of conjugate cells the standard statistical procedures (which are commonly used in this situation) could be used. But in many applications it is enough just to find the intensity peaks in both fragments under consideration.

It should be stressed that we do not assume that the representations of the object X in E l, E2 could be transformed to one another by some shift or, more generally, by some affine mapping. This transformation is assumed to be projective. But according to our assumptions there will still exist a range of typical shifts between conjugate points (see Figure 2). It is because of the assumption that the rows and columns of both matrices E l , E2 approximately represent horizontal and approximately vertical planes in the space (consequently images El and E2 are not rotated by a large angle one with respect to other). Otherwise our algorithm requires a certain modification to take into account the angle of rotation. Let us discuss briefly the problem of the statistical determination of significant spikes of the difference ( f l - fo) - i.e. such spikes which correspond to the typical shifts between conjugate points. To do this we need to eliminate random spikes of the difference (fl - fo). In order to analyze random spikes let us assume that there are no conjugate points in the given images El and E2. Then it is natural to assume that El and Ez are independent random elements defined on some probability space ( Q l , E l , P l ) . In this case the random variable 6 and event A (see (2.2), (2.3)) could be defined on the product ( Q I ,C1, P I )x (a,C , P ) .If El and E2 are independent then 6 is independent from A. Consequently the

327

X

Q&)

4-.

Figure 2.

distributions fo and f1 considered on 521 x 521 will coincide. It follows from lemma 2.2 that fo in this case will be determined by the same formulas (2.7)-(2.10). Consequently, the distribution f1, defined on the product 521 x 52, will be determined by formulas (2.7)-(2.10). For a concrete w1 (i.e. conditionally for concrete matrices El and E2) the distribution f1 can differ from fo, but if n. is large enough then this difference will be small for a wide class of distributions PI (according to the central limit theorem). In applications the value of n. is usually greater then 1000 which is enough to make random spikes of the difference f1 - fo much smaller than the spike corresponding to conjugate points. Lemma 2.3 below suggests a possible method to eliminate random spikes of f1 - fo. We will prove it under the following assumptions. Assume that the given pair of square matrices (El,&) of size n x n is a result of some (arbitrary) stochastic experiment (01,C1, P I ) . Without loss of generality we will assume that (E1,Ez) = wy E 521 is an elementary event of 521 and that an arbitrary w1 E 52 could be represented by a pair of square matrices of size n x n which contain natural numbers in the range 1,..., m as their matrix elements. Consider the above described construction (R, C, P) of the uniform random choice of two cells from the matrices w1 = ( E l ,Ez) Then on the product (R1, C1, PI) x (52, C, P) we can define random variable 5 and event A by

328

the same relations (2.2) and (2.3). Let us define the functions fo, f1 (i.e. unconditional and conditional distributions of <) by the relations similar to (2.4), (2.5) with the only difference that we take the product measure PI x P instead of the measure P . It follows from lemma 2.2 that the function fo in this situation will still be defined by the same formulas (2.7)-(2.10). Below C J ~and w will denote not only elements of the probability spaces R1 and 01, but also subsets of the form w1 x R and w x R1 in 01 x RI. Let us fix some E > 0, by

1 - n 5 i, j 5 n - 1 and define anevent on R1 x R 1 (2.12)

A t = { ( w ~ , w:)P {[ = (i7j)lA7u1}- P {< = (i,j)lA} 2 E } .

Let us define the functions f l ( i , j ) - fo(i,j) by the same formulas (2.4), (2.5) as above. If and A are independent then the event A:j consists of such elementary events (El,E2) x w, that the difference f l ( i , j ) - fO(i,j), calculated from ( E l , E2) at the point ( i , j ) is not less then E. Then the following lemma holds.

<

Lemma 2.3. A s s u m e that 1- n 5 2 , j 5 n - 1 :

E and A are independent. T h e n

QE

> 0, (2.13)

Proof. Fix some E > 0, 1 - n 5 i , j 5 n - 1. Denote by B the event B = {E = ( i , j ) ) . According to the assumptions of the lemma, events B and A are independent. Therefore using (2.12) we have:

329

The events B and A are independent, so the above formula implies:

This completes the proof of the lemma.

0

3. The stability of determination of the projective mapping using a configuration of conjugate points Let us recall some notions from projective geometry. For any nonzero vector x E R3 denote by 2 the straight line parallel to x and passing through 0. Any such line determines uniquely a point on the projective plane RP2 and could be considered as an element of RP2 : 2 E RP2. For a 2 E RP2 its homogeneous coordinates are coordinates of x in R3.The multiplication of homogeneous coordinates by an arbitrary coefficient X # 0 does not change the point in RP2. Homogeneous coordinates are denoted by ( X I : 2 2 : 53). The affine coordinates of a point 2 E RP2 are defined as follows. Fix some plane n in R3 such that 0 6 n and fix some basis on it. We say that 2 E RP2 belongs to the affine map n if and only if the line 2 intersects the plane n in R3.The affine coordinates of the point 2 E RP2 in the affine map n are defined as the coordinates of their point of intersection in the basis which was fixed on n. Assume that some orthonormal basis {el, e 2 , e 3 ) was fixed in R3.We will fix the affine map S3 which is generated by the plane 2 3 = 1 with basis {el, e 2 ) on it. Without loss of generality we assume that the domains which are mapped into one another by the unknown projective mapping F belong to the map 5’3. The affine coordinates in S3 of a point 2 = ( X I : x2 : 23) E 5’3 are (xl/x3,x2/X3)Consider four points in the space RP2. We will say that they are in general position if and only if any three of them represented by straight lines in IR3 passing through 0,do not belong to any plane.

Theorem 3.1. (1211 Assume that {PI, P2, P3, P4} and (91, Q2, Q 3 ,Q 4 ) are two sets of f o u r points from RP2 each. Assume that both these sets are in general position in RP2. Then there exists a unique projective mapping such that it maps P i into Qi for all i .= 1 , 2 , 3 , 4 . In order to analyze the stability of the computation of this mapping we need an explicit form of a system of linear equations for it.

330

We will denote a point from RP2 and any vector corresponding to it in R3 by a same letter. Assume that two sets of four points {P,Q, R , T } c RP2 and {P’,Q’, R’, T’} c RP2 in general position are given and all these points belong to the affine map S3. Denote by F = (fij) ( 1 5 i , j 5 3) the 3 x 3 square matrix which defines a linear mapping R3 R3 which corresponds to the projective mapping which maps Pi into Qi for all i = 1,2,3,4. Such a matrix is not uniquely defined - it could be multiplied by an arbitrary nonzero coefficient. In order to define it uniquely assume that

-

and F ( P ) = P’,

F ( Q ) = Q’,

F ( R ) = R’,

F ( T ) = T’.

(3.2)

We write:

The four vector relations (3.2) represent a system of 12 linear scalar equations with 9 unknown coefficients f i j of the matrix F and 3 unknown coordinates q;, r;, t;. It could be written in the following form:

AX = y, where

(3.4)

33 1

A=

p1p210 0 0 0 0 0 0 0 0 41q21 0 0 0 0 0 0-a, 0 0 r l r 2 1 0 0 0 0 0 0 0 -ar 0 t l t 2 1 0 0 0 0 0 0 0 0 -at 0 OOp1p210 0 0 0 0 0 0 0 Oql 42 1 0 0 0 - b , 0 0 0 0 0 7 - 1 ~ 2 10 0 0 0 -br 0 0 0 O t l t 2 1 0 0 0 0 0 -bt 0 0 0 0 01p1p21 0 0 0 0 0 0 0 OOq1q21-1 0 0 0 0 0 0 O O r l r 2 1 0 -1 0 0 0 0 0 00tlt21 0 0 -1

The proof of the following lemma is due t o E. S. Skripka.

Lemma 3.1. The determinant of the matrix A is proportional to the product of the areas of triangles A P Q R , A P R T , A P Q T , and AQ'R'T'. The relation holds:

Proff. for a Proff. for a Proff. for a =

(Q'D

+ R'F) b

-br 2

+ (T'E + R'F) b

-bt - (T'E + Q ' D ) 2

2

332

The remaining tree formulas can be proved similarly. This completes the proof of the lemma. 0

t"'

El Figure 3.

It follows from this lemma that in order to perform a robust calculation of the projective mapping with help of given conjugate points one should try to choose such a configuration of conjugate points that the product of the areas of 4 triangles built on them is as large as possible. References 1. R. Hartley, A. Zisserma, Multiple view computer geometry, Cambridge Univ. Press, Cambridge, 2000. 2. R. Hartshorne, Foundations of Projective Geometry, Lecture Notes Harward Univ., W. A. Benjamin, Inc., NY, 1967.

333

3. A. T. Fomenko, G. V. Nosovskiy, Recognition of original structures in mixed sequences, Trudy seminara PO vektornomu i tenzornomu analizu, Vol. 22, Moscow Univ. Publishing House, MOSCOW, 119-131, (1985), (Russian). 4. A. T. Fomenko, G. V. Nosovskiy, Recognition of original structures in mixed sequences, Trudy seminara PO vektornomu i tenzornomu analizu, Vol. 23, Moscow Univ. Publishing House, Moscow, 104-121, (1988), (Russian). 5. Photogrammetry, Ed. L. N. Kel’ et al., Nedra, Moscow, 1989 (Russian). 6. B. P. Horn, Vision of robots, Mir, Moscow, 1989 (Russian).

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ON THE TOPOLOGICAL STRUCTURE OF THE SET OF SINGULARITIES OF INTEGRABLE HAMILTONIAN SYSTEMS *

A. A. OSHEMKOV Moscow State University, Faculty of Mechanics a n d Mathematics, Department of Differential Geometry a n d Applications, Vorob'evy gory d. 1 , 119992 Moscow, E-mail: oshemkovQmech.math.msu.su

1. Introduction In this paper we consider topological properties of singularities of integrable Hamiltonian systems. Let us recall some definitions (compare [ 11). Let M2" be a symplectic manifold with symplectic form w and H a smooth function on M . The vector field w = sgrad(H) corresponding to the 1-form dH under the duality given by w will be called a Hamiltonian system on the phase space M with Hamiltonian H . We will call a function F an integral of the Hamiltonian system 21

= sgrad(H)

if it commutes with the Hamiltonian H under the Poisson bracket corresponding to the symplectic form w, i. e.

{ H ,F } = w(sgrad(H), sgrad(F)) = 0. * MSG.2000: primary 70H06; secondary 57R17, 57R20. Keywords :Hamiltonian system, symplectic manifold, characteristic classes. This paper was published in Russian in Topological methods in the theory of Hamiltonian systems, ed. A. V. Bolsinov, A. T . Fomenko and A . I. Shafarevich. Faktorial, Moscow 1998, ISBN 5-88688-040-2, and was translated by: S. Terzit, Faculty of Science, University of Montenegro, Cetinjski put bb, 81000 Podgorica, Yugoslavia, email: [email protected] and D. Kotschick, Mathematisches Institut, Ludwig-Maximilians-Universitat Miinchen, Theresienstr. 39, 80333 Miinchen, Germany, email: [email protected] 335

336

We will say that the Hamiltonian system v = sgrad(H) is integrable (in the sense of Liouville) if there are n functionally independent pairwise commuting integrals F1,. . . ,F, for which the vector fields sgrad(Fl), . . . ,sgrad(F,) are complete (note that because sgrad({F;, F j } ) = [ sgrad(Fi), sgrad(Fj)] the vector fields sgrad(Fl), . . . ,sgrad(F,) commute). An integrable Hamiltonian system defines an action of the group Rn on the phase space which can be described as follows.

Definition 1.1. Suppose that on a symplectic manifold M2" we are given an integrable Hamiltonian system with integrals F1,. . . ,F,. For any element X = (XI,. . . ,A,) E R" define a diffeomorphism p(X) : M -+M by the time one map of the flow of the vector field A 1 sgrad(F1)

+ . . . + A,

sgrad(F,) .

Since the vector fields sgrad(Fl), . . . ,sgrad(F,) commute and are complete on M, the map p : R" + Diff ( M ) is a homomorphism and defines an action p of the group R" on the manifold M.We call this action the Poisson action of the group R" defined by the functions F1,. . . ,F,. For any point 5 E M its stabilizer I', under the Poisson action p is a Lie subgroup of the group R". It is well known that any Lie subgroup of R" has the form Rk x 25'. Therefore the orbit 0, of any point x E M with stabilizer rz has the form Tzx R F k - ' . In particular, dim 0, = n - dim r,. The orbits of the Poisson action which are of dimension n will be called nonsingular orbits. The other orbits are called singular orbits. The union of singular orbits of a Poisson action defined by an integrable Hamiltonian system will be called the set of singularities of that integrable system. The set of singularities of an integrable Hamiltonian system can be described by the following map.

Definition 1.2. Let p be a Poisson action of the group R" on a symplectic manifold (M2",w ) defined by the functions F1,. . . ,F,. The map p : M2" -+. R" defined by p(x) = (Fl(x), . . . ,F,(x))is called the moment map of the Poisson action p. We say that a point x E M is a critical point for the moment map p if rank(dp(z)) < n. The image under the moment map of the set of critical points is called the bifurcation diagram. From the above definitions it is clear that rank(dp(x)) = dim 0,. Therefore the set of singularities of an integrable Hamiltonian system coincides with the set of critical points of its moment map.

337

Definition 1.3. ( [ 11) The decomposition of the phase space of an integrable Hamiltonian system into the connected components of the level sets of the moment map is called the Liouville foliation of the integrable system. The functions Fl, . . . ,F, which define a Poisson action of R" are constant on the orbits of the action. Therefore every orbit is contained in a leaf of the Liouville foliation. Any regular leaf of the Liouville foliation (i. e. a leaf which does not contain any critical points of the moment map) consists of exactly one nonsingular orbit; a singular leaf can consist of several orbits of different dimensions (in particular it can contain nonsingular orbits). In this paper we are going to consider 4-dimensional compact symplectic manifolds, although some of the results below can be formulated in much greater generality (see Remark 3.1 after Theorem 3.2 in Section 3). For Hamiltonian systems on 4-dimensional symplectic manifolds, i. e. for systems with two degrees of freedom, the integrability in the sense of Liouville means the existence of an additional integral F which is functionally independent of H . In this case the functions H and F define a Poisson action of the group R2 on the phase space M 4 . The stabilizer rz of any point z E M 4 under this action is isomorphic to one of the following six subgroups: R2,R x Z, W,Z x Z, Z, {e} of the group R2.The corresponding six types of orbits are: nonsingular orbits T2, S1 x R,R2;1-dimensional singular orbits S1,R; 0-dimensional orbits consisting of one point. We will call the 0-dimensional orbits singular points of the Hamiltonian system, and we will call the 1-dimensional orbits singular orbits. For integrable Hamiltonian systems with two degrees of freedom a leaf of the Liouville foliation is a connected component of a common level set of the functions H and F . Therefore we can consider the Liouville foliation on a fixed regular level set of the Hamiltonian H (i. e. on an isoenergy surface). The topology of Liouville foliations on isoenergy hypersurfaces has been studied in detail in the framework of the Liouville classification of integrable Hamiltonian systems initiated in papers of A. T. Fomenko (this theory is presented in detail in the books [ 1, 21). The main result of that theory is the construction of a topological invariant for integrable Hamiltonian systems which are nondegenerate in a natural sense (see the definitions in Section 2 ) . That invariant (the FomenkcZieschang invariant) classifies such systems on isoenergy surfaces up to homeomorphisms preserving the Liouville foliation. In fact, the FomenkoZieschang invariant describes firstly the topology of the Liouville foliation

338

in the neighborhoods of l-dimensional singular orbits and secondly it encodes the decomposition of the isoenergy surfaces in standard blocks which contain exactly one singular leaf of the Liouville foliation. Regular isoenergy hypersurfaces (by definition) do not contain singular points of the integrable Hamiltonian system. Therefore these points can not be described directly by the Fomenko-Zieschang invariant. Singular points of integrable Hamiltonian systems are studied in the papers of L. M. Lerman and Ya. L. Umanskii [ 51. They introduced a natural notion of nondegeneracy for such points (see Section 2) and described the structure of the Liouville foliation in the neighborhood of a singular leaf containing exactly one singular point. The case when a singular leaf of the Liouville foliation contains several singular points has been investigated in detail by A. V. Bolsinov and V. S. Matveev [ 31. In this way there is a description of the topology of the Liouville foliation in the neighborhoods of regular isoenergy surfaces and also in the neighborhoods of the leaves of the Liouville foliation which contain singular points. The present paper contains some results about the topology of Liouville foliations in the neighborhoods of the whole set of singularities of integrable Hamiltonian systems. More precisely, the paper describes some properties of the set of singularities and of its embedding in phaie space. In Section 2 we define the class of integrable Hamiltonian systems with nondegenerate singularities and formulate some well known properties of the set of singularities of such systems. In Section 3 we describe the classical construction relating degeneracy loci of sets of sections with the Chern classes of complex vector bundles. After that we use this construction to prove a statement about the “homology” of the set of singularities of integrable Hamiltonian systems. In Section 4 we prove some further properties of this set. 2. Nondegenerate singularities In this section we discuss the definition of nondegeneracy of singular points and singular orbits for Poisson actions of R2 on 4-dimensional symplectic manifolds. Let us consider first the singular orbits. A Poisson action of the group R2 is given by some integrable Hamiltonian system with Hamiltonian H and an additional integral F . Any singular orbit is contained in some isoenergy surface of that system. Therefore we can define the notion of nondegeneracy of a singular orbit in terms of the function F restricted to the corresponding

339

isoenergy surface. Let us recall the following definition.

Definition 2.1. ( [ 11 ) We say that the integral F is a Morse-Bott function (or Bott integral) on a regular isoenergy surface Q3 if the critical points of F restricted to Q form critical submanifolds in Q which are nondegenerate (in the sense of Morse theory). In fact this definition describes a class of singular orbits which can be called nondegenerate (since the set of critical points of the function F restricted to a regular isoenergy surface coincides with the set of singular orbits which are contained in that surface). Nevertheless this definition is not suitable for isoenergy surfaces which contain critical points of H . In addition it is convenient to formulate the definition of nondegeneracy in terms of the Poisson action, i. e. without choosing one of the functions as a Hamiltonian. Let us do this in the following way. For any point z E M consider the action p, of its stabilizer I?, on the manifold M4. This action is the restriction of the Poisson action p : R2 --t Diff(M) to the subgroup r, of R2. Since z is a fixed point for p,, this action induces a homomorphism of r, into the group of linear symplectic transformations of the tangent space T, M4. Denote this group by Sp(T,M4) and its Lie algebra by sp(T,M4). Since the differential of a homomorphism of Lie groups is a homomorphism of Lie algebras we have for any point z E M a homomorphism from the Lie algebra T,r, into the Lie algebra sp(T,M4), where e is the neutral element of the group R2. The construction described above can be applied to Poisson actions of any group. In the case when we have an action of the group R2 and the stabilizer r, is isomorphic to R or R x Z we obtain a linear map of R into the Lie algebra 5p(T,M4). The image of that map is an operator A, E sp(T,M4) which is defined up to multiplication by nonzero constants. Since the group R2 is Abelian, all the points of a singular orbit 0, containing x E M are fixed points of the action p,. This implies that the tangent space to the orbit 0, is contained in the kernel of the operator A,. It is well known (see for example [ 11) that the characteristic polynomial f ( t ) = det(A-tE) of any operator A E 5p(V), where V is a linear symplectic space, is a polynomial in t2. Therefore, in our case the set of eigenvalues of the operator A, has the form {O,O, 7 , - T } or (0, 0, i7,-ZT}, with 7 E R.

Definition 2.2. A singular (one-dimensional) orbit 0, is called nondegenerate if the corresponding operator A, has two nonzero eigenvalues. We say that a nondegenerate orbit 0, is elliptic if these nonzero eigenvalues

340

are purely imaginary and hyperbolic if they are real. Obviously, the above definition does not depend on the choice of the point x on the singular orbit Ox,since for any two points x and y belonging to the same singular orbit the operators A, and A , are conjugate.

Remark 2.1. Note that the critical circles of a Bott integral are nondegenerate singular orbits of the Poisson action in the sense of the definition given above (the minimal and maximal circles are elliptic orbits and the saddle circles are hyperbolic orbits). Nevertheless the trajectories of the integrable Hamiltonian system which lie on the minimal and maximal tori and Klein bottles of the Bott integral are degenerate in the sense of the above definition. Now let us consider the singular points of a Poisson action. If x E M4 is a singular point of a Poisson action of the group R2 then its stabilizer is the whole group R2. Applying the above construction to this case we get a homomorphism of the commutative Lie algebra R2 into the Lie algebra 5p(T,M4). The image of this homomorphism is a commutative subalgebra in 5p(TxM4).Let us denote it by C,.

Definition 2.3. ( [ 51 ) A singular point x E M4 is called nondegenerate if the corresponding subalgebra C, in the Lie algebra sp(TzM4)is a Cartan subalgebra. There are four types (up to conjugation) of Cartan subalgebras of 5p(TxM4) (which is isomorphic to the classical Lie algebra sp(2,R)). To these four types of Cartan subalgebras correspond four types of nondegenerate singular points of the Poisson action which can be described in the following way.

Proposition 2.1. (151) Let x E M be a nondegenerate singular point of a Poisson action of the group R2 given by functions H and F (assuming H ( x ) = F ( x ) = 0). T h e n in some neighborhood of the point x there are coordinates (p1,q17p2,q2)such t h a t x = (O,O,O,O), w = dplAdql+dpzAdqn, and the functions H and F have one of the following forms: (1) a singular point of type center-center

H = (P? + 47) H1 + (P; + 422) H2

7

341

(2) a singular point of type center-saddle

+ (pi+ d )H2 , F = Pl q 1 Fl + ( P i + 4 3 F2 ;

H

=pi

41 Hi

(3) a singular point of type saddle-saddle

H

F

(4)

= P l Q1

H1+ P2 q 2 H2

= P i q1 Fl

7

+ P2 q 2 F2 ;

a singular point of type focus-focus

F

= (P1q 1

+ P2 q2) Fl + (Pi42 - P2 41)F2 .

Here HI,H2,F1,F2 are smooth functions which in the point x satisfy the condition Hl(x)F2(z)- H2(x)FI(x) # 0.

Definition 2.4. We say that a Poisson action of the group R2 is nondegenerate if all its singular orbits and singular points are nondegenerate. Let us formulate some well known facts (see [ 1, 51) about the set of singularities of integrable Hamiltonian systems, which we shall use, in the following form:

Theorem 2.1. Let p be a nondegenerate Poisson action of the group R2 o n a compact symplectic four-manifold ( M , w ) . The set of singularities K of this action (i. e. the union of all singular points and singular orbits) can be represented in the form

K where

= (Pi U . . . U Pi) U (QiU . . . U Qm) U

TI, . . . ,T, are isolated singular points

(Ti U . . . U T,) ,

of type focus-focus, and

P I , .. . , f i ,Q1,. . . ,Q m are closed immersed 2-dimensional submanifolds in M4 satisfying the following properties:

ui,l

(1) all elliptic singular orbits are contained in 2 Pi, all hyperbolic singular orbits are contained in Qj; (2) all intersections of the submanifolds P i and Qj (including selfintersections) are transverse and occur only in singular points, the intersections of the form Pi n Pj contain only singular points of type center-center, the intersections of the form PinQj contain only singular points of type center-saddle, and the intersections of the form Qi n Q j contain only singular points of the form saddle-saddle;

Uy=l

342

(3) the restriction of the symplectic form w to the submanifolds Q j is nondegenerate.

Pi and

3. Gauss-Bonnet formula for complex vector bundles

Following the book [ 41, let us recall a classical result about the geometric interpretation of Chern classes of complex vector bundles. (This result is referred to as a Gauss-Bonnet formula.) Let n/r be a compact oriented manifold, E --t M a complex vector bundle of rank k, and D = ((TI,. . . ,D k ) globally defined smooth sections of the bundle E. Define the degeneracy set D,(a) to be the set of points 2 E M in which the sections u1, . . . ,up are linearly dependent, i. e.

D , ( u ) = { ~ I u l ( z ) A. . . A

~ ~ ( z ) = o } .

A collection of sections is called generic if for every p the section D,+I intersects transversely the subspace in E which is spanned by the sections 0 1 , . . . , up, so that away from D p ( a )the set DP+1(u)is a submanifold of codimension 2 (Ic - p ) , and if, moreover, integration over DP+l(o)\ D p ( a ) defines a closed current. In this case one can define an orientation on the smooth manifold D, \ D,-1. In a neighborhood of a point xo E D, \ D,-1 the vectors el = ul,... ,ep-l = up-l are completed to a frame ej for E and we write UP(2)

=

Cf&,

.j(.)

.

j

Near ICO the set Dp is an analytic submanifold defined by { f p = . . . f k = O } . Let aP be the orientation of D, near 20 for which the form

is positive with respect to the given orientation of M. The subset D, together with the orientation @, on D, \ Dp- 1 represents a cycle in homology which is called the degeneracy cycle of the sections u.

Theorem 3.1. ( [ 4 ] , Gauss-Bonnet formula) For a generic collection of sections u the Chern class c T ( E )is Poincare' dual to the degeneracy cycle Dk-r+l

Ofu.

The construction described above makes the link between the degeneracy sets of sections of a complex vector bundle and its Chern classes. Formally this construction does not apply to the set of singularities of an integrable

343

Hamiltonian system. Here (in the case of two degrees of freedom) we have two vector fields sgrad(H) and sgrad(F) on a real symplectic manifold M4, i. e. in our case the bundle under consideration (the tangent bundle of M4) is not complex. Nevertheless, on this bundle we can always define a complex structure compatible with the given symplectic structure. Thus, the vector fields sgrad(H) and sgrad(F) can be considered as sections of a complex vector bundle. We are interested in their linear dependence over R, not over @. But it turns out that for integrable Hamiltonian systems there is no difference. In this way we can apply the previous construction in our situation. Let us formulate the necessary statement. First we describe the complex structure on the tangent bundle of M4.

Definition 3.1. An almost complex structure on a symplectic manifold ( M 2 n ,w ) (i. e. a tensor field J of type (1,l)which at every point z E M satisfies J," = - I d z ) is said to be compatible with the symplectic structure ? if the bilinear form ( u ,u)on tangent vectors to M defined by (u,v ) = w ( u , J v ) is symmetric and positive definite (i. e. it is a Riemannian metric on M ) .

It is known that for any symplectic manifold there exists an almost complex structure compatible with the sjmplectic structure. Moreover, that complex structure is uniquely defined up to homotopy (i. e. the set of almost complex structures compatible with the given symplectic structure is contractible). A proof of this can be found for example in [ 71. In this way we can consider the tangent bundle of a symplectic manifold as a complex bundle by choosing an almost complex structure compatible with the symplectic structure. The Chern classes of this complex vector bundle are independent of the choice of almost complex structure since the set of such structures is contractible. Further, when speaking about Chern classes of a symplectic manifold we are going to assume that its tangent bundle is endowed with an almost complex structure compatible with the symplectic form. Let us prove the following simple statement:

Lemma 3.1. Let ( V , w ) be a linear symplectic space and J a (constant) complex structure o n V compatible with the form w , i. e. an operator J:V V such that J 2 = -Id" and w ( v , J v ) > 0 for every nonzero vector u E Consider a collection of vectors u = { u l , . . . , u k } in v. Denote by rkR U the dimension of the span of the vectors ~ 1 , .. . , U k and

-

v.

344

by rkc U the dimension of their span over C, where we consider V as a complex vector space with the complex structure given by J . Then, i f the span L of u1,. . . ,u k over IW is an isotropic subspace in V (i. e. wIL = 0 ) then rkw U = rkc U .

Proof. By definition of rkw U , we have rkwU = dimw L. Further, r h U coincides with the complex dimension of 2, where L is the real subspace of V spanned by u1,. . . , u k , Ju1,. . . ,Juk, i. e. rkc U = dime L

1

=-

2

dimw

.

In that way, t o prove the Lemma it is enough t o show that L = L @ J ( L ) , which is obviously equivalent t o the condition L n J ( L ) = (0). Assume that there is a nonzero vector v in the intersection L n J ( L ) . Then the vector J ( v ) also belongs t o this intersection since J 2 ( L ) = L. In particular we see that the vectors v and J ( v ) belong t o the isotropic subspace L. This is impossible because of the condition w ( v , J v ) > 0. The Lemma is proved. 0 Thus we can apply the construction of degeneracy loci to a pair of commuting Hamiltonian vector fields on a four-dimensional symplectic manifold. By Theorem 2.1 from Section 2 the set of singularities of an integrable Hamiltonian system with two degrees of freedom consists of a twodimensional complex which is the union of closed two-dimensional submanifolds immersed in the phase space M4 and some number of isolated points of type focus-focus. Let us temporarily ignore the isolated singular points of type focus-focus and refer t o the set of singularities without these points as the complex of singularities. On the submanifolds forming the complex of singularities we define an orientation using the symplectic form w in the following way: on the submanifolds P i filled by elliptic orbits we take the orientation defined by w , and on the submanifolds Qj filled by hyperbolic orbits we take the orientation defined by -w. The submanifolds Pi and Qj with these orientations realize two-dimensional integral homology classes of the manifold M4. Denote these classes by [Pi1, [ Qj ] E H 2 ( M 4 Z). , Denote the sum of all these classes by [ K 1:

+

+

[ K ]= [ P I ] ... + [ 9 ]

+ . .. + [ Q m ]E H2(M4,Z) .

[Ql]

345

We define the orientation of M4 by the form w A W .

Theorem 3.2. Let K be the set of singularities of a nondegenenzte Poisson action of the group R2 o n a compact symplectic four-manifold (M4,w ) . The class [ K ] E Hz(M,w ) is Poincare' dual to the first Chern class c 1 ( M 4 ) E H2(M4). Proof. We need only to check if the genericity condition is satisfied. In other words, we need t o check that if H and F are Poisson commuting functions defining a nondegenerate Poisson action of R2, then the vector fields sgrad(H) and sgrad(F) form a generic collection of sections of the tangent bundle T M endowed with an almost complex structure compatible with w . This check is not hard to do in a convenient choice of coordinates. For example, we can assume that in some symplectic coordinates (PI ,pa, q1 ,q 2 ) the Hamiltonian H is simply a function of pl and the integral F does not depend on the coordinate q1. By a direct computation it is easy to conclude that if the singular orbit is nondegenerate, then in all of its points the section sgrad(F) intersects the subspace spanned (over C) by the section sgrad(H) transversely. The Theorem is proved. 0 Remark 3.1. The construction given at the beginning of this Section applies to complex bundles of arbitrary rank. Therefore Theorem 3.2 can be generalized to the case of integrable Hamiltonian systems with more degrees of freedom. It is necessary only to define correctly the notion of nondegeneracy of a Poisson action (so that the transversality condition which is necessary in the proof of the Theorem is satisfied). Roughly speaking, with the correct definition of nondegeneracy Theorem 3.2 is true in the high-dimensional case. Remark 3.2. We can obtain another generalization of Theorem 3.2 by weakening the assumption of nondegeneracy of the Poisson action. We can allow the Poisson action to have so-called simple degenerate onedimensional orbits (see [ 5 ] ) . In that case some of the two-dimensional submanifolds which form the complex of singularities will be filled with singular orbits of different types (elliptic and hyperbolic). Nevertheless the rule given above for defining the orientation can be applied in this case. In other words, the orientation on the whole submanifold will be correctly defined if at the points belonging to elliptic orbits we define it by the form w , and at the points belonging to hyperbolic orbits we define it by -w.

346

4. Some properties of the complex of singularities of integrable Hamiltonian systems

Consider the immersion of the submanifold Qj which is filled by hyperbolic orbits of the Poisson action.

Theorem 4.1. Let K be the set of singularities of a nondegenemte Poisson action of the group R2 o n the symplectic manifold ( M 4 , w ) . A n y submanifold Qj (which belongs to the complex K and which is filled by hyperbolic orbits) has trivial normal bundle in the manifold M 4 . Proof. The restriction of the symplectic structure w to the submanifold Qj is nondegenerate (see Theorem 2.1 in Section 2). Therefore we can assume that in any point x E Qj the fiber of the normal bundle to the submanifold Q j in M is the skew-orthogonal complement (with respect t o the form w ) of the tangent space T,Qj. Recall (see Section 2) that for any point z which belongs to a nondegenerate hyperbolic orbit there is an operator A, E 5p(TxM4)defined up to multiplication by nonzero constants which has two zero and two nonzero eigenvalues of opposite signs. To the nonzero eigenvalues correspond two one-dimensional eigenspaces. Let us prove that these eigenspaces are contained in the skew-orthogonal complement of the tangent space T,Qj. Indeed, the tangent space T,Qj is symplectic, invariant under the operator A,, and contains an eigenvector corresponding to the zero eigenvalue (a tangent vector tangent to a singular orbit). From this it follows that the tangent space T,Qj corresponds to the eigenspace for the pair of zero eigenvalues of the operator A,. It is also easy to see that for any operator from 5p(T,M4) the subspaces corresponding to its two eigenvalues X and p such that X + p # 0 are skew-orthogonal with respect to w. Thus, in any fiber of the normal bundle to the submanifold Qj in the manifold M 4 there exist two one-dimensional subspaces. On the whole manifold Qj they form one or two subbundles of rank one of the (rank two) normal bundle. Passing to a four-fold covering 7r: Qj Qj we may assume that the bundles which are the pullbacks of the given rank one bundles over Qj are orientable (see for example [ 81). (It is enough to consider a four-fold covering since the normal bundle which contains these rank one subbundles is orientable.) Then the rank two bundle over Qj induced from the normal bundle of Qj by the same covering is trivial since it is orientable and has nowhere zero sections. From this it follows that the

-

347

normal bundle is trivial (see [ 81).

0

Let us formulate some statements about the intersection index of the submanifolds Pi and Qj. More precisely, we are going to consider the intersection index of the corresponding homology classes [ Pi],[ Qj ] E H 2 ( M 4 ) under the intersection form H 2 ( M , Z)x H2(M, Z) Z.We are going to denote the intersection index of homology classes a,P E H 2 ( M , Z ) by a.P E Z. Also let us denote by npi and nQj the numbers of selfintersection points of the (immersed) submanifolds Pi and Qj respectively. We are going to denote the Euler characteristic of a manifold X by x(X).

-

Theorem4.2. L e t K = ( P I U ... U P l ) U ( Q I U . . .UQ,)U(TlU ... UT,) be the set of singularities of a nondegenerate Poisson action of the group R2 on a compact symplectic manifold ( M 4 , w ) (with the notation of Theorem 2.1 in Section 2). For the intersection indices of the homology classes [Pi], [ Qj ] the following conditions are satisfied: (1) [ P i I . [ P j I > O , [ P i I . [ Q j ] 1 0 , [ Q i ] . [ Q j ] 2 0 ; (2) [ K I . [ P i ]= X(Pi) [ P i ] .[ P i ]- 2npi;

+

(3) [ K I .[ Q j l =

-x(Qj);

(4) x ( ~ I+. ) . . + ~ ( f i-)x(Q1) - . . . - ~ ( Q r n )= 2 ( x(M4)- s)Proof. Let us prove the statement (1). These inequalities follow from the fact that for every case the sign of the intersection of the corresponding submanifolds at any point of their intersection is exactly as claimed. Consider for example the intersection of the submanifolds Pi and Pj (the other cases are similar). By Theorem 2.1 from Section 2 the submanifolds Pi and.Pj intersect transversely in points of type center-center. Consider the tangent spaces to the submanifolds Pi and Pj at their intersection point x as subspaces of the space TxM4. Obviously these subspaces are invariant under the Poisson action. They correspond to the different pairs of nonzero eigenvalues of the operators from a Cartan subalgebra which corresponds to the singular point z under consideration. Therefore, similarly to the proof of Theorem 4.1 one can prove that these subspaces are skew-orthogonal with respect to the form W.

The intersection index of the submanifolds Pi and Pj at the point x can be defined (locally) as follows. Choose pairs of vectors e l , e2 and e3, e4 in the subspaces T,Pi and TxPj respectively, which define the positive orientations of these subspaces (i. e. for the vectors e l , e2, e3, e4 the conditions

348

> 0 and w(e3,e4) > 0 are satisfied since the orientations on the submanifolds Pi and Pi are given by w ) . Then the intersection index of the submanifolds Pi and Pi at the point x is equal to the sign of the expression w A w(e1, e2, e3, e4). Computing, we obtain w(el,e2)

1 2

- w A w(el, e2, e3r e4) = w(el7 e2) w(e37 e4) - w ( e l , e3) w(e2, e4)

+ w(e1, e4) 4 e 2 , e3) = 4 e 1 , e2) 4 e 3 , e4) ,

since the vectors e l , e2 are skew-orthogonal to the vectors e3, e4. The last expression is positive because of the choice of the vectors e l , e2, e3, e4, and it follows that at any intersection point of the submanifolds Pi and Pi the sign is equal to +l. Summing these signs over the intersection points, we obtain the claimed inequality. Let us prove statement (2). Assume first that npi = 0. Then the equality we want to prove can be written in the following way:

The Euler characteristic x(Pi) of the manifold Pi is the sum of the indices of the singular points of a vector field along the singular orbits. This vector field is 'zero only in the singular points and possibly on some orbits (this vector field can be zero at a point on an orbit only if it vanishes along the whole orbit). The zeros of the vector field along an orbit do not contribute to the Euler characteristic. It remains to note that in the equality we want to prove both the right hand and the left hand sides are sums of the same numbers (equal to fl) indexed by the singular points contained in Pi. Indeed, any singular point of type center-center contained in Pi contributes +1 to the right hand side of the equality as an intersection point- with some submanifold Pit. At the same time it contributes +1 to the left hand side as an elliptic singular point of the vector field. The points of type saddle-center contained in Pi analogously contribute -1 to each side of the equality. The manifold Pi can not contain points of type saddle-saddle (see Theorem 2.1 in Section 2). In the case when npi # 0 it is easy to understand that to any selfintersection point correspond two elliptic singular points of the vector field, and in this way any selfintersection point contributes two times +1 to the right hand side of the equality. The equality (3) can be proved in a similar way keeping in mind that in that case the normal bundle is trivial, which implies the equality

[ Q i l ~ [ Q i l = 2 ~ ~ ~ -

349

The equality (4) can be proved in the following way. Summing all the equalities in (2) and (3), some simple transformation reduces the equality (4) to the statement that the Euler characteristic of the manifold M 4 is equal to

Jcenter- center)

+ )saddle- saddle) + Ifocus - focus1 - Jsaddle- center] ,

where by I * 1 we denote the number of singular points of that type on the whole manifold M4. The last claim can be proved using the Morse theorem that the Euler characteristic of the manifold is the alternating sum of the numbers of critical points of given index for any Morse function on that manifold. In order to apply the Morse theorem consider for example the Hamiltonian as a Morse function on the manifold M4. Obviously all singular points are critical points of the Hamiltonian. Besides, from the explicit formulas given in Proposition 2.1 it is easy to see that all of them are nondegenerate (in the sense of Morse theory as critical points of the Hamiltonian) and that odd index arises only for points of type centersaddle. 0

Remark 4.1. Of course there are more restrictions on the homology classes and the Euler characteristics of the submanifolds P i and Q j than we have given here. For example not every twedimensional integral homology class of a four-dimensional manifold can be realized by a smoothly embedded sphere. Moreover, there exist estimates for the genus of twodimensional manifolds realizing a given two-dimensional homology class of a four-manifold. Some results on this topic can be found in [ 61.

References 1. A. V. Bolsinov and A. T. Fomenko, Introduction to the topology of integrable Hamiltonian systems, Nauka, 1997, (in Russian). 2. A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian systems. 1, 2. Geometry, topology, classification, Izdatelskii Dom “ Udmurtsk; Universitet l’, Izhevsk, 1: 444 pp.; 2: 447 pp., 1999, (in Russian). 3. A. V. Bolsinov and V. S. Matveev, Integmble Hamiltonian systems: topological structure of saturated neighborhoods of nondegenerate singular points, “Tensor and vector analysis. Geometry, mechanics, and physics ed. A. T. Fomenko, 0. V. Manturov and V. V. Trofimov Gordon and Breach, 31-56, (1998). 4. P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, 1978.

350

5. L. M. Lerman and Ya. L. Umanskii, Classification of four-dimensional integrable systems and Poisson actions of R2 o n extended neighborhoods of simple singular points I, 11, III, (in Russian). I: Mat. Sb. 183,No 12, 141176, (1992) (translation in English: Acad. Sci. Sb. Math. 77, No 2, 511542, (1994)); 11: Mat. Sb. 184,No 4, 105-138, (1993) (translation in English: Acad. Sci. Sb. Math. 78, No 2, 479-506, (1994)); 111: Mat. Sb. 186,No 10, 89-102, (1995) (translation in English: Sb. Math. 186,No 10, 1477-1491, (1995)). 6. R. Mandelbaum, Four-dimensional topology: a n introduction, Bull. Amer. Math. Society 2, 1-159, (1980). 7. D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford University Press, 1995. 8. J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, 1974.

Translators’ Notes:

Except for obvious misprints, we have translated the original article without any changes. In the following notes we comment on the contents of Oshemkov’s paper. 1. The second and third statement in Theorem 4.2 follow from the adjunction formula for symplectic submanifolds. For the third statement one uses, as in Oshemkov’s proof, that the normal bundle of every Q j is trivial. 2. Concerning the final Remark, there are many more results about embedded surfaces proved using gauge theory, which go beyond Mandelbaum’s survey.

LIErCARTAN PAIRS AND CHARACTERISTIC CLASSES IN NONCOMMUTATIVE GEOMETRY *

TH. YU. POPELENSKY AND YU. P. SOLOVJEV t Moscow State University, Faculty of Mechanics and Mathematics, Department of Differential Geometry and Applications, Vorobyovy Gori, 119992 Moscow, RUSSIA E-mails: [email protected], [email protected]

Various versions of characteristic classes based on Chern-Weil construction in noncommutative geometry are considered. We define characteristic classes for 2/2graded L i e c a r t a n pairs and prove their basic properties.

1. Introduction

This paper is devoted to certain constructions of characteristic classes in noncommutative geometry. In usual differential geometry characteristic classes are defined for vector bundles of particular type. To be more precise: Chern classes are defined for complex vector bundles, Stiefel-Whitney classes - for real bundles, Pontryagin classes - for oriented real bundles. In these cases characteristic classes are elements of the cohomology of the base. The theory of characteristic classes origins from the papers [ 2, 13, 14, 181. Now one uses several ways to construct characteristic classes (see [ 11, 12, 71). Also there are versions of characteristic classes with values in some extraordinary cohomology theories, such as K- theory and cobordism theories (see [ 15, 171).

It is possible to define Chern classes for smooth complex vector bundles over smooth manifold A4 by means of differential geometry. For this purpose one uses so called Chern-Weil construction which is described in Section 2. It * MSC2000: 81R60, 81T75, 58B34, 46L87. Keywords: characteristic classes, Chern-Weil construction, L i e c a r t a n pair, noncommutative geometry. t The authors were partially supported by RFBR grants N 02 - 01- 00578 and N 03 - 01- 0629.

35 1

352

is easy to see that Chern-Weil construction uses only the algebra of smooth functions C”(M) on manifold M instead of points of manifold M , and the space of smooth sections E = I?(€) of the bundle E M instead of points of bundle E , and so on. This observation allows to define characteristic classes of projective modules (that is an analog of vector bundles) over an abstract associative algebra A which is not an algebra of smooth functions on a manifold. Moreover, the algebra A has not to be commutative. The using of finitely generated projective modules as an analog of vector bundles is explained by Serre-Swan theorem [ 161. It states that there is an equivalence of the category of finite dimensional vector bundles over compact manifold M and the category of finitely generated projective modules over the algebra C - ( M ) of smooth functions on M . The equivalence is established by correspondence:

-

”

bundle E ”

H ”

space of smooth sections I?(€) ”.

The theory of characteristic classes for noncommutative algebras was developed by A. Connes, M. Karoubi, B. L. Feigin, B. L. Tsygan, Yu. J. Zhuraev, Yu. P. Solovjev, A. S. Mishchenko [ 3, 4, 9, 10, 8,5 , 6 , 19, 201. Let A be an associative algebra with unit over the field of zero characteristic. Also let E be a finitely generated projective module over A. To use ChernWeil construction it is necessary to define notion of connection on projective module E. Also there is natural condition: this definition should coincide with usual one for the case A = C”(M), E = I?(€) there E is vector bundle over compact manifold M. In ”classical” differential geometry there are two different definitions which are equivalent. Definition 1.1. A connection on complex vector bundle € is a C-linear

-

map

D : r(E)

r(E)@c-(M,@)W M ,el,

suchthat D ( s . f ) = D ( s ) . f + s @ ~ d f f o r a l l fECW(M,C)a n d s € I ’ ( € ) . Definition 1.2. A connection on a complex vector bundle E over manifold M is a map VX : I?(€) ---f I?(€) defined for all vector fields X on M . The map VX should satisfy following identities: (1) VfX+gY = f v x

+g v y ,

(2) Vx(sf)= VX(S).f

+ s . Xf,

353

for all f , g , X , Y , s, where f , g are smooth functions on M , and X , Y are vector fields, and s is a section of E. Here X f denotes the derivation of the function f along vector field X .

It is interesting that in the noncommutative case these two definitions give different theories of characteristic classes. Construction based on Definition 1.1 also uses notion of quasiresolution (differential calculus) of an algebra A. The quasiresolution SZ*(A) is an analog of de Rham complex of smooth manifold. There is no unique way to define quasiresolution of a given algebra A. Characteristic classes of projective module are elements of cohomology of quasiresolution factorized by some elements. This approach is considered in Section 2. To use Definition 1.2 it is necessary to define an analog of Lie algebra of smooth vector fields on the manifold. In Section 3 for this purpose one uses Lie algebra D of derivations of A. In this case characteristic classes are elements of cohomology of Lie algebra D. In Section 4 general Lie algebra equipped with mappings (i) A x L - L

(ii) L x A

-

A,

is used. These mappings should have properties which are similar to the properties of (i) C m ( M )x V e c t ( M ) by a function,

-

(ii) V e c t ( M )x Cm(M) a vector field.

V e c t ( M )-multiplication

-

of a vector field

Cm(M)- derivation of a function along

Such pairs ( A ,L ) are called Lie-Cartan pairs. We are developing theory of characteristic classes for Lie-Cartan pairs ( A , L ) equipped with Z/2grading. Characteristic classes are elements of cohomology of Lie algebra L. Contrary to the Section 2 and 3 we are developing theory for Z/2-graded projective modules. Also we are stressing exact analogy between theory of Chern characteristic classes for complex vector bundles and characteristic classes theories in noncommutative geometry. One denotes degree of the element z by 1x1. In all sections (except Section 5) Zgrading is used, that is 1x1 E Z.In Section 5 (which deals with Lie-Cartan pairs) Z/Zgraded objects are used, that is 1x1 E Z/2.

354

2. Characteristic classes of complex vector bundles In this section we are describing plan which is used in the following sections t o construct various versions of characteristic classes. Almost all proofs are omitted because they are just the same as proofs of similar statements from Section 3. Also the proofs for this section can be found in [ 121. Let M be a smooth closed manifold and let A be an algebra C"(M, C)of smooth complex valued functions on M . Denote complexification of the de Rham complex R* ( M )@ C as R* ( M ,C) . Also let E be a finite dimensional complex vector bundle over M and let E = I?(&) be a space of its smooth sections. The module E is a finitely generated projective A-module.

Definition 2.1. A connection on a complex vector bundle E is a C-linear map D : E + E @ A a l ( M ,C), such that D ( s . a ) = D ( s ) . a

+ s @ A da for all a E A and s E E .

Example 2.1. Let E = M x C" be the trivial vector bundle over M . Then the space of smooth section E is the space of smooth functions on M with values in C", that is E = A @ C" = A". Also it is easy to see that E @ A R1(M,C) = (R'(M,C))". Define D : A" (R1(M,C))" by formula D(f1, ...,fn) = ( d f i,... df,), where f k E A and d is the differential of de Rham complex. It is easy to check that D is a connection on the trivial bundle E .

-

The set of all connections on a given bundle I has no natural structure of linear space. Nevertheless, if D1 and 0 2 are connections on E then linear combination f D1 (1 - f) DZwhere f E A is also a connection on E.

+

Proposition 2.1. Let E be a finite dimensional vector bundle over manifold M . Then where exist a connection o n E . The proof of the proposition is based on the following observation. Every finite dimensional vector bundle E' is a subbundle of some trivial vector bundle E = M x C" of appropriate dimension. Also there is a fiberwise projection P such that fibers of the bundle E' are images under this projection of the fibers of the bundle &. Operator P maps the sections of the bundle E to the sections of subbundle E'. Therefore one can consider P as a linear endomorphism of E. It is not difficult to check that P is a projection in E and that im P = E'. Following statement finishes proof of Proposition 2.1.

355

Lemma 2.1. Let E be a vector bundle over manifold h4 equipped with connection D and let E' be a subbundle of E . T h e n the map D' : E' E' @ A R1(M,C ) given by formula D' = P o D i s a connection on E l .

-

The proof is by direct checking of the definition.

Proposition 2.2. There is a unique extension of the map D t o a C-linear mapping

D :E

a*(kf)

@A

E

4

@A

a*+l(M),

such that D ( x .e) = D ( z ) . O for all z E E

@A

R*(M), 0

+ (-1)121

x . do,

E R*(M).

The proof is similar to the proof of Proposition 3.4.

-- -

Definition 2.2. A curvature of connection D is the map

R = D o D : E @A R*(M)

E

G*+'(M).

Proposition 2.3. T h e map R : E @ A R*(M) homomorphism of right R*( M )-modules.

E

@A

R*+'(M) is a

The proof is similar to the proof of Proposition 3.5. Now let us consider compositions

RQ: E

@A

R*(M) --+ E

@ A o*+'(M).

Taking traces (see Appendix) one obtains elements tr Rq E R2q(M). It should be noted that elements t r RQ depend on the connection D on the vector bundle E.

Theorem 2.1. (a) Element tr RQ is a cocycle of degree 29 in the complex R* ( M ) . (b) Suppose that R' is a curvature of another connection o n the same vector bundle E . Then tr RQ-tr R" is a coboundary in the complex

R*(M). Therefore there is correctly defined cohomology class Ch,(&) = [(l/q!)tr Rq] in H 2 Q ( k f ) One . can show that up to multiple of 1/(27ri)q the characteristic class Ch,(E) coincides with homogeneous component of degree 29 of Chern character of the bundle E .

356

To be more precise we give corresponding formulas for small q :

and so on. Here Q ( € ) is Chern class of the bundle E . 3. Characteristic classes of projective modules over an

associative algebra In this section we define characteristic classes of finitely generated projective right modules over an associative algebra. We are following Karoubi book [ 81.

Definition 3.1. Let R be a field of characteristic zero. And let A be an associative algebra with unit over the field R. A differential quasiresolution of the algebra A is a differential graded algebra R*(A) over R such that: (1) Ro(A) = A, ( 2 ) d(w1 w2) = (dWI)W2 ( 3 ) d2 = 0.

+

(-l)IW1l

w1(dwz),

In particular, homogeneous components R'(A) are bimodules over the algebra A .

Example 3.1. Let M be a smooth closed manifold. Also let R = C, A = C"(M,C), R*(A) = R*(M) @I C. All necessary conditions are easy to check. Moreover in this example the equality w1 w2 = (-l)lwlllwzI w2 w1 holds for arbitrary w1,w2 E R*(A). In general case this equality does not hold. Another important example is an universal differential quasiresolution Otni,(A). It is defined as follows.

-

A, where m is the Let 0tniV(A) = A, Rt,,,(A) = ker m : A @IcA multiplication in the algebra A. It is obvious that Qiniw(A)is an A-bimodule. For all n 2 2 define Rz,,,(A) as the tensor product OAniw(A)@ A . . . @ A Rtniv(A),where Rt,,,(A) is taken n times. It is clear that RZniw(A)= @,>o REni,(A) is a graded algebra.

-

357

Now let us define differential d : RZn,,(A) RZZ:,,(A). For n = 0 define d(a) by formula d ( a ) = 1 @ a- a 8 1. It is easy to see that Leibnitz identity d(ala2) = d(al)az - a1 d(a2) holds for all a l , a2 E A. For other elements of R;,,,(A) differential d is defined as follows. Any element w E Rt,,,(A) can be expressed as a finite sum w = C bi dai, where ai, bi E A and C biai = 0. Indeed by definition w = C bi @ ai, where ai, bi E A and C biai = 0. Then

=

x

bi(1 8 ai - ai @ 1 ) =

bi dai,

as was stated. Note that this expression is not unique. Define dw where w = C bi dai E Rtni,,(A) by formula dw = C dbi dai in Rini,,(A). For other elements REni,,(A) differential d is defined by Leibnitz identity d(wlw2) = (dW1)wz (-l)lwll w1 dwz. It is not difficult to check that this definition is correct.

+

The quasiresolution Rtniv( A ) is called universal because of the following statement.

Proposition 3.1. Let R*(A) be a digerential quasiresolution of the algebra A . Then there exists unique homomorphism of diflerential graded algebras I : Rtni,,(A) R*(A), which is identity on Rtni,,(A) = Ro(A) = A .

-

Proof. Here to be more precise duniv denotes differential of Rtni,,(A), and (2 denotes differential of R*(A). Let us check uniqueness of I . By assumption I ( a ) = a. As I is a homomorphism of differential graded algebras, then, in particular, identity d o I = I o duniv holds. Consequently

I ( w )=

C bi, dai,

@A

. . . @ A bik h i , ,

where w = C bil dZLniVail@ A . . . @ A bi, dunivaik, and homomorphism I is unique. In order to prove existence one should define I by the last formula and then to check correctness of this definition. 0 Let us define connection, curvature of connection and characteristic classes we define for projective modules over the algebra A. The algebra A is not to be commutative. Therefore we consider right A-modules. Let W(A) be a differential quasiresolution of the algebra A.

358

-

Definition 3.2. A connection on finitely generated projective right Amodule is a R-linear map D : E E @ A R1(A), such that D ( s a ) = D(s)a f

S @ A da

for all s E E , a E A.

Remark 3.1. It is interesting that if there exist a connection on the right A-module for universal quasiresolution then E is projective. Proposition 3.2. Let E be a right A-module equipped with map D : E -+ E @ A RAniv(A), such that D ( s u ) = D(s)u s @ A da for all s E E , a E A. Then module E is projective over A.

+

-

--

Proof. Consider short exact sequence of right A-modules

o

E

R;,,,(A)

.L~ 6 9 A5 z E

0.

Here homomorphism j is defined by formula j ( s @ A d f ) = s 69 a - sa 69 1, and homomorphism m by formula m ( s @ u )= sa. Our aim is t o show that E is a direct summand in E @c A. It is sufficient t o construct splitting of the considered short exact sequence. Consider a map S : E E 695 A given by S(s) = s @ 1 + j ( D s ) .

-

Let us check that S is a homomorphism of right A-modules. Indeed,

+

+

69 1 j ( D ( s f ) )- [ s 69 1 j D ( s ) ]f = sf 69 1+ j ( D ( s ) f s 69df) - s 69 f -jD(s)f = sf691 - s 6 9 f + j ( s 6 9 d f ) + j ( D ( s ) f ) - (jD(s))f=O.

S ( S f ) - S(s)f = sf

+

Finally it is easy to show that mS = i d E , therefore short exact sequence.

s is a splitting of our

Example 3.2. Let E be a free A-module, that is E = A”. Then E @ A R*(A) = A” @ A R*(A) = (R*(A))”. Under this isomorphism define map Do : E E @ A R*(A) by formula Do(f1,. . . ,fn) = (dfi,. . . , d f n ) , where fi, . . . ,fn E A. Obviously Do is a connection on the module E .

--

The set of all connection on a given module E has no natural structure of a linear space. Nevertheless, if D1,DZ are connections on the module E then for every f E A the map f Dl (1 - f) 0 2 is also a connection on the module E. Also difference D1- Dz is a homomorphism E E @ A R1(A) of right A-modules.

+

-

359

-

In particular it means that any connection on a free module is equal to DO+I?, where I? is a homomorphism A” (R1(A))%of right A-modules. Note that I? can be written as a n x n matrix with entries in R1(A).

Proposition 3.3. Let E be a finitely generated projective right A-module. Then there is a connection o n E . Proof. As every finitely generated projective A-module is a direct summand in a free A-module of finite rank, it is sufficient t o prove the following statement. 0 Lemma 3.1. Let E be a finitely generated projective right A-module equipped with the connection D. And let E’ be a submodule of E and let P E EndA(E) be a projection in E such that P2 = P, E’ = im P . Then D’= P o D is a connection o n E’. Proof. Consider s’ E El, f E A. As s’ E El, then s’ = Ps for some element s E E. Let us check the equality D’(s’f) = D’(s’)f f s’ @ df. Indeed, D’(s’f) = PD(s’f) = P(D(s’)f S’ @ d f ) = P(D(s’))f Ps’ @ d f = D‘(s’)f S’ @ df,

+

+

+

because P(s‘) = P P ( s ) = P ( s ) = s’ .

D

The connection of form PDo is called Gmssmannian connection. Let us given an algebra A, a quasiresolution R*(A), a finitely generated projective right A-module, a connection D on E .

Proposition 3.4. There is unique extension of the connection D to a Rlinear map

D :E

@A

R*(A)--+ E @ A R*+l(A),

such that

D(“.e) = D ( ~. e) + ( - I ) I ~ ‘ ~ . de for a l l x E E @ AR*(A), 0 E R*(A).

Proof. Define a sequence of R-linear maps

360

where j 2 0, as follows. Let D(O) = D. For j 2 1 and define D ( j ) by formula

D(j)(S@ A W ) = ( D s ) W

5

E

E,w E @ ( A )

+ s @ A dw.

It is easy to check correctness of this definition:

D ( ~ ) ( s@uA W )

+ + su @ A d w = (DS)UW + s @ A d(aw) = D(’)(s @ A a w ) .

= (DS)UWs @ A (da)w

Let us check that equality D(i+j)(xw)= D(a)(z)w+ ( - l ) i a : d w is valid for all x E E @ A Ri(A), w E Rj(A). Without loss of generality one can suppose that x = s @ A 8, where x E E , 8 E R i ( A ) . Indeed, ~(i+j)((s

e)w) = ~ ( a + j ) ( sg Aew) = ( D s ) e w + s

d(8w)

+ s @ A (d8)w + ( - I ) ~ s 8 d w - ~ ( z ) ( @s A elW + (-1li(s @ A 8)dw = ( D ( ~ ) z )+ w( - 1 1 ~xdw.

= (Ds)8w

Now we shall denote all the maps D ( j ) as D ;it makes no confusion.

Definition 3.3. A curvature of a connection D on a module E is the composition R = 0’ : E @ A S2*(A) --+ E @ A R*+’(A).

-

Proposition 3.5. The curvature R : E @ A R*(A) E right R* ( A )-module homomorphism, that is the equality R ( x w )= R ( x ) w holds for all x E E

@A

R*(A), w

@A

R*(A) is a

E R*(A).

Proof. Let x E E @ A Ri(A) and w E Rj(A). Make the following calculation: D2(xwW)= D ( D x . w ( - l ) a x . d w ) = D 2 x .w ( - l ) i + l D x . dw 0 + ( - l ) i ( D ~dw . (-1)Zx. d ’ ~ = ) 0’2. W .

+

+

+

Therefore there is an endomorphism R of the right R*(A)-module E @ A R*(A). It is easy to show that E @ A R*(A) is a projective R*(A)module. Consequently traces of the compositions R are well defined, that are elements tr RQE fi*(A). Definitions of the trace and an algebra fi*(A) are given in Appendix.

361

Theorem 3.1. Let E be a finitely generated projective right A-module with a connection D,and let R be a curvature of the connection D. Then tr RQ is a cocycle in Q*(A). Proof. First we consider the case of Grassmannian connection on E . This means that D = PDo, where P is a projection in a free A-module A" of an appropriate rank and E = im P. One can consider P as a matrix with entries in A. Taking into account the isomorphism A" @ A R1(A) = (O1(A))", one can consider the connection D as a composition D = P o d. Let us Calculate the extension of D to a map D : E @ A R*(A) + E @ A R*+l(A). By definition for s E E and w E R*(A) we have

D(s @ A W ) = DS @ A w + s @ A d~

= Pds @ A w

+ PS@ A dw = P d ( s

@AW).

Now let us calculate the curvature of D:

O*(A))) = P ( d P ) d ( S @ A W ) f PPd2(s@ A W ) = P(dP)d(s@ A W ) .

R ( S @ A W ) = D ( P d ( S @ A W ) ) = Pd(Pd(S @ A

Let us note that s @ A w E E @ A R*(A), therefore s @ A w = P ( s @ A w ) . Consequently d( s @ A W ) = (dP)(s @ A W ) -I- Pd( s @ A w ) . Therefore R ( S @A W)=

P(dP)(dP)(S @ A W ) + P(dP)Pd(s@ A W).

Now check that P(dP)P= 0. Indeed, P2 = P , hence (dP)P+P(dP)= dP. Apply to the both sides of the equality projection P and obtain

P(dP)P + PP(dP) = P(dP), hence P(dP)P = 0,

and

R ( s @ A w ) = P(dP)(dP)(s@ A W ) . Now calculate the compositions RQ.Calculation

P(dP)(dP)P= P(dP)(d(PP)- P ( d P ) ) = P(dP)(dP)- P(dP)P(dP)= P(dP)(dP), shows that RQ= P(dP)2Qand d(RQ) = (dP)2Q+1. Now we can show that d(tr Rq) = 0. Consider homomorphism Q = 2P - 1. We have equalities Q2 = 1 and QdP = -(dP)Q. Then (dP)2qf1 = Q2(dP)2Q+1 = -Q(dP)2Q+1Q.

Therefore d(tr RQ)= -tr &(dP)2Q+1Q.

362

As cyclic permutation of operators under the trace sign does not affect the value of trace then d(tr R4)= -tr Q(dP)24f1Q= -tr Q2(dP)2Q+1 = -d(tr RQ). Hence d(tr R Q )= 0. Now let us consider the case of a free module E . As it was shown before arbitrary connection on a free module E has the form D = Do r = d I?. It is easy to obtain the expression for curvature (here z E E @ A R*(A)):

+

+

R ( ~=) D ( ~ + Z rZ)= ( d + r)(& + rZ)= ddrc+ r(dZ) + d(rZ) + r2Z = r(d5)

+ (dr) z - r(dZ)+ r2Z= (a+ r2)

Z.

a

Hence dR = a . r - r . = Rr - r R . By induction it is easy to show that d(RQ)= RQ. I? - I' . RQ.Therefore d(tr Rq)= tr dRq = tr (Rq. r

-

I?. Rq) = 0.

Finally, let us consider case of an arbitrary module E and an arbitrary connection D on E. Let E' be a projective right A-module such that E @ E' = A". Consider Grassmannian connection D' ,on El. Is is easy to show that D @ D' is a connection on E @ El = A". Denote the curvatures of connections D , D', D@D" by R , R', R" correspondingly. Obviously R @ R' = R". Hence tr RQ tr R" = tr R'". Is was shown before that tr R" and tr R'" are cocycles in n*(A), therefore t r RQis a cocycle also. This finishes proof of the theorem.

+

Theorem 3.2. Cohomology class of tr RQ in cohomology of the complex fi*(A) is independent of connection D on E . Proof. Consider differential graded algebra L* = @LZ,where Lo = R [ t ] is a polynomial ring, L1 = R [ t ]dt, and Li = 0 for all others i. Define differential d : Lo L1 by usual formula d ( P ( t ) )= P'(t)dt, where P'(t) is a formal derivative of the polynomial P ( t ) . Define map HO : L R as follows. Put H&o = 0 and

-

-

1

P ( t )dt .

Ho(P(t)dt)= 0

It is easy to check that

(dH0

+ Hod)(P(t))= P(1)

-

P(0) and

(dH0 + HOd)(P(t)dt) = 0 .

363

Now definemap H:L*@RR*(A)-SZ*(A) byformulaH(O@w)= Ho(B)w, where 8 E L* and w E SZ*(A).Here one considers L* @R S1*(A) as a tensor product of a graded algebras. Suppose x E L* @R R*(A) is equal t o the 8 @ w, where 6 E Lo, and put ~ ( 1= ) 8 ( l ) w and x(0) = e(0)w. Otherwise, that is x = 0 @ w, where 8 E L', put x(1) = x ( 0 ) = 0. Then it is easy to see that

(dH

+ H d ) z = ~ ( 1-)~ ( 0 ) .

-

The map H induces a map H : L* @fi*(A)

( d H + Bd) z

fi*(A) with similar property

= z(1) - x(O),

where z E L* @ fi*(A). Let us make following observation. Suppose z is a cocycle in L* @ fi*(A). Then z(1) - x ( 0 ) is a coboundary in fi*(A). Indeed, z(1) - x ( 0 ) = ( d H

+ Bd)z = d A ( z ) .

Let E' be a right projective A-module such that E @,Elis free A-module A". Consider on E' an arbitrary connection D'. Denote its curvature as R'. Direct sum D" = D @ D' is a connection on the free A-module E @ El. The curvature of this connection is equal to R" = R @ R'. Hence t r Rq = tr R"" - t r R'". Taking this equality into account we are left to prove that cohomology class of tr R"" is independent of D because tr R'" is independent of D by construction. Let us show that cohomology class of tr R'" is equal to zero for an arbitrary connection D" on the free A-module A" = E @ El. Let D" = d r. Consider free module (A[t])" over the algebra A[t]. Suppose r' = tr. Consider L* @R R* (A) as a differential quasiresolution for A[ t]. The map d tr is a connection (A[t])". Consider R = dr' E L* @R fi*(A). Then 77 = t r @ is a cocycle in L* @R fi*(A), with ~ ( 1 = ) t r R'" and ~ ( 0= ) 0. Taking into account observation made before we see that tr R'" = ~ ( l ) - q ( O )is a coboundary in fi*(A). This finishes the proof of the theorem

+

+

+

Let E be a finitely generated projective right A-module and let R be a curvature of some connection on E . Denote Ch,(E) = l / q ! [ tr RQ ] in H(fi*(A)). Theorems 3.1 and 3.2 claim that cohomology classes Ch,(E) are well defined and are independent of a connection on E . That means that Ch,(E) depends only on the module E itself.

364

4. Mishchenko-Solovjev-Zhuraev construction

This section is devoted to the construction of characteristic classes developed by A.S. Mishchenko, Yu. P. Solovjev and Yu. J. Zhuraev in [ 19, 201. Omitted proofs can be found in these two papers. In the Section 3 we have defined connection on a projective right A-module as a R-linear map

D:E

-+

E@~Q~(A),

+

such that D ( s . f) = D ( s ) * f s @ A df for all f E A and s E E . In this section we use generalization to noncommutative case of the Definition 1.2 (see Introduction). For this purpose we have to find a noncommutative analog of the Lie algebra of vector fields on a smooth manifold. Here we use the following well-known fact. The set of all smooth vector fields V e c t ( M ) on a smooth manifold M coincides with the space of all derivation of the algebra C " ( M ) . Recall that a map X : C " ( M ) C"(M) is called a derivation of C"(M) if X(f. g) = X(f). g f . X(g)for all f,9 E CO"( M I . Suppose A is an associative algebra with unit over a field R of characteristic zero. Let 2 be a subalgebra in the center of A, D be a subalgebra of Lie algebra of derivations of A closed with respect to the action of 2. Here action of 2 on D is defined by formula ( h X ) ( f )= h ( X ( f ) ) ,where f E A, X E D, h E A c A. It is obvious that D is a 2-module.

+

-

Definition 4.1. A connection on finitely generated projective right A-module E is a R-linear map

V :E

-

Homz(D, E),

such that

Vx(sf) = Vx(s)f + s . Xf, for all s E E, X E D f E A. Here VX(S) denotes V(s)(X) E E. Example 4.1. (1) Suppose M is a smooth closed manifold, A = C"(M, C ) is the algebra of complex valued smooth functions on M , 2 = C"(M) is the subalgebra of real valued functions, D is the Lie algebra of smooth vector fields on M . Let & be a complex vector bundle over M and E = r(&) be the space of its smooth sections. Then connection on the bundle & in the sense of usual differential geometry is a connection on A-module E in our definition.

365

(2) Suppose E is a free A-module, that is E = A". Put by definition V$(fi,. . . ,fn) = (Xfi,. . . ,Xf").It is obvious that Vo is a connection on E = A". Proposition 4.1. Suppose E i s a finitely generated projective right A-module. T h e n there is a connection o n E. The proof is based on the following statement.

Lemma 4.1. Suppose V i s a connection o n a finitely generated projective right A-module E and P i s a projection in E , that is P2 = P . T h e n the map P o V is a connection o n the module im P = P ( E ) . Define a curvature R of a connection V by formula

R ( X ,Y ) s= VXVYS- VYVXS- V [ x , y ] ~ , where X , Y E D , s E E .

-

Now one should define analog of de Rham complex in this setting. Let V be a 2-module equipped with an action of D. Let r : A V be a 2- and D-homomorphism. Suppose also that equalities X(fv) = (Xf)v ~ ( X W ) , (fX)w= f(Xv), r(ab) = r ( b a ) hold for all X E D , f E 2, a , b E A and v E v.

+

Define a complex R*(D,V) of V-valued differential forms as follows. Denote as A",(D) the factor of A"(D) by all possible relations

f(X1A .. . A XI, A .. . A xn)= (xiA .. . A XI, A . . . A xn), where XI,. . . ,X, E D , f E 2. It is clear A",D) is a 2-module. Put by definition Ro(D,V) = V, Rn(D,V) = Homz(AE(D), V) if n 2 1. Now define a differential d : R"(D,V) P ( D ,V) and XI,. . . ,X,+l E D. Put

-

R"+'(D,V).

Suppose w in

n+1

dw(X1A .. . A xn+l)= c(-l)i+'xi w(X1A .. . A X i

A

.. . A xn+l)

i=l

+ C

(-l)i+j~([Xi,Xj] A.. .A X i

A . . .A X j A..

. AXn+1).

I
It is easy to check that dw is an element of R"+l(D, V). Straightforward calculation shows d o d = 0. Let us denote cohomology of R*(D,V) with respect to the differential d as @(D, V). Suppose module V is an algebra. Then direct sum R*(D,V) = @ P ( D ,V) can be equipped with an operation of multiplication of forms. R*(D,V)

366

is an algebra with respect t o this operation. w1 E Rn(D,V ) and w2 E R"(D, V ) by formula (

~

Define product of forms

A1W Z ) ( XA~.. . A Xn+") = =

c

(-l)'wl(xd

. A X m ) .W2(X++l)

A..

A . . .A

xu(m+n))

1

uES(n,m)

where sum is taken over all (n,m)-shuffles u Suppose multiplication in algebra V is compatible with the action of the Lie algebra D , that is X(wlw2) = (Xv1)212 v l ( X w 2 ) . For example V = A is the case. Then the differential d is a graded differential with respect t o the multiplication of forms:

+

d(w1 A ~

2 =) ( d w l ) A

w2

+ (-1)lw1'w1 A dw2.

The following basic property of the curvature R is straightforward.

Proposition 4.2. Suppose E is a finitely generated projective right A-module with connection V. Then the curvature R of the connection V is an EndA(E)-valued 2-form. In particular for all X , Y E

D ,s

E

E, f

E

A the following equality holds

R W , Y ) ( S f ) = ( R ( X ,Y ) ( S ) ) f . Therefore the elements

R"q = R A

. . . A R E R2q(D,EndA(E)) are defined.

-

V is a map with properties mentioned above. Then Suppose r : A there is an extension of T to the trace EndA(E) V which we shall denote by the same sign T . Hence elements r(R"9) E R 2 q ( D , V ) are defined. --.)

Theorem 4.1. (a) Element r(R*'Q) is a cocycle of degree 29 in O*(D,V ) . (b) Suppose R' is a curvature of another connection o n the same modis a coboundary in R*(D,V ) . ule E . T h e n r ( R f A q-) T(R"'Q) For proof see [ 19, 201. Therefore cohomology class [ T (R"4) ] is well defined and is independent of connection on E . Denote class [ T ( R " ~ as ) ] Ch,(E). The following example establishes connection of classes Ch,(E) with Chern classes of complex vector bundles. As in Example 1 of this section let M

367

be a smooth closed manifold, A = C"(M,C) be an algebra of complex valued smooth functions on M , 2 = Cm(M)be a subalgebra of real valued smooth functions] D be a Lie algebra of smooth vector fields on M . One can show that cohomology group H,"(D,A) is isomorphic to de Rham cohomology H"(M, C ) of the manifold M with coefficients in C. Let & be a complex vector bundle over M and E = I?(€) be a space of smooth sections of €. Then there are characteristic classes Ch,(E). Taking into account isomorphism H,"(D,A) = H"(M,C) we see that Ch,(E) E H2q(M,C). The relation between Ch,(E) and Chern classes Q ( € ) of the bundle & is established by formula Chq(E) = Q q ( 2 7 r i c ~ ( &... ), , ( 2 ~ i ) ' ~ ~ ( € ) ) .

+ + tL

Here as usual Qq is an expression of a polynomial t; . . . mentary symmetric polynomials. For proof see [ 19, 201.

via ele-

5. Z/a-graded Lie-Cartan pairs

In this section we take as base of our construction the definition of connection in which instead of an Lie algebra D of derivation some Lie algebra L with additional mapping is used. Let R be a field of characteristic zero. Suppose A is a commutative algebra with unit over R, L is a Lie algebra over R. Suppose there are given Rlinear maps ABRL-L

and

LBRA-A

which define an action of A on L and an action of L on A with following properties:

(1) [X,YIf = X(Yf)- Y(Xf); ( 2 ) (fg)x = f(gx), 1 A x = (3)

x;

X(fg)= (Xf). 9 + f . xg, f(Xg)= (fX)g, [X,fYI = fIX,YI+ (Xf)Y.

In this case we shall call such pair (A, L) Lie-Cartan pair. In particular] property (1) means that A is a L-module, and property ( 2 ) means that L is an A-module. Motivating example is as follows. Suppose M is a smooth manifold] A = C " ( M ) is an algebra of smooth functions on M and L = V e c t ( M ) is a Lie algebra of smooth vector fields on M . The action of A on L is

368

multiplication of a vector field by a function, and the action of L on A is a differentiation of a function along a vector filed. We shall define characteristic classes in more general situation: for Z/P-graded Lie-Cartan pairs. Remind that in this section la1 denotes Z/2-grading of an element a.

Definition 5.1. We shall say that (A, L ) is a 2/2-graded Lie-Cartan pair if A is an superalgebra with unit and L is a Lie superalgebra such that following properties hold: (1) algebra A is supercommutative, that is f g = ( - 1 ) ~ f ~gf ~ g for ~ all f , g E A; (2) L is a graded A-module; (3) A is a graded L-module;

(4) f(Xg) = (fX)g for all f , g E A, X E L; ( 5 ) [ X , f Y ] = (Xf)Y+(-l)lxllflf[X,Y] f o r a l l f E A a n d X , Y E L. This definition can be reformulated as follows. Supercommutative (super)algebra A with unit and Lie superalgebra L form an Z/2-graded LieCartan pair if there are maps A @R L L and L @R A A such that for all f,g E A, X, Y E L following equalities hold:

-

-

(1) f ( g x ) = ( f g ) x , 1 A x = x; (2) X ( f g ) = (Xf)g (-l>'x'lf' f(Xg), [ X , Y ] f = X ( Y f ) - (-l)'X"Y'Y(Xf);

+

(3) f(Xg) = (fX)g,

[X,f Y I

=

( X f Y + (-l>'X"flf[ x,y I .

Let (A, L ) be a Z/2-graded Lie-Cartan pair. Suppose E is a finitely generated projective Z/2-graded right A-module.

-

Definition 5.2. We shall say that there is a connection on E if for all X E L a R-linear map Vx : E E with the following properties is given: 1) Vfx+gy = f V x + g V y for all f , g E A, X,Y E L 2) Vx(sf) = (VXS)f

+ ( - l ) ~ xs(Xf), ~ ~ sfor ~ all s E E , f E A

Remark 5.1. As algebra A is supercommutative one can consider a right A-module as a left A-module. Left action of A on module E is defined by formula sf = (-l)lsllflfs where s E E , f E A.

369

Example 5.1. Suppose E is a free A-module, that is E = A". Then formula Vg(f1,. . . ,fn) = ( X f l , . . . , X f n ) defines a connection on E . Proposition 5.1. Let E be a finitely generated projective 212-gmded right A-module. T h e n there is a connection o n E . The proof is based on the following statement.

Lemma 5.1. Suppose V i s a connection o n a finitely generated projective Z/2-graded right A-module E and P is a n endomorphism of E such that P2 = P . T h e n the map P o V is a connection o n the module im P = P(E).

-

Definition 5.3. A curvature of a connection V on a module E is a map R(X,Y) : E E given by formula R(X,Y)s = Vx(Vys) - (-1)1X"Y' Vy(Vxs) - V [ X , Y ] S , where X , Y E L.

It is easy to check that the curvature of connection Vo is zero. Proposition 5.2. The curvature R(X,Y) : E of right A-modules for all X , Y E L.

-

E i s a homomorphism

Proof is by straightforward calculations.

--

Put by definition A'(& E ) = E , A1(L, E ) = HomR(L, E ) . Let LP(L,E) be the set of all R-linear maps L@' E . Define Ap(L,E) to be a submodule of LP(L,E ) such that w : L@P E belongs to Ap(L,E ) if and only if following equalities hold:

w(X1,. . . ,xi,Xi+l,. . . , X,) = - (-l)l+IxilIxi+ll w(X1,. . . , Xi+l,x i , . . . , X,),

-

If there is given a connection on a module E with zero curvature then one can define a differential S : AP(L,E ) AP+l(L,E ) . Let w E AP(L,E ) .

370

+. . . + ~ x i - l ~ ) + ~ x j ~ ( ~ x.z.++IXj-11). l~+.

.(i,j) = i+j+(lXil+ IXjl)(1x21

Straightforward calculation shows that 6 o 6 = 0. Suppose module E is an algebra. For example E = A is the case. Then one can define product of two forms w E AP(L,E ) and 0 E AQ(L,E ) . The value of the product w A 0 on elements X1,. . . XP+, is defined by formula )

(w A -

wx1,.. . ,Xp+q) =

-y

(-1)"(-1)C%<. lx%llx.lxu(~d ( - q l ~ l ( l X U ( lI+...+lXO(,) ) I) x

" W P d

x W(Xcr(l),. . .,Xu(p))* 0(Xp+l, *

. .,xp+q>,

where S ( p ,q) is a set of all ( p ,q)-shuffles. Function x u ( i ) j ) is defined for all ( i , j ) such that 1 5 i < j 5 p + q in the following way. If a ( i ) > a ( j ) then put xu(i,j)= 1 and put x u ( i , j ) = 0 otherwise. This function appears as usual: if Xi and X j changed their order under the shuffle n then there is an additional sign (-l)~x~~lx~~.

It is easy to check that for this multiplication hold associativity:

~1 A (wq

the Leibnitz rule:

S(w1 Aw2) = ( 6 ~ 1 A) w 2

A WQ) = ( ~ A1WZ) A ~

3 .

+ (-1)l"'lwl

A

(6~2).

Proposition 5.3. Curvature R is a EndA E-valued differential form, that is R E R2( L ,EndA E ). Taking into account Proposition 5.2 t o prove this proposition one should check that map

R ( X , Y ): s H vxvys - (-1)'X"Y'

vyvx

-

has following properties:

R(X, Y ) = -(-1)1X"Y' R(Y,X), R(X, f Y ) = (-l)'X"f' f R ( X , Y ).

V[X,Y]S

371

It is straightforward computation. Consider elements R"q = R A . . . A R E A2q(L,End~(E)). Let tr : EndA(E) A be a trace. Then elements tr (R"q) E A2Q(L,A) are defined.

-

Theorem 5.1. (a) An element tr(R"\Q) is a cocycle of degree 29 in the complex A*(L,A). (b) Suppose R' is a curvature of another connection o n the same module E . T h e n the element tr(R'") - tr(R"q) is a coboundary in A*(L, A) f o r any q . Proof is similar to the proof of Theorems 3.1 and 3.2. Therefore cohomology classes [ tr (RAP)] are well defined and are independent of a connection on E. Denote class [.(RAq)] as Ch,(E), it is clear that Ch,(E) E H(A*(L,A ) ) .

Appendix: The trace of an endomorphism of a projective module Suppose K is a (noncommutative) ring with unit. Let F be a finitely generated projective right K-module. In this appendix we shall define trace of a K-endomorphism of the module F . We are following [ 81.

- -

Denote HomK(F, K ) as F*. It consists of all K-linear homomorphisms f :F K . For arbitrary right right K-module G define a linear map I ( G ) : G@KF* HomK(F, G) as follows. For an element g @ K f E G@F* map I ( G ) assigns homomorphism which takes z E F to g . f (z) E G.

Proposition. Suppose F is a finitely generated projective right K-module and G is a right K-module. T h e n the map I ( G ) is an isomorphism. The claim is obvious for F = K . Also it is easy to check it for F = K". Now it is not difficult to check it for arbitrary finitely generated projective module F as it is a direct summand in an appropriate K". Denote as [ K , K ] a subgroup in K , generated by commutators A p - p A (if the ring K is graded then take graded commutators A p - (-l)lxll"l PA), Let K = K / [K , K ] . Formula d [A, p ] = [ dA, p ] (-l)lxl [A, d p ] shows that if the ring K is equipped with differential satisfying Leibnitz rule then there is an induced differential in K. Define map 6 : F @KF* K by formula 2 @ K f H f(z).

+

-

372

Definition. A truce is the composition

B 0 I(F): HomK(F, F) 3F @K F*-+

K.

Denote the trace of A E HomK(F, F ) as trA. Following properties explain why the map t r = B o I(F)is a generalization of a trace of linear operator. (1) Suppose F = K" and endomorphism A is given by matrix Then t r A = C u i i mod [ K , K ] .

-

(aij).

-

(2) Suppose F and G are finitely generated projective right K-modules and A : F G and B : G F are K-homomorphisms. Then tr AB = tr BA.

-

+

(3) Suppose C : F G is an isomorphism of finitely generated projective right K-modules and A : F F is a K-endomorphism. Then tr A = tr C A C-l.

(4) Suppose F = Fl @ F 2 and A : F F splits into a direct sum A = A1 @ A2. Then tr A = tr A1 tr Az. (5) Suppose F is extracted from K" as direct summand by a projection p and A is an endomorphism of F given by a matrix M such that p M = M p = M . Then t r A = t r M .

References 1. R. Bott, S. S. Chern, Hermitian vector bundles and the equidistributions of the zeroes of their holomorphic sections, Acta Math., Vol. 114, 71-112, (1965). 2. S. S. Chern, O n the multiplication in the characteristic ring of sphere bundle, Ann. Math., Vol. 49, 362-372, (1948). 3. A. Connes, C*-algebres et geometrie diflerentielle, C. R. Acad. Sci. Paris. Ser. A, Vol. 290, 599-604, (1980). 4. A. Connes, Noncommutative differential geometry, Publ. IHES 62, 41-144, (1985). 5. B. L. Feigin, B. L. Tsygan, Additive K-theory and crystalline cohomology, Funk. analysis and its applications, Vol. 19, No 2, 52-62, (1985). 6. B. L. Feigin, B. L. Tsygan, Additive K-theory, Lect. Notes in Math. 1289, 67-209. 7. D. Husemoller, Fibre bundles, N. Y., 1966. 8. M. Karoubi, Homologie cyclique et K-the'orie, Asterisque 149, 1983. 9. M. Karoubi, Homologie cyclique et K-the'orie algebrique I, C. R. Acad. Sci. Ser. 1, Vol. 297, 447-450, (1983). 10. M. Karoubi, Homologie cyclique et K - the'orie algebrique. 11, C. R. Acad. Sci. Ser. 1, Vol. 297, 513-516, (1983).

3 73

11. S. Kobayashi, K. Nomizu, Foundations of differential geometry, Interscience, N. Y., 1963. 12. J. Milnor, J. Stasheff, Characteristic classes, Ann. of Math. Studies, Vol. 76, Princeton, 1976. 13. L. S. Pontryagin, Characteristic cycles of differential manifolds, Matematicheskij sbornik 21, 233-284, (1947). 14. E. Stiefel, Richtungsfelder und Fernparallelismus in Mannigfaltigkeiten, Comm. Math. Helv., Vol. 8, 3-51, (1936). 15. R. Stong, Notes on cobordism theory, Princeton University Press, 1968. 16. R. Swan, Vector bundles and projective modules, Trans. Amer. Math. SOC., VO~.105, 264-277, (1962). 17. R. Switzer, Algebraic topology -homotopy and homology, Springer, 1975. 18. H. Whitney, Sphere spaces, Proc. Nat. Acad. Sci. U.S.A., Vol. 21, 462-468, (1935). 19. Yu. J. Zhuraev, A. S. Mishchenko, Yu. P. Solovjev, On characteristic classes in algebraic K - theory, Tiraspol symposium on general topology and its a p plication, Kishinev, 91-92, (1985). 20. Yu. J. Zhuraev, A. S. Mishchenko, Yu. P. Solovjev, On characteristic classes in algebraic K - theory, Vestnik MGU, Ser. 1, Mathematics, Mechanics, No 1, 75-76, (1986).

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ON RATIONAL HOMOTOPY OF FOUR-MANIFOLDS *

SVJETLANA TERZIC+ Faculty of Sciences, University of Montenegro, Cetinjski put bb, 81000 Podgorica, YUGOSLAVIA E-mail: [email protected] u

We give explicit formulas for the ranks of the third and fourth homotopy groups of all oriented closed simply connected four-manifolds in terms of their second Betti numbers. We also show that the rational homotopy type of these manifolds is classified by their rank and signature.

1. Introduction In this paper we consider the problem of computation of the rational homotopy groups and the problem of rational homotopy classification of simply connected closed four-manifolds. Our main results could be collected as follows.

Theorem 1.1. Let M be a closed oriented simply connected four-manifold and b2 its second Betti number. T h e n (1) I f

b2 = 0

then rkr4(M) = rkry(M) = 1 and r p ( M ) is finite f o r

P # 477, (2) I f b2 = 1 then rkrz(M) = rkrS(M) = 1 and 7rp(M) is finite f o r

P

# 275,

(3’) If bz = 2 then rkrz(M) = rkrS(M) = 2 and r p ( M ) is finite f o r

P

# 273,

* MSC2000: 53C25,57R57, 58A14,57R17. Keywords: four-manifold, rational homotopy groups. t The paper is written while the author was postdoc supported by the DFG graduiertenkolleg “Mathematik im Bereich ihrer Wechselwirkung mat der Physik” at the Mathematical Department of the Ludwig-Maximilians University in Munich. The author is a member of EDGE, Research Training Network HPRN-CT-2000-00101, supported by The European Human Potential Programme.

375

376

(4) If b2 > 2

then dimn,(M) @ Q = 00, and

rk7r2(M) = b 2 ,

When the second Betti number is 3, we can prove a little more.

Proposition 1.1. If bz = 3 then rk7r5(M) = 10. Regarding rational homotopy type classification of simply connected closed four-manifolds, we obtain the following.

Theorem 1.2. T h e rational homotopy type of a closed oriented simply connected four-manifold i s classified by its rank and signature. The stated results are new, although as it will be clear later, we obtained them easily by known methods. Namely, as far as we know, they can not be found in the well known publications presenting the results on topology of four-manifolds [ l o ] , [ 61, [.3],nor in those presenting the results on application of rational homotopy' theory [ 51.

1.1. Some applications

Remark 1.1. The first four-manifolds we woulJlike to apply Theorem 1.1 are homogeneous spaces. For them the ranks of the homotopy groups are already well known. More precisely, simply connected four dimensional Riemannian homogeneous spaces are classified in [ 71 and the only such ones are Iw4, S4, @ P 2 ,S2 x S2, Iw x S3 and Iw2 x S2.

Example 1.1. A smooth hypersurface S d in @P3 is the zero set of a homogeneous polynomial of degree d in four variables. It is simply connected and bZ(Sd) = d ( 6 - 4 d d 2 ) - 2, see [ 141. Thus, for d # 1, using Theorem 1.1, we get

+

+ d 2 ) - 2, + d 2 )( d ( 6 - 4 d + d2) - 3) 2 d ( 6 - 4 d + d 2 )( d 2( 6 4 d + d 2 ) 2- 6 d ( 6 - 4 d + d 2 ) + 8 ) rk 7r4( Sd) =

rk7rz(Sd) = d (6 - 4 d d (6 - 4d rk7r3(Sd) =

1

-

3

377

Example 1.2. It is known, see [ 141, that

S4

is an example of a K 3 surface

and, thus, rkTz(K3) = 22,

rk7r3(K3) = 252,

r k ~ 4 ( K 3 )= 3 5 2 0 .

The same is also true for the ranks of homotopy groups of any logarithmic transform L,(m) Lb(n)(S)of an elliptic K 3 surface S, where m and n are odd and relatively prime. This follows from the results of Kodaira [ 81, since he proved that such surfaces are (and only them) homotopy K 3 surfaces.

Example 1.3. Let us consider complete intersection surfaces, i.e., surfaces M in @Pn+2which are transversal intersections of n hypersurfaces Y1,.. . ,Y, that are smooth at the points of intersections. If deg Y, = di then (d1, ..., d,) is said to be the type of M and M is usually denoted by S ( d 1 , . . . , &). Then, see [ 141, M is simply connected and bz(M) = e ( M ) - 2, where

Theorem 1.1 implies that for e ( M ) 2 4 e(e - 2 ) ( e - 4 ) e(e - 3) r k ~ 2 ( M= ) e - 2 , rk7r3(M) = -, rkT4(M) = 3 2

1.2. The method of the proof Our proof is based on Sullivan’s minimal model theory. It is a well known that Sullivan’s minimal model of a simply connected space X of finite type contains complete information on the ranks of its homotopy groups, and furthermore classifies its rational homotopy type. If the space X is formal in the sense of Sullivan, then its minimal model coincides with the minimal model of its cohomology algebra. By [ 111 all closed oriented simply connected four-manifolds are formal, and, thus, their cohomology algebra contains complete information on their rational homotopy. Following this, in Section 2 we first recall some results on real cohomology structure of simply connected closed four-manifolds and then in Section 3 state the necessary background from Sullivan’s minimal model theory and prove Theorem 1.1 and Proposition 1.1. In Section 4 we prove Theorem 1.2.

378

2. Real cohomology structure of closed oriented simply connected four-manifolds We denote by M a closed oriented simply connected topological fourmanifold. The symmetric bilinear form Q M : H 2 ( M , Z )x

H 2 ( M , Z )+Z,

defined by Q M ( ~ b) , = (a u b, [MI)

is called the intersection form of M . Poincark duality implies that (for b2 # 0) the form QM is non-degenerate, and, furthermore, it is unimodular (det QM = 1). For simplicity, we will denote the cup product a u b by ab below. The intersection form Q M can be diagonalised over R,with f l as the diagonal elements. Following standard notation (which comes from Hodge theory in the smooth case), we denote by b z the number of (+1) and by b, the number of (-1) in the diagonal form for Q M . Then b2 = b$ b, and (T = b i -by are the rank and the signature of the manifold M , respectively. Using the intersection form one can easily get an explicit description of the real cohomology algebra of a closed oriented simply connected fourmanifold.

+

Lemma 2.1. Let M be a closed oriented simply connected four-manifold, such that b2(M) 2 2. T h e n

H* ( M ,R)2 IR [ X I , . . . ,x + ,xb; ,. . . ,xb2]/relations , b2

where degxi = 2 and the relations are as follows:

x:= . . . = x;; = - x + 2 b,

+1

--x;2,

= * ' .

-

xzxj = 0, i # j .

(2.1) (2.2)

Proof. We recall here the standard proof. Let xi, 1 5 i 6 b2, be the cohomology classes in H 2( M ,R),representing the basis in which the intersection form Q M is diagonalisable. This means that Q ( x i , x i )= 1, for 1 6 i 5 b2+ , Q ( x i ,x i ) = -1, for b z Q ( x i , x j )= 0 , for i

#j

(2.3)

+ 1 5 i 5 b2 ,

(2.4)

.

(2.5)

379

Denote by V the generator in H4(M,R) such that (V, [ M I ) = 1. Since xixj = c . V, for some c E R,then (2.3) implies that x! = V for 1 5 i 5 b g , and (2.4) implies that x: = -V for bg 1 5 i 5 b2. Also, (2.5) implies 0 that xi xj = 0 for i # j.

+

For

b2

= 0 or b2 = 1, obviously

M has the following cohomology structure.

Lemma 2.2. Let M be a closed oriented simply connected four-manifold. (1) I f b 2 ( M ) = l thenH*(M,R) = R [ x ] , w h e r e d e g x = 2 a n d x 3 = 0 . (2) I f b z ( M ) = O t h e n H * ( M , R ) = R [ x ] , w h e r e d e g x = 4 a n d x 2 = 0 .

Remark 2.1. As we will see in Section 4, the above statements on the cohomology structure of four-manifolds are true over Q as well. But, it will be clear from below, that for the purpose of computation of the ranks of the homotopy groups or determining formality, it is sufficient t o work with real coefficients. 3. The ranks of the homotopy groups of closed oriented

simply connected four-manifolds 3.1. General remarks We refer t o [ 51 for a comprehensive general reference for rational homotopy theory. Let ( d , d A ) be a connected ( H o ( d , d A ) = k) and simply connected ( H (d , d A ) = 0 ) commutative N-graded differential algebra over a field k of characteristic zero. Let us consider the free N-graded commutative differential algebra ( A V,d) for a N-graded vector space V over k. We say that ( A V, d ) is a minimal model for (A, dA) if d(V) c A'2 V and there exists a morphism

which induces an isomorphism in cohomology. Let X be a simply connected topological space of finite type. We define the minimal model p(X) for X t o be the minimal model for the algebra d p L ( X ) . One says that two simply connected spaces have the same rational homotopy type if and only if there is a third space t o which they both map by maps inducing isomorphism in rational cohomology. Then the following facts are well known. The minimal model p ( X ) of a simply connected topological space X of finite type is unique up t o isomorphism (which is

380

well defined up to homotopy), it classifies the rational homotopy type of X and, furthermore, it contains complete information on the ranks of the homotopy groups of X . More precisely, rkn,(X) = dim(p(X)/p+(X) . p+(X)),,T- 2 2,

(3.1)

where by p+(X) we denote the elements in p(X) of positive degree and . is the usual product in p ( X ) . One says that X is formal in the sense of Sullivan if its minimal model coincides with the minimal model of its cohomology algebra ( H *(X,Q), d = 0) (up to isomorphism). It follows that, in the case of formal simply connected topological spaces of finite type, we can get the ranks of their homotopy groups from their cohomology algebras, by some formal procedure. This formal procedure is, in fact, a procedure of constructing of the minimal model for the corresponding cohomology algebra.

Remark 3.1. For some spaces with special cohomology one can easily compute their minimal models. Namely, using the terminology of [ 11, one says that X has good cohomology if

where the polynomials P I , . . . ,P k are without relations in Q [ 2 1 , . . . ,zn ] (i.e. (PI,.. . ,Pk) is a Bore1 ideal). Then in [ 1] it is proved that such a space X is formal and its minimal model is given by p(X) = Q [ Z l , . . .

dxi = 0 ,

,%I

8 A(Y1,.

. ., Y k )

7

dyi = Pi.

Clearly, (3.1) implies that these spaces are rationally elliptic, i. e. dimn,(X) 8 Q < 0 0 . Unfodunately, most of four-manifolds (or precisely those with bz not have good cohomology.

> 2)

do

But, the results stated in the above Remark are, in fact, consequences of a general procedure for the construction of the minimal model for a simply connected commutative differential N-graded algebra. This procedure is given by the proof of the theorem which states the existence (and also the uniqueness up to isomorphism) of the minimal model for any such algebra. We will briefly describe this procedure here, since we are going to apply it explicitly.

381

3.2. Procedure for minimal model construction In the procedure for the computation of the minimal model for a simply connected commutative differential N-graded algebra (A,d) one starts by choosing m2 : ( p 2 , O ) (A,d) such that mg) : p2 H2(A,d) is an isomorphism. In the inductive step, supposing that pk and mk : ( p k , d ) ( A , d ) are constructed we extend it to p k + l and mk+i : ( p k + i , d) ( A d ) with

-

-

-

-+

pk+l = pk '8 L(Ui,wj)

7

(3.2)

where C(ui,wj) denotes the vector space spanned by the elements ui and wj corresponding to yi and zj respectively. The latter are given by

H k + l ( A ) = Qmf+l)

cl3 -c(Yz)

(3.3)

and

(3.4) Then we have that m k ( z j ) = dwj for some w j E A and the homomorphism mk+l is defined by mk+l(ui) = yi, mk+l(wj) = wj and dui = 0, dwj = zj . Ker mr+2)= L ( z j ) .

Remark 3.2. In general, for a simply connected topological space X we have that A = A p L ( X ) and, obviously, by (3.1), we see that rkrk+l(X) is the number of generators in the above procedure we add to p k ( X ) , in order t o obtain p k + l ( X ) . Note that for a formal X, the algebra p ( X ) @Q k coincides with the minimal model of the cohomology algebra ( H * ( X , k ) , d= 0) for any field k of characteristic zero. The converse is also true. If there exists a field k of characteristic zero for which p ( X ) ' 8 k~ is the minimal model for the cohomology algebra ( H * ( X ,k),d = 0), then X is formal.

Remark 3.3. Obviously, (3.1) implies that for the purpose of calculating the ranks of the homotopy groups of X we can use p ( X ) ' 8 R~ as well. In the case of formal X it means that we can apply the above procedure to

H* (X, R). 3.3. Computation of the mnlcs of the homotopy groups Before we proceed to the computation of the ranks of the low degree homotopy groups of four-manifolds, let us note the following important facts.

Remark 3.4. All closed oriented simply connected four-manifolds with b2 > 2 are rationally hyperbolic, i.e. dimr,(M) '8 Q = 00. One can

382

see that using the fact that for M rationally elliptic must be satisfied C21crk7rzk(M) 5 d i m M (see [5]). Then from Hurewicz isomorphism it follows that 2b2 5 dim M . In particular, it implies that for bz > 2 these spaces do not have good cohomology.

Remark 3.5. All closed oriented simply connected four-manifolds are formal in the sense of Sullivan. One can see it using the results of [ 11] which say that any compact simply connected manifold of dimension 5 6 is formal. The Remarks 3.4 and 3.5 together with Procedure 3.2 for minimal model computation and the knowledge of cohomology structure of four-manifolds, make us possible t o prove the Theorem 1.1.

Remark 3.6. Note that, using Hurewicz isomorphism, we already know that rk7rZ(M) = bz. Proof of the Theorem 1.1. Because of formality the minimal model of a closed oriented simply connected four-manifold M is the minimal model of its cohomology algebra. Therefore, t o compute the minimal model p ( M ) @ q R, we can apply Procedure 3.2 t o the algebra (A,d) = ( H * ( M R), , d = 0). For simplicity we denote p ( M ) @Q R by p ( M ) and H * ( M , R ) by H * ( M ) below. As stated in the theorem we distinguish the following cases. 1. For bz = 0 Lemma 2.2 immediately implies that p ( M ) = A(x,u), where deg x = 4, deg u = 7 and dx = 0, du = x2. Thus, rk7rd(M) = rk7r7(M) = I and 7rp(M)are finite for p

# 4,7.

2. For b2 = 1 it follows from Lemma 2.2 that p ( M ) = A(x,u), where degx = 2, degu = 5 and dx = 0, du = x3. Thus, rk7r2(M) = rk7r/Tg(M)= 1 and

7rp(M)are

finite for p

# 2,5.

3. Let the second Betti number of M be 2. Lemma 2 implies that M has good cohomology, since the polynomials xs *x;, ~ 1 x are 2 without relations in IR [ x 1 , x 2 ] .By Remark 3.1, the minimal model for M is given by p ( M ) = R [ x i , x 2 ] @ A ( y i , y 2 ) , dXi=O, d y i = x ? f x ; , d Y z = x i x 2 . Thus,

7 r p ( M )are

finite for p

# 2 , 3 and

rk7r2(M) = rk7r3(M) = 2 .

383

4. Let b2 > 2. Remark 3.4 implies that in this case dim T* ( M ) @ Q = 00. We will use here the results on cohomology of M proved in Lemma 2.1. According to Procedure 3.2 for minimal model construction, it follows that p 2 ( M ) = R [ x 1 ,..., X b 2 ] , r n z ( x i ) = [ x i ] .Therefore,rknp(M) =b2. A t t h e next step in the application of Procedure 3.2, we know that H 3 ( M ) = 0 a n d K e r m ~ ) = L C ( x ~ f x ~ , z i x j ) , 2 ~ i < b z , l IWetakehere i<j~b~. b$ - 1 times sign (-) and b, times sign (+). Since the elements x: fz : , x i x j are linearly independent, we obtain that p 3 ( M ) = M ( M )@ L ( v i , u i j ) , 2 5 i

I b2, 1 5 i < j 5 b 2 ,

This implies that

In order to continue this procedure, let us note the following. For k 2 3, p k + l ( M ) is given by p k + I ( M ) = p k ( k f ) 8 L ( w j ) , where wj correspond to basis for H " ' ( p k ( M ) , d ) . We get this from (3.2) using the following two observation. First, (3.4) implies that Kermf")(M) = H k + ' ( P k ( M ) , d ) for k 2 3, since then Hk+'(M) = 0. Second, since Smy) 2 8 m g ) / K e r m g ) 2 H4(M),it follows that, for k 2 3, there are no yi's in (3.3). Thus, in order to construct p 4 ( M ) ,we need to find the basis for H 5 ( p 3 ( M ) ) . Since in p g ( M ) we have no nontrivial 5 dimensional coboundaries, this is equivalent to finding the basis for the 5 dimensional cocycles in p 3 ( M ) . Any cochain of degree 5 in p 3 ( M ) is of the form

c2=l

cF==,

a: X k and Pij = a; x k . Computing the coefficients where Pi = forx;, x:, 2 5 i I b2, x:xj, 2 5 j I b2, x:xj, i # j , 2 I i 5 b2, 1 I j I b2, xi X j x k , 1 I i < j < k 5 b2 in the expression for d ( c ) , one obtains that the equation d(c) = 0 gives rise to the following system of linear equations (respectively). b2

Ca,l= 0 , i=2

(3.5)

384

+

The number of variables in the above system is b2(b2 - l)(b2 2 ) / 2 . By the inspection one sees that the equation ( 3 . 5 ) eliminates 1 variable, each of the systems (3.6) and (3.7) eliminates b2 - 1 variables, the system (3.8) eliminates (b2 - 2)(b2 - 1 ) b2 - 1 variables and the system (3.9) eliminates variables. Thus, the dimension of the solution space for the above system is

+

(t)

b2(b2 - l)(b2 2

+ 2 ) - (”:+ (;)

= b2(% - 4 )

3

So, at this step in the construction of the minimal model, we extend p 3 ( M ) by adding the generators w k , 1 5 k 5 b2(bi - 4 ) / 3 of degree 4,i. e.

The differentials dwk are given by some basis for the solution space of the above system. Thus, we have

One can continue the above procedure for the construction of the minimal model and calculation the ranks of the homotopy groups, but it is obvious that at each step the vector space H k + ’ ( p k ( M ) , d ) for which we want to get a basis becomes bigger and more complicated. Clearly, rk 7rk+l ( M ) is given by the dimension of the H k + 2 ( p k ( M ) d, ) and it is some polynomial Pk+l(b2) in b 2 ( M ) . In order to continue the process we need to have a basis for H k + ’ ( p k ( M ) , d ) as well.

Remark 3.7. On each step in construction of the minimal model we should solve some system of linear equations, whose dimension of the solution space determines the number of generators of corresponding degree in the minimal model and the differentials for these generators are given by some

385

solution of that system. The calculation procedure done in the proof of Theorem 1.1 suggests that the dimension of the solution space for such system does not depend on the signature of the manifold, but only on its rank, while its explicit solution does. In other words, it suggests that the ranks of the homotopy groups of simply connected four-manifold are completely determined by the rank of its intersection form. Our attempt to obtain explicit proof for it following the proof of Theorem 1.1 involved complicated calculations, which we were not able to carry out.

Remark 3.8. Note that for a simply connected topological space X of finite type and finite rational Lusternik-Schnirelman category, hyperbolicity of X implies that its rational homotopy groups grow exponentially (so-called rational dichotomy, [ 41). This explains why one should expect the computations in the above procedure to be more and more complicated. This also gives that we can not expect to control the degrees of the polynomials P k ( b 2 ) with the growth of k. Proof of the Proposition 1.1. If we continue the procedure we started with in the proof of Theorem 1.1, we need to extend p 4 ( M ) by adding generators of degree 5 that correspond to basis of H 6 ( p 4 ( M ) ) . First note that any cocycle of degree 6 in p s ( M ) is cohomologous to the cocycle of the form (3.10) To simplify the calculations we can assume that b; is also 3 , since it is clear that the dimension of the solution space for the equation d(c) = 0 does not depend on b g . According to the proof of the above theorem we can take the differentials dwi to be as follows.

On that way, for c of the form (3.10), the differential d(c) is given as follows.

386

Using the above expression one can easily see that all 's can be expressed in terms of 's. Besides that we see that p23

= p11 7

p15

= p22

p32 = 0 1 17

+ p33

- p14

and

p34 = 013 - p2i p25 = p31

,

+ p12

while the other P’s are linearly independent. It implies that for b2 = 3 we have that rkn(2M) = 10.

4. On homotopy type classification

Proof of the Theorem 1.2. As we already mentioned, Sullivan’s minimal model theory provides a bijection between rational homotopy types of simply connected spaces of finite type and isomorphism classes of minimal Sullivan algebras over Q. Thus, in our case, two closed oriented simply connected four-manifolds have the same rational homotopy type if and only if they have isomorphic minimal models over Q. Any such four-manifold is formal and its rational cohomology structure is determined by its intersection form over Q. It follows that the rational homotopy type of a closed oriented simply connected four-manifold is classified by its intersection form over Q. In other words, two closed oriented simply connected four-manifolds

387

have the same rational homotopy type if and only if their intersection forms are equivalent over Q. Using the Hasse-Minkowski theorem one can see that any quadratic form over Q which has integral unimodular lattice is equivalent (over Q)to some diagonal form with -fl diagonal elements [ l a ] . It follows that two intersection forms are equivalent over Q if and only if they have the same rank and signature. By a result of Pontryagin-Wall (see [lo]), the homotopy type of a simply connected closed oriented four-manifold is determined by its intersection form. Thus, for closed oriented simply connected four-manifolds, we have the following homotopy type classification.

(1) Rational homotopy type (2) Homotopy type

-

-

rank and signature.

intersection form.

Note that not every homotopy type of a closed oriented simply connected four-manifold can be realized by a such smooth manifold. Namely, by a result of Freedman [ 61, for any unimodular symmetric bilinear form Q there exists a closed oriented simply connected four-manifold having Q as its intersection form. On the other hand the theorems of Rokhlin [ 13 J and Donaldson [ 21 give the constraints on the intersection form of smooth four-manifold. This implies the existence of the intersection forms (like Eg) which can not be realized by some smooth four-manifolds. We see that, in contrast to homotopy type, every rational homotopy type of closed oriented simply connected four-manifolds has a smooth representative. More precisely, any closed oriented simply connected four-manifold is rationally homotopy equivalent to a connected sum of @P2' s and @'s.

Remark 4.1. One can define R-homotopy type of a simply connected space X of a finite type to be the equivalence class of the algebra p ~ w ( X= ) p ( X ) @Q R,where p ( X ) is the minimal model for X , see [ 91. Then obviously we have that two spaces have the same R-homotopy type if and only if their minimal models are equivalent over R. Theorem 1.2 implies that in the class of closed oriented simply connected four-manifolds there is no difference between the rational and the real homotopy types.

Remark 4.2. Note that, in general, in the class of simply connected spaces of finite type we have strict inclusion among the spaces having the same rational and the same real homotopy type. To see that it is enough to construct two minimal algebras starting from two rational quadratic forms

388

which have the same rank and signature, but which are not equivalent over the rationals. Since for any minimal algebra p over Q there exists the simply connected space of finite type having p as its minimal model, we get on this way two simply connected spaces of finite type which have the same real but different rational homotopy type.

Acknowledgements

I am grateful to Yuri Petrovich Solovyov for getting me interested into this problem. I would also like to thank Dieter Kotschick for useful conversations. References 1. A. K. Bousfield and V. K. A. M. Gugenheim, O n PL de R h a m theory and rational homotopy type, Mem. Amer. Math. SOC. 8, No 179, 1-94, (1976). 2. S. Donaldson, An application of gauge theory to f o u r dimensional topology, J. Diff. Geom. 18,279-315, (1983). 3. S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, Oxford University Press, 1990. 4. Y. Fklix, S. Halperin, Rational LS category and its applications, Trans. Amer. Math. SOC. 270, 1-37, (1982). 5. Y. Fklix, S. Halperin, J. C. Thomas, Rational Homotopy Theory, SpringerVerlag, 2000. 6. M. Freedman, T h e topology of four-dimensional manifolds, J. Diff. Geom. 17, 357-453, (1982). 7. S. Ishihara, Homogeneous Riemannian spaces of f o u r dimensions, Journ. of the Math. SOC.of Japan 7,345-370, (1955). 8. K. Kodaira, O n Homotopy K 3 Surfaces, Essays on Topology and Related Topics (Mkmoires dkdiks Q Georges de Rham)., 58-69, (1970). 9. D. Lehmann, ThQoriehomotopique des formes diffhrentielles (d’aprb D. Sullivan), Sociktk Mathkmatique de France, Asterisque 45, (1977). 10. R. Mandelbaum, Four-dimesional topology: a n introduction. Bull. Amer. Math. Society 2, 1-159, (1980). 11. T. Miller, J. Neisendorfer, Formal and coformal spaces, Illinois J. Math. 22, 565-580, (1978). 12. 0. T. O’Meara, Introduction to Quadratic Forms, Springer -Verlag, 1963. 13. V. Roklin, N e w results in the theory of f o u r dimensional manifolds, Dokl. Akad. Nauk USSR 84, 221-224, (1952). 14. 1. R. Shafarevich (Ed.), Algebraic Geometry 11, Encyclopaedia of Mathematical Sciences, Springer - Verlag, 1996.

DUAL MAPS AND KOBAYASHI DISTANCE OF BOUNDED CONVEX DOMAINS IN C" *

STEFAN0 TRAPANI Universith degli Studi di Roma "Tor Vergata" Dipartimento di Matematica Via della Ricerca Scientifica 1, 1-00133 Roma, ITALY E-maiktrapani @mat.uniroma8. i t

We describe the Kobayashi distance and Busemann functions of bounded strictly convex domains in terms of boundary values of critical discs, their derivatives and their dual maps.

1. Introduction In the fundamentaI paper [ 91, Lempert studied the Kobayashi metric on a bounded strictly convex domain with smooth boundary. He described the extremal holomorphic discs of this domains. In 1161 we introduced a bridge between the CR geometry of the boundary of the domain and the Kobayashi geometry of the interior. We called this bridge the "eversive map". In the present paper we continue the description of the relation of Kobayashi and CR geometry, by describing Kobayashi Busemann functions and horospheres in .terms of boundary data coming from the extremal discs.

2. Preliminaries 2.1. Basic notation The standard complex structure of R2" = C",given by the multiplication with i = & is i, denoted by J . For any two elements v , w E C", we denote by < u,w >= C i u i z the standard Hermitian product and by Ivl = -./, The vectors (€1,. . . ,E k ) will always denote the elements of the standard basis of Rk. By A we denote the unit disc * MSC2000: primary 32H15, 32C16; secondary 53C60. Keywords :convex domains, Kobayashi metrics, CR structures, complex Finder metrics. 389

390

A = { z E C : IzI < 1} c C. We always consider A endowed with the Poincark metric d s i = dCd(/(l - IC12)2, with constant Gaussian curvature -4. Let D c C" be a domain. Let Tl'D and T'lD be the holomorphic and anti-holomorphic tangent spaces, respectively, i.e. T;'D = { v E l $ M

: v = V - i J V for some V E

T,D } , TZ'D = T J o D .

Similarly let .*lo(,) and T*O1(D) be the subbundles of T * @ Dof 1forms vanishing on the 01, respectively 10, vectors. Accordingly, if v E T x D c T,@D(resp. 8 E T,D c T * @ D )we , denote by v = vl' + v o l (resp. B = 01' Ool) the canonical decomposition into holomorphic and anti-holomorphic part.

+

Let S

c

-

C" be a smooth real hypersurface in C" and assume that p : U c C" lR is a local defining function for S , i.e. a function with dp, # 0 for any z E S n U and such that S n U = ( x E U : p ( x ) = 0 ) . By V c T S and V ' c T'S we denote the real holomorphic distribution and the complex holomorphic distribution, respectively. They are the distributions defined by

} = T,S n J(T,S) , V ~ o = { v ~ T , ' o:Re(v)EVz)='D,@nT,''Cn. S

V, = { v

E T,S

: JV E T,S

The complex structure J of Cn induces on each subspace V, a complex structure, which we may denote by J,. The pair ( V ,J ) formed by V and the smooth family of complex structures J = { J, : 23, V, , IC E S } is an (integrable) CR structure of codimension one (for the definition and main properties of CR structures, see e.g. [ 41).

-

Let S c C" be a smooth real hypersurface. We call (local) defining 1-form any l-form B (defined on a subset U c S) such that KerB, = V, at any point x E U . The holomorphic part 0 = 0'' will be called (local) holomorphic defining 1-form. If p is a defining function for S, then we call def B = -l/2 (dp, o J ) I T Sthe defining l-form associated with p.

For a given defining l-form 0, we denote by LE the bilinear form on V x

L:

:

V xx V,

-

1

R , L ! ( X , Y )= -2 dO,(X, J Y ) = Re (-idO(X'',Y'O))

.

Notice that, in case 6' and 6" = X(x)6' are two distinct defining l-forms, then Lf = X(x) . L: for any point x E S. If ( z ' , . . . , z " ) is a system of complex coordinates centered at IC and if we consider a (local) defining

39 1

function p with associated defining 1-form 6, = -1/2 (dp, any XIO = w i ( a / a z i ) )Y , l o= wi (alazz), we have

o

J ) , then for

In other words, Lz is (up to a scalar multiple) the classical Levi form of S at z and it is an Hermitian form. In the following, we will say that Cz i s the Levi form rescaled by the defining form 6. Recall that if S = aD is the boundary of a strictly convex domain, then Lz is positive definite. It follows that any defining form 6 is a contact form and V is a contact distribution.

-

Let D c @" be a bounded domain with a smooth boundary S = aD. We call disc attached t o S = aD any holomorphic map f : A @" such that f (aA) c S. We call disc properly embedded in D any holomorphic embedding f : A @" such that

-

(4 f ( A ) c D ; (b) f is attached t o S = aD; (c) f'(<) @ c T@Sfor any

D:??)

< E aA.

Throughout the following, D is always assumed t o be a bounded, strictly linearly convex domain in C" with smooth boundary.

2.2.

Extremal and critical discs of a bounded domain

Let S =

-

dD and ( D , J ) its induced CR structure.

-

Denote by IT : T*lo(S) S the pull-back on S of the bundle IT : T*lO@" @" via the natural immersion map z : S C". Let also E'O(S) c T*"S be the subbundle of all holomorphic defining 1-forms 0 for D. Then for any properly embedded disc f : A D , we consider the map

-

-

where we denote by Of(f(C) the unique element in r-'(f(<)) C E''(S) such that

392

If p is a defining function for S = a D , we have that, for any

ap

where The definitions of Of and of (see also [ 121).

C E dA,

afi

f are clearly independent on the choice of

-

Definition 2.1. A disc f : A if (1) f extends smoothly to

p

D is called critical (or stationary) disc so to give a properly embedded disc in

D;

(2) there exists an holomorphic embedding f : A that f l a ~coincides with the embedding (2.1).

-

T*1°C7L

such

I f f is a critical disc, we call f the lift of f in T*l0D (f is also called dual map of f , see [ 13, Def. 2.101). The family of complex subspaces at the points of f (A) c D defined by

e

Bfci)= { V E Tf(C)C"

: VIO E Ker f f ( ~n TfFC)Cn}

c Tf(C)C"

is called inner extension along f of D. By construction, $'Jf(aa)= Dlf(aa), where we denoted by DD(~(~A) the restriction of D to the circle f(aA) c

S=dD. In his fundamental paper [ 91, L. Lempert showed how critical discs are crucially related with the Kobayashi distance of D. In the following theorem, we recall just a few of their properties, which will be constantly used in the sequel.

Theorem 2.1. (191,[6]) Let D be a smoothly bounded, strictly convex domain in C", endowed with the Kobayashi metric K . T h e n : (1) a disc f : A D is critical i f and only i f it is a holomorphic, totally geodesic, isometric embedding of the unit disc A; (2) for every x E D the exponential map gives a diffeomorphism exp, : T,(D) D (3) for any (x,y) E D x D with x # y, there exists a unique critical D such that disc f,,y : A

-

x = f,,y(0),

Y = fi,y(t)

for Some t E (0,1] ;

(2.3)

393

-

and any 0 # v E T x D there exists a unique critical D such that

(4) f o r any x E D disc

fx,v

:A

z = f (0) ,

W1O

E R f'(0)

,

(2.4)

where Rf'(0) denotes the real span of the vector f'(0); (5) f o r a fixed x, the map w H fX,,, is C".

-

-

In the following, for any pair (x,y) E D x D with x # y , and for any 0 # v E T x M , we will denote by fx,y : A D and fx,v : A D the critical discs which verify (2.3) and (2.4), respectively. 2.3. Levi f o r m s , holomorphic symmetric forms, and normalized defining functions

We need now to recall some concepts introduced by M. Y. Pang in [ 131, which are crucial tools for studying Jacobi vector fields in D.

-

Definition 2.2. Let f : A D be a critical disc and f the corresponding lift in T*l0D. Assume also that p is a (local) defining function for d D on a neighborhood U of f(dA) and that = (zo, . . . ,z") : U @" is a system of complex coordinates on U. (1) The Levi form of f is the 1-parameter family of Levi forms L:, parameterized by the points C E dA, defined by

-

<

-it&).

Lf - Lef I f ( < ) , B f = R e ( G f l c )= R e ( (2.5) (2) The Holomorphic Symmetric form off in the system of coordznates

< is the 1-parameter family of symmetric forms f?[>', by the points

parameterized

C E dA, defined by

BCf'C(X,Y)= -( X 4d C )

-iJX)

((p - i J Y ) p )

Ic

(2.6)

where q is the function defined in (2.2) and where the symbols X and P denote any two vector fields with values in the real holomorphic distribution V that are equal to X and Y at the point

f (0In coordinates, L: and

B;" are given by the expressions

394

Since { C : , E aA } is a family of Levi forms, it does not depend on the choice of the system of coordinates nor on the defining function p. On the other hand, it can be directly checked that the family { S:" , 4- E a A } does not depend on the choice of the defining function p but does depend on the given system of coordinates. For explicit formulas, which express the dependence on the system of coordinates of Sf,€, see [ 13, Thm. 2.17(3)].

-

We conclude with the following lemma.

D be a Lemma 2.1. ([13, Prop. 2.36 and Thm. 2.451) Let f : critical disc. Then there exists a defining function p of D , defined o n a neighborhood U of f (A), such that for any 4- E A

and for any

E

aA, the function q defined in (2.2) and

f verify

A function p which verifies (2.8) and (2.9) is called a strongly normalized defining function w.r.t. f .

-

Lemma 2.2. (191, [ l o ] , [ l l ] )Let f : A D be a critical disc. Then there exists a field of holomorphic frames ( e l , . . . ,en)<, C E A, for the subspaces B:' c T;rC,Cn such that f

Lei+(ea e p ) = sap , 1

for any ei+ E aA.

It follows that, for any C E A, the set of holomorphic vectors (f'(<), e l ( < ) ,. . . ,en(<)) constitutes a holomorphic basis for T;rC,C". We will call (f'((), e l ( < ) ,. . . ,en(<)) the unitary frame field adapted to the disc f. We will denote eo(C) = f'(<) and we will use the symbol ei(<) to denote any vector of the unitary basis; we will use the symbol e,(C), with an index denoted by a Greek letter, t o denote the vectors which span the subspace

B:'.

395

2.4. The symplectijication and the hyperplane

bundle o f a D Consider the real hypersurface S = aD c @" endowed with its induced CR structure (V, J ) . The symplectijication of ( S ,72) is the subbundle Es of 7-r : T * S S , defined by

-

-

E s ~ f { 6 ' c T ~XSE, S: KerO=V,}.

(2.10)

S is a principal bundle with structural group R*. The bundle 7-r : Es Moreover, since S = aD is the boundary of a convex domain, there exists a globally defining function p for S, with grad p pointing outwards D at all points. This implies that there exists a (global) associated defining 1-form 6'P = -1/2 ( d p o J ) for the distribution V ,which represents a global section of 7-r : E s S. It follows that Es has exactly two connected components, say E: and E,, each of which with structural group R,o. For simplicity of notation, we will use the symbol Es to denote just the connected component containing the 1-forms P l Z , z E S.

-

The tautological 1-form of Es is the 1-form defined at any 6' E Es as def

wle = ~ * 8 .

(2.11)

Definition 2.3. The hyperplane bundle of S is the subbundle

H ( E s )= where

u

f&%& OEEs

cTEs

-

H(Es)e = { w E TOES : we(.)

(2.12)

I

=l}.

-

If we denote by .ir : H ( E s ) Es the natural projection of H ( E s ) onto E s , one can realize that the projection no? : H ( E s ) S makes H ( E s ) a fiber bundle over S , with standard fiber diffeomorphic to EXzn. 3. Jacobi fields of bounded convex domains

-

Let D c @" be a strongly convex bounded domain. Let f : A D be a critical disc. We want to determine for any value V, W E Tf(o)D the unique holomorphic Jacobi field 9 such that Re(J(0)) = V ,

Re(J'(0)) = W .

First of all, observe that for any 1-parameter family V of extremal disc centered at f there exists a 1-parameter family of embedded discs in T*l0Cn defined by %I

0= Pk)

396

where

f(')(c)

is the lift in T*"C"

of the embedded disc f(") = V ( s ,*).

We call lift in Tl0(T*loC") of a holomorphic Jacobi field J the holomorphic map

It is clear that if

7r

: T*l°C"

-

C" is the standard projection map, then

7r*(k))

=

Jr(0 7

c

for any E A. Notice also that, since T*Z,)C" is a vector space, we have the natural identification

and hence

-

is naturally identified with a map

5 :A

T*l°C",

<

H

@)

E T*Z,,C".

Let f : A D be a critical disc and J a holomorphic Jacobi field. Let also p be a normalized defining function (see Lemma 2.1) and f(") a 1parameter family of extremal discs associated with 9. Then, if we denote by qs : A --C i the 1-parameter family of functions

we have that the values j(c) for of p, qs and J as follows:

= ei+ E

dA may be, expressed in terms

Consider now the following decomposition of the vectors

J(C)

JJ(0 = W C ) + J W , W C ) E ef'(c),JL(C)

as

E ql.

397

Such unique decomposition can be done by expressing I({) in terms of an adapted unitary basis (eo = f',e l , . . . en)<,as given in Lemma 2.2, and by taking as .U1 the projection into span(e1,. . . ,en}. Let us express J// as

Jl'(C)

=m)f'(O.

(3.1)

Now, observe that by Lemma 2.1,

and hence, recalling the definition of Cf and B f ~ fwe , may write j(C), for any C E aA as

Since

:A

--f

T*l°C" is holomorphic, for any holomorphic section of

el0which vanishes at t.he origin To

=

c . 7- : d +3 1 0 c TlOC",

we have that

Notice that, since the function

is holomorphic for any holomorphic function X J and any holomorphic section r : A -+ and since j ( r ) = 0 for any such section r , we have

elo,

398

that (3.2) and (3.3) implies the following condition on the component J' of the holomorphic Jacobi field 9: 1 0=2r

/

27r

{

eiZ4 Lfi+(7,J')

+ S:;:

(7,

J')} d 4

(3.4)

0

for any holomorphic section r : A

-+

@lo

We may now prove the following

-

-

Theorem 3.1. Let D c @" be a smoothly bounded, strongly convex domain and let f : A D be an extremal disc. Let 4 = XJ f ' + 9' : d TlO@ be a holomorphic Jacobifield, of which 'J : A ,lo is the corresponding holomorphic section 8". Let also (ei)c a unitary frame field adapted to f (see Lemma 2.2) and, for any holomorphic map H : A -+ D ' O of the form H = Haeu, with Ha holomorphic, let us denote by V H = HLe". Then:

-

-

a) for any pair of vector u = u"e,(O) and w = w"ea(0) E denote by jv,w: A @',lo the section of the f o r m il,w(I)

= (uQ

+ Iw") ea(<)

7

then there exists a unique holornorphic section ji:i : A that

c ) the function A J ( < ) is of the form

where

Q

E @ and b E Iw

en7let us

-

(3.5) V l 0 such

399

Proof. a) Observe that the left hand side of (3.6) can be rewritten as

where Hess(p)f(ei+)(Re(ei2'7> 7 Re(eiz4~.(z) v ,1) w denotes the real Hessian of the defining function p evaluated at the point f ( e i 4 ) along the vectors Re(ei24r), Re(ei24j$) E Tf(,++))(C.". By hypothesis of strict linear convexity, the Hessian Hess(p) is positive definite at all tangent spaces Tf(,i+)aD.From this it follows that the left hand side of (3.6) is vanishing for any section r if and only if the section ei24ji:L is identically equal to zero. Same conclusion holds for the section j?v w and this proves the uniqueness part. For the existence part, see [ 131. b) It is clear that (3.4) holds for any holomorphic section

T

if and only if

(3.10) for any holomorphic section r. From the hypothesis, it follows that J'(C) is of the form (3.7) where jv,w (2) is the unique solution of (3.6). c) Let p be a normalized defining function for aD and let f(") be a 1parameter family of critical discs, centered at f , associated with the Jacobi field J. This implies that for any s we have that

This can be rewritten as

By developing XJ(<) into Fourier series, we obtain from (3.11) that AJ(<) is exactly of the form (3.8). The proof consists in a suitable adaptation to a more general case of a few arguments of the proof of Thm. 3.6 in [ 141.

400

4. Busemann functions and horospheres of strongly convex domains Let D c C" be a smoothly bounded, strongly convex domain in C". By Lempert's results, we know that the Kobayashi metric k~ makes D a complex Finsler manifold. Let us denote by S ( D ) = {u E T ( D ) : k ~ ( u= ) 1) and by Sx(D) the fiber of z for the natural projection proj : S ( D ) -+ D. Let us also denote by 9 the (finite) Kobayashi distance of D,and let 9x: D R be the function y -+ 9 ( z 7 y ) . Let y :lo, m[ be a unit speed geodesics which from now on will be called an oriented half line. By triangular inequality the functions D ( y ( t ) ,z) - t is decreasing in t and uniformly bounded by D(y(O),z), therefore it has a limit B,(z) as t goes to 03. The correspondence z B,(z) defines a function on D which is called the Busemann function of y. We clearly have

-

-

BY(Y(S)) = --s

(4.1)

and

The level sets B-, = c are called horospheres or limit spheres, and the sublevel sets B, < c are called horoballs or limit balls. Let us denote by O,(T) the ball of radius T centered at z. The terminology is justified by the following equality which is valid for every real number c (see [ 51)

u

+ c) = {

3:

E D : B,(z)

tEW,t>-c

Let C ( D ) be the space of continuous real functions on D with the topology of uniform convergence on compact sets, let us take the quotient space C*(D) of C ( D ) modulo the constant functions with the quotient topology. There is a continuous embedding i of D into C,(D) given by z -+ [ Dx] here ([ ] denotes the equivalence class). According to Gromov, see [ 31, we compactify Dby embedding it into i ( D ) and define the boundary of D by B ( D ) = i ( D ) \ i ( D ) . Note that the equivalence class of a Busemann function is in B ( D ) ,in fact B,(z) = lims--too D(y(s),z) - s. On the other hand, as one can see from formula (4.1), the function B, does not have a minimum, hence it does not belong to i ( D ) . When this construction is applied to a simply connected complete Riemannian manifold with non positive curvature, it can be shown that the Busemann functions are C2 and that the boundary B ( D ) coincides with the

401

set of Busemann functions. Moreover, it also coincides with the usual definition of boundary as the set of equivalence classes of geodesics, where two geodesics are considered to be equivalent if their distance is bounded [ 31. We do not know whether the Kobayashi metric on a strictly convex bounded domain D enjoys some reasonable notion of non positive curvature, however it is proved in [ 11 (notes pp. 280) that, given a fixed 20 E D the map from d D to B(D) given by y By, is bijective. Here y is the half-line connecting 20 and y. We will give an explicit expression of the Busemann functions in terms of the dual maps of critical discs and this will show that the Busemann functions are C".

-

Proposition 4.1. Let us recall that for any x E D, we denoted by Dz : D --t R the distance function ZJz(y) def = ZJ(x,y). Then, for any y E D , with y # x,

-

(4.3)

Proof. Consider a vector 2, E TyD and let u : [ - E , E ] D be a smooth curve with uo = x and u; = u and consider the 1-parameter family of critical discs ft

:A

-

D

ft

7

From the definition of the critical discs for any t , and hence

ft

= .fm,,z.

it follows f t (tanh (Bz(ut))) = 2,

where we denoted by Jl the holomorphic Jacobi vector field for fo defined by

ft

: A -+ T*l0D and recall that by Consider now the family of lifts definition, f t ( f l ) = 1 for any t. Moreover J = A f' +J1, where JL is a holomorphic Jacobi vector field such that fo(JL) = 0 and A is a function of the form

A: A

--+

D,

A(() = c r + i b [ - ~ [ ~

402

where

Q

E C and b E

W.From (4.4) we find Jl(tanh(d,(y))

= 0 and

which is a real number. This implies that

which is -Re(a) = ( d D zdu( g u ) lU=o). Notice now that, since f t ( 0 ) = at for any t, then we have

do= J(0) E T;OD. Therefore f,,z(v) = &(vlO) = fo(J(0)) This concludes the proof. Note that if y : R

-

= Q.

(4.6) 0

D is an oriented straight line then lim 73 = fyo,y&(l).

3-00

(4.7)

For this reason, for any oriented straight line y and for the corresponding ray yl[o,+m), in what follows we will call the point fyo,y;(l) the point at infinity of y and we will denote it by ym.

Lemma 4.1. For any oriented straight line y, the associated Busemann function B, is C1 and, for any x E D dBy)z = -Re(.L,y_lo).

-

(4.8)

Proof. Consider a point x E D , a vector v E T,D and a regular curve a : [ -E, + E ] D such that uo = z and v = ire. We want t o prove that B7 is C1 and

By (4.3), we know that for any s E

Iw

403

and fz,,s(0)= IC for every s E R.From Theorem 2.1 and well known facts on exchanging of limits under uniform convergence, we may infer that B, is C1 and that

Lemma 4.2. Let y be an oriented half line of D with $0) = ym = y. Then the function 2

takes real positive values.

-

ICO

and

fzo,Y(f;,yll)

It follows that for every u E T y ( d D )the number (-i f i o , y l l ) ( w ) is real. On the other hand i fL,yll never belongs t o the holomorphic distribution, so that IC fzo,g(fA,y/l) is real never vanishing. Since fzo,y(f~o,y]l) = 1,

-

the statement follows by continuity.

0

Theorem 4.1. Let y be an oriented half line of D with y(0) = ym = y. Then 1 B,(z) = 5 1% (fzo,Y(f:,yll)) *

20

and

Proof. We have 1/2 log (.fz,,,y(f~o,y~l)) = 1/2 log(1) = 0 = B,(zo). Therefore, by Lemma 4.1, it is sufficient to show that for every z E D 1 d ( 5 lo~(fzo,Y~f;,yll)))z = dB,Iz = -Re(fz,ylo).

-

Consider a vector w E T z D and let (T : [ - E , E ] D be a smooth curve with (TO = z and (T; = w and consider the 1-parameter family of critical discs ft

:A

+

D,

ft

= fgt,y.

404

Let J be the holomorphic Jacobi vector field for fo defined by

Let us write

J=

(a+ib(-~(2)f~,yI-Jl(().

Since ft(1) = y for every t E and J*(l)= 0. Now

[-E,E]

Now by formula (4.9)

we find J(1) = 0, i.e. a + z b - 6 = 0

0

fy,z(u) = a.

Remark 4.1. Since for every fixed x E D and y E b D the linear map -i fz,y is real on T y ( d D ) vanishes on Dyand it is positive on the tangent vector to critical discs passing through y, we see that by changing x and keeping y fixed the Busemann function of the geodesics half-line connecting x with y changes by a constant. Remark 4.2. Let xo E D be fixed then for y and z in D we have 1 D(Y,z ) = 2 wmax E a D 11% (Lo,w(f;,wld) - 1% ( L o , w ( f : , w I I ) )

1

1

8P

1

~ ~dP ~ ~ ~ . l ~ ~

=2 wmax E a D l o g ( ( l ~ ( ~ ) ( f ; , w ) ( l )l)

Where p is a defining function of aD. In fact the inequality 1 D(Y,). 2 2 wmax E a D log (Lo,w(f;,wI1)) - log (Lo,w(f:,wld)

1

I

comes form the Theorem 4.1 together with formula (4.1). To find equality we assume y # z and use formula (4.2) applied to the geodesics half line connecting y to z.

405

5. The eversive map and horospheres

In [ 161 we prove the following : Let E ~ and D H ( E ~ Dbe) the symplectification and the hyperplane bundle of d D , respectively, and let a2,, : R -+ (T*dD)@be the curve

a2,,(t)=

[@fZ,.

lTaD

Ifi,u(eit)]

=

[Qfi+*p(Jt)v

I f ~ , e x p ( ~ t ) ~ (lTaD l)]

. (5.1)

takes values in E ~ and D the tangent vector a;,,It=~ Then the curve ax,,, is in H ( E ~ D ) . If moreover we define the map E : S ( D ) H ( E ~ Dby )

-

E H(E0D)

E ( v ) = QIk,,Io

then E is a biholomorphism equivariant diffeomorphism. The reason for the name eversive map, is that E maps the fibers of proj onto section of 7r o ii and some sections of p r o j onto the fibers of 7r o ii. The foliation of S ( D ) given by the fibers of p r o j defines via E a foliation F1 of H ( E ~ Dby ) spheres. Similarly the foliation of H ( E ~ Dgiven ) by fibers of ii : H ( E ~ D ) E ~ D defines via E a foliation Fz of S ( D ) by closed submanifolds diffeomorphism to R2n-1. We are going t o describe the foliation F2 in more details. For a fixed y E d D consider the map sy : D S ( D ) given by sy(x) = fz,y’(0). This is a section of proj. We have the following

-

-

Proposition 5.1. Let v E S ( D ) , xo = proj(v) and y = 7r o i?(E(v)) = fZo,,(1) E dD. T h e n the leaf of 35 through v i s the image by sy of the horosphere in D passing through xo and with point at infinity y. Proof. Let y be the geodesics half-line connecting xo and y , let F be the leaf of F2 through the point v and let I‘ = { J: : By(x)= 0} be the horosphere through xo with point at infinity y . Clearly F = s , ( p r o j ( F ) ) , so we need to show that p r o j ( F ) = I?. Let x E p r o j ( F ) , then -ifio,y(l)= -ijz,y(l).Therefore 1= L , Y ( f ’ 2 , y ) ( 1 )

= fzo,Y(f’2,y)(1),

so B,(x) = 0 because of Theorem 4.1. Vice versa let x E I?, since 7r(-z.fz,y-(l)) = ~(-if3co,y)(1)) = y , there exists a positive real X such that -i f2,y(l) = -i X f 2 0 , y ( l ) . On the other hand, since IC E I?, by Theorem 4.1 again we have 1= L o , Y ( f ’ 2 , y ) ( 1 ) I

=

L,Y(f’2,y)(~)

I

so that -i f2,y(l) = -i fZoIy(1) and

IC

E p r o j ( F ).

0

406

Acknowledgement

I would like to t h a n k t h e referee for his/her careful reading of t h e manuscript.

References 1. M. Abate, Iteration theory of holomorphic maps on taut manifolds, Mediterranean Press, 1989. 2. M. Abate and G. Patrizio, Finsler Metrics-A Global Approach, Lecture Notes in Mathematics 1591,Springer -Verlag, 1994. 3. W. Ballmann, M. Gromov and V. Schroeder, Manifolds of non positive curvature, Birkhauser, 1985. 4. A. Boggess, CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics- CRC Press, Boca Raton, 1991. 5. H. Busemann, The Geometry of Geodesics, Academic Press-New York, 1955. 6. C. H. Chang, M. C. Hu and H. P. Lee, Extremal Analytic Discs with Prescribed Boundary Data, Trans. Amer. Math. SOC.310(1), 355-369, (1988). 7. C. Fefferman, The Bergmann kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26, 1-65, (1974). 8. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. I & 11, Interscience Publ., 1963-1969. 9. L. Lempert, La me'trique de Kobayashi et la reprbentation des domaines sur la boule, Bull. SOC.Math. France 109, 427-474, (1981). 10. L. Lempert, Holomorphic retracts and intrinsic metrics in convex domains, Analysis Math. 8, 257-261, (1982). 11. L. Lempert, Intrinsic Distances and Holomorphic Retracts, Complex Analysis and Applications '81, Sofia, (1984). 12. L. Lempert, Metamorphoses of the Kobayashi Metric, Lectures notes of three lectures at the Dae Woo workshop on Geometry and Topology at PostechPohang (South Korea), published by Seul National Univerity of Korea, 1993. 13. M. Y. Pang, Smoothness of the Kobayashi Metric of Non-convex Domains, Internat. J. Math. 4 (6), 953-987, (1993). 14. M. Y . Pang, Pseudoconvex Domains in C2 with Convexifiable Boundary around an eztremal disk, Math. Z. 226(4), 513-532, (1997). 15. A. Spiro, The Structure Equations of a Complex Finsler manifold, to appear in Asian J. Math. 16. A. Spiro and S . Trapani, Eversive map of bounded convex domain, J. Geom. Anal. 12 (4), 696-715, (2002).

OSCULATING SPACES AND HIGHER ORDER CURVATURE TENSORS OF SUBMANIFOLDS IN R'" *

K. TRENCEVSKI Institute of Mathematics, St. Cyril and Methodius Uniuersity, P. 0. Box 162, 1000 Skopje, MACEDONIA E-mail: [email protected]

This paper is a continuation of the papers [ 2, 3, 4, 51 and the results are obtained using the recent paper [ 11. In this paper are considered the osculating spaces of a submanifold in W" of the first, second and higher order. Principal curvatures and directions of higher order are also studied, together with their geometrical interpretations. This construction naturally yields to the higher order curvature tensors of submanifolds of the Euclidean spaces.

1. Preliminaries In this paper a geometric interpretation of the normal vectors and normal curvatures introduced in [ 2 ] , of an n-dimensional manifold M , imbedded in Rn+k is given. Namely, this paper is continuation of the previous papers [ 2, 3, 4,51. The basic concept in this paper is the application of the recent results obtained in [ 11. In this section we give a brief view of the results obtained in [ 11. Let us consider the Euclidean space IE" with an inner product ( , ) and two subspaces C and II of dimensions p and q , ( p 5 q ) and suppose that C and II are generated by the bases al, . . . , ap and bl, . . . , b, respectively. It is proved in [ 11 that det(MMT) I rlrz, where A4 is a p x q matrix with elements Mij = (ai,bj), rl and l?2 are the Gram's determinants obtained by the vectors a l , .. . ,ap and bl, . . . ,b, respectively. Thus the angle 'p between C and 11is defined by coscp =

Jzqziq JK.dG

*

* MSC2000: 53A05, 53A07. Keywords :osculating spaces, normal curvatures, high order curvature tensors, principal values, principal directions. 407

408

Theorem 1.1. If C and I2 are arbitrary subspaces of the Euclidean vector space En and C* and II* are the corresponding orthogonal subspaces, then

' p ( C 1 , C 2 ) = ' p ( C ; , C;). The following theorems yield us to the main results.

Theorem 1.2. Let C and II be two vector subspaces of the Euclidean space En of dimensions p and q ( p 5 q ) , and let A1 and A2 be n x p and n x q matrices whose vector rows generate the subspaces C and II respectively. Then the eigenvalues of the matrix

are p canonical squares cos2'pi, (1 5 i 5 p ) and moreover P

cos2 'p =

I-Icos2

'pa,

i=l

where cp i s the angle between the subspaces C and II.

If we consider the case p 2 q, we should consider the matrix f ( A 2 , A l ) which is of type q x q, instead of the matrix f ( A l ,Az). Analogously to the theorem 1.2 the eigenvalues of f (A2,A l ) are q canonical squares of cosine functions but the product of them is equal to zero if p > q. Theorem 1.3. A n y nonzero scalar X is a n eigenvalue of f ( A 1 ,Az) if and only i f it is a n eigenvalue o f f (Az,A l ) and moreover the multiplicities of X for both matrices f ( A l , A 2 ) and f ( A z , A l ) are equal. Note that X = 0 is eigenvalue of the matrix f ( A 2 , A l ) if q > p , but X = 0 may not be eigenvalue of the matrix f (Al,A2). The common eigenvalues will be called principal values. There are decompositions of the subspaces C and TI into the orthogonal eigenspaces for the common non-negative eigenvalues and for the zero eigenvalue if such exists. These eigenspaces are called principal subspaces, or principal directions for the eigenvalues with multiplicity 1.

Theorem 1.4. T h e function cos2 'p, where 'p is the angle between any vector x E C and the subspace II, has maximum i f and only i f the vector x belongs to a principal subspace of C which corresponds to the maximal principal value. The maximal value of cos2 'p is the maximal principal value.

409

Note that analogous statement like theorem 1.4 is true also if we consider x as a vector of II and ‘p is the angle between x and C. Hence the geometrical interpretation follows. Among all values cos2 ‘p where ‘p is angle between any vector x E C and any vector y E II, the maximal value A: is the first (maximal) principal value. Then C1 = {x E C I cos2(x,II) = A:} is the principal subspace of C. Analogously

n, = {y E II I cosyy, C) = xq } is the principal subspace of II and moreover dimC1 = dimII1. Now let us consider the subspaces C’ and II’ where C’ is orthogonal complement of C1 in C and 11’ is orthogonal complement of II1 in II. Among all values cos2‘p where ‘p is an angle between any vector x E C‘ and any vector y E II’, the maximal value A; is the second principal value. Then C2 = {x E C’ I cos2(x,II’)

= A;}

is the principal subspace of C‘. Analogously II2 =

{y E

II’ I cosyy, C’)

= A;

}

is the principal subspace of II’ and moreover dim C2 = dim I I 2 . Continuing this procedure we obtain the decompositions of orthogonal principal subspaces

+ . .. + CS+l II = I31 + TI2 + . . . + IIS+l C = C 1 + C2

where dimCi = dimIIi for 1 5 i 5 s. The subspaces Cs+l and IIs+l correspond to the possible value 0 as a principal value. Note that the same conclusions for the eigenvalues and principal subspaces (principal directions) are also true for the subspaces C* and II*. Theorem 1.5. If C and II are a n y subspaces of the Euclidean vector space IE” and C* and II* are their orthogonal complements, then the nonzero and different f r o m 1 principal values f o r the pair (C,II) are the same f o r the pair (C*,II*) with the same multiplicities and conversely. (a) If p+q 5 n, t h e n the multiplicity of 1 f o r the pair (C*, II*) i s bigger f o r n - p - q than the multiplicity of 1 f o r the pair (C, n). (b) If p q 2 n, t h e n the multiplicity of 1 f o r the pair (C, II) i s bigger f o r p q - n than the multiplicity of 1 f o r the pair (C*, II*).

+ +

410

Corollary 1.1. According to the notations of the theorem 1.5, i) The number of nonzero and nonunit principal values (each value counts many times as its multiplicity) of the pair (C,II) is less or equal to 4 2 ; ii) If n is odd number and p = q, then at least one of the pairs (C, II) and (C*,II*) has a principal value 1, i.e. they have a common subspace of dimension 2 1. Now we are able to give the canonical form of two subspaces together with their orthogonal subspaces. Without loss of generality we assume that p 5 q and moreover we also assume that p q 5 n, because if p q > n then ( n - p ) ( n- q ) < n and we can consider the subspaces C* and II*. Assume that 1 = q, > c1 > c2 > ... > c, > c,+1 = 0 be the principal values for the pair (C, II) with multiplicities ro, r1, . . . ,r,+1 respectively, such that p = ro r1 ... r,+1. Let C be generated by the following orthonormal vectors

+

+

+

+ + +

ao1,. . . ,aoro, a111 .. . a1r1 1

. ,as1,. . . ,asr, as+1,1,.. . ,as+l,rs+l, 7

7 . .

such that the vectors a i l l .. . ,airs generate the principal subspace for the principal value ci, (0 5 i 5 s 1). The pair of subspaces (C*,II*) have the same principal values 1 = Q > c1 > c2 > ... > c, > c,+1 = 0 with multiplicities rb = TO n - p - q, r1, . . . ,r,+1. Assume that C*is generated by the following orthonormal vectors

+

+

4 17

...

7

7

4 17

...

7

,. . . a,., , . . . ,a,.rs 7

7

where the vectors a;l,. . . ,azri generate the principal subspace for the principal value ci, (1 5 i 5 s l),a&, . . . , a&, generate the principal subspace for the principal value 1 and a;, . . . ,a:-: be the rest q - p orthonormal vectors. Now we choose the orthonormal vectors of II as follows. We choose

+

b01,...,b~ro, b l l , . - . , b l r l , . . . , b s l , . . - 7 b s r s , b s + 1 , 1 , . . . , b s + ~ , r , + ~ , b l .,- . , b q - p

such that boz coincides with a ~ for i 1 5 i 5 T O , b,l,. . . ,birt generate the principal subspace for the principal value cz1(1 5 i 5 s ) and such that (aiu,bzw) = 5,, ci. The vectors b,+l,l,. . . , bs+l,r,+l generate the same subspace like the vectors a:+,,,,-.. and for 1 5 i 5 rs+l, we can choose b,+l,, = a:+l,,. The vectors b l , . . . ,bQ-,generate the same

411

space like the vectors a;, . . . ,a&p and we can choose bi = a;-p+l--i for 1 5 i 5 q - p . Finally we determine the orthonormal vectors of II*

Gl, . ,b;,; * *

7

b;, . . . 7 qT1 ,. . * ,b,*l,* . . 7 b,*,.7 b;+l,l,* . . 7 b;+l&+, 7

as follows. The vectors b&,.. . ,b;,;, can be chosen such that b&= a& for

1 5 i 5 T&. The vectors bzl, . . . ,bi*,igenerate the principal subspace for the principal value ci, (1 5 i 5 s), and the vectors b:l, . . . ,b:,< can uniquely be chosen such that (a:u,b:,,) = b,,ci. The vectors b:+,,,, . . . ,b:+l,,s+l generate the same subspace like the vectors az+l,l, . . ,a:+l,,a+,and thus we can choose bf+l,i= for 1 5 i 5 ~ , + 1 . Moreover, the vectors a;, , .. . ,a;,,, . . ,afl,. . . ,a,*,s can be chosen such that

-

-

(ai*,, bi,) = -buu

41

-

c:

for 1 5 i 5 s. Now we know some of the inner products between the base vectors of C and C* and the base vectors of II and II*. The matrix P of all such n x n inner products must be orthogonal and can uniquely be obtained from the above inner products. Considering the base vectors of C in the mentioned order together with the base vectors of C* in the opposite order and on the other side the base vectors of II in the mentioned order together with the base vectors of 11*in the opposite order we obtain the following

. . . + T,

+

+ (q -p ) +

(TO

+T1+

(To

+ r l + . . . + r , +r,+1+ ( q - p )

T,+1

T,+1

+r,+1

+ + . .. + + +r, + ... + +.A) 7-1

T,

T&) x

T1

matrix as a canonical matrix for the subspaces C and II: ‘I

0 0 C1I 0 0

P=

0 0 0 0

0 0 0 0

0

0

0 0

... ...

c ~ I...

0 0 0 0 0

0 0 0

0 0 0 0 0 0 0 0 0

0 0 0

...

0

... d2I‘ 0 0

... csI 0 0 0 dsI’ ... 0 ... 0 001’ 0 ... 0 ... 0 OIO o . . . o ... 0 1 ’ 0 0 0 ... 0

... -d,I’ 0 0 0 csI ... 0

0 0 -d2I‘ ... 0 -d,I’ 0 ... ,o 0 0 ...

0 0 0

0 0 0 0 0 0 0 0 0

0 0

... 0 d1I’O

0 0

0 0

. o o 0 0 0 0

0 ... c2.l 0 0 0 ... 0 ClI 0

o...o

O I

I

412

d q ,

where di = (1 5 i 5 s) and I' denotes the matrix with 1 on the opposite diagonal of the main diagonal and the other elements are zero. Note that the principal values of the pair (C,ll*) (also (C*,II)) are the numbers d: = 1- C: = sin2 pi with the same multiplicities as c:. Moreover the previous canonical matrix P is also canonical matrix for the pair (C, ll*) (also (C*,ll)) if we permute its rows and columns. Then the order q - p converts into n - p - q and vice versa.

+

Theorem 1.6. Let n , p , q be positive integers such that n 5 p q and p 5 q. Then for any p values c1, ... ,cp (0 5 ci 5 l), there exist two subspaces C and II of En with dimensions p and q such that c1,.. . ,cp are principal values for the pair (2,ll). The existence of the subspaces C and ll is uniquely up to orthogonal motion in En. 2. Canonical normal vectors and normal curvatures of the first order Let us consider an n-dimensional manifold M' imbedded in the Euclidean space R" where m = n + k and choose an arbitrary point on M'. We choose arbitrary orthonormal n vectors Y1,. ,Y n of the tangent space at the chosen point on M' and choose also arbitrary k orthonormal vector fields N1 ,.. . ,NI, which are orthogonal to the tangent spaces in a neighborhood of the chosen point. Our aim in this section is to obtain a subspace of the space generated by the vectors N1 ,. . . ,NI, at the chosen point such that its sum with the tangent space gives the (first order) osculating space at the chosen point of the imbedded submanifold M'. We convenient to denote the vectors N I , . . . ,NI,by Y(,+1), . . . ,Y ( mand ) without loss of generality we assume that the coordinates Yh, of Y ( j satisfy ) = b& , (1 5 i ,j 5 n) at the considered point. Using the Taylor's series

yj,

where the partial derivatives are calculated at the considered point, the condition of orthonormality of Y ( n + l.).,. ,Y ( myields ) to m

n

dxj + i=l

j=1

12

n

n

u=l v = l

d2Y dzu dxv + . . . axu ax"

___

413

Proff. for a Proff. for a Proff. for a

414

Using that

rjk = -

8Zl"'

m C Y

& ) T where Z(2) for i

= 1 , 2 , . . . , m are the

u=l

yj,

1-forms dual to the vector fields Y ( i )in , the special case = 6; at the considered point, and using the orthonormality of the vector fields Y ( j )we , get

Hence we obtain n

A,, = ,a,

n

n

qu ,,:?I

-

dx" dz"

+ . .. =

i = l u=l v = l n n n

i = l u=l v = l

We are interesting for the non coinciding subspaces of the normal spaces of two close points of M', i.e. for the eigenspaces of eigenvalues different of 1. Thus we are interesting for the eigenspaces for the non-zero eigenvalues of the matrix n

B(') = ST

n

7,

n

i = l u=l v = l

dx" dxv rg, r;, ds ds

and the sum of these eigenspaces for all directions together with the tangent space yields to the (first order) osculating space at the considered point. Indeed we are looking for such a new matrix whose sum of eigenspaces coincides with the sum of all eigenspaces of of all directions, i.e. together with the tangent space gives the (first order) osculating space. Note that B ( l ) is a matrix depending on the choice of the direction $, .. . , in the tangent space. Note also that B ( l )is a non-negative definite matrix. We choose any n orthogonal directions p l , . . . ,p n on the tangent space of M' and define matrix

B,;(I

-

B:;)(pl)

+ . . + B:;)(pn) . '

One can verify that the matrix B(l) does not depend on the choice of the orthogonal directions and it is given by n

n

i = l j=1

We prove now that BL:) is the required matrix that we are looking for. Indeed, B;:) as a sum of non-negative matrices, the sum of its eigenspaces

415

C contains each eigenspace of a matrix B,,( 1 )( p i ) , (i E 1,.. . ,n}). On the other hand C contains each subspace of any matrix B,, ( q ) where q is an

L)

arbitrary tangent direction, because

BL:) = BI;:)(q) + B$)(ql) ... + B::)(qn-~), where 41, ...,qn-l are orthogonal tangent directions which are orthogonal to the tangent direction q. Finally we proved that the space C together with the tangent space gives the (first order) osculating space at the chosen point. - ( 1 ). Now we are going t o give more convenient form of the matrix B,,

Lemma 2.1. The matrix B ( l ) is equal to the following matrix

Proof. Note that rn u= 1

and hence n

n

It is obvious that P(') does not depend on the choice of the orthonormal base. The eigenvalues of P(') are non-negative numbers and its eigenvectors are orthogonal. Let kl = rank ( P ( l ) )and let A!'), . . - ,Akl (1) be the positive eigenvalues and for any eigenvector ( P I ,. . . ,p k ) we consider the vector p l N l + . . .+pkNk and hence we obtain the following vectors N f ) ,. . . ,N g ) as eigenvectors from the normal space. These vectors do not depend on the choice of the base N,. The positive eigenvalues A?), . . . ,) :A are called the first normal curvatures and the corresponding vectors N P ) ,. . . , N ): are called the first normal vectors. Note that 51 5 n2 because in the right side of (2.2) there appear n2 summands. In this section will be considered the geometrical interpretation of the first normal curvatures and vectors. Finally we proved the following theorem.

Theorem 2.1. The vectors Y1,.. . ,Y n ,N f ) ,. .. ,N g ) generate the ('rst order) osculating space at the considered point.

416

Note that the Christoffel symbols ri, are antisymmetric with respect to i and j , and they satisfy also another property. Namely, using that the distribution generated by Y( 1).,. . , Y(,)is integrable, we obtain that y r23. = rr. 32,

(n+1IrIm, 1 I i , j I n )

and hence it is also true that

rz. T3 = r3. T2,

(n

+ 1 I r I m, 1 I i , j I n).

Finally we give the geometrical meaning of the eigenvalues of P ( l ) .For any unit orthogonal field of vectors N = q"N, where the coefficients qa are constants, we consider the covariant derivatives V,N in the unit direction p = (x, dx' ... Let its projection on the tangent space be denoted

,%).

g.

Now we choose n orthonormal directions pl , . . . ,p, symbolically by in the tangent space and consider the expression

dN aN dPl aP1 dP2 dP2 apn apn' It is easy to prove that the expression (2.3) does not depend on the choice of the orthonormal tangent vectors p l , ...,p,. Moreover, it easy to see that

-dN . - + dN - . - + .dN . . + -dN .-

and hence we obtain the following theorem.

Theorem 2.2. The eigenvalues of the operator P ( ' ) , i.e., B ( l ) are the values Cy=, for the corresponding eigenvectors q" N,, 1 5 cy I k.

E.

Moreover, the eigenvectors (eigenspaces) of P(') have such directions where En 3=i m.m apj apj has critical value (specially achieves its maximal and minimal value).

Note that in [ 51 it is given the following geometrical interpretation of the principal directions on the tangent space and the corresponding principal curvatures.

Theorem 2.3. The principal directions are such directions o n the tangent space where the expression ( takes its critical value (especially it takes its minimal or maximal value), where A'p is the angle between the tangent spaces of two close points, or equivalently the angle between the normal spaces of the two close points. The principal curvature in a chosen principal direction is just the value ( in that direction.

g)2

g)2

417

According to the above notations, the principal curvatures and principal directions are determined by the following symmetric matrix n

n

m

i=l p=n+l

m

i = l p=n+l

Hence we obtain that traceC(l) = traceB(l),which means that the sum of the principal values (i.e. the trace of C(I)) is equal to the sum of the first normal curvatures (i.e. the trace of B(l)).These principal values and principal directions will be called the first principal values (curvatures) and the first principal directions or principal values (curvatures) and principal directions of the first order. Further will be introduced principal values and principal vectors of higher order.

3. Canonical normal vectors and normal curvatures of the second order Without loss of generality we can choose the orthonormal system of normal vector fields N1,. . . ,N k such that N1 = N,(1) , . . . ,N k , = Ng’. Now it is

Ni . N:’) = 0 for i > k l and j 5 k l . Analogously t o the matrix P(’) we can consider the following ( k - k l ) x ( k - kl) matrix n+kl

n

i=l j=]

+

for k l 1 5 a , p 5 k , and where we have denoted Yn+l = N1, Yn+2 = Nz, . . . ,Y n + k l = N k l . According to the choice of N f ) , . . . ,Ng) we get the simpler form

i=l j=1

If

k2 = rank

P(’)

= 0 at any point of the submanifold, then the manifold

+

locally can be imbedded in n k1-dimensional affine subspace of Rm. If k2 > 0 , let ( A 1 , . . . ,A k - k l ) be an eigenvector of P(2), then we consider the . . & - k l N k as an eigenvector. According to this idenvector A I N k l + l tification, the eigenvectors of P(’) and the principal directions do not depend on the choice of the basis N,. The positive eigenvalues A,(2) ,.. . , Xkz (2)

+. +

are defined to be the second normal curvatures and the corresponding eigenvectors NY), . . . ,Nk2 (2) are defined to be the second normal vectors.

418

Now we are going to prove that the vectors

..

Y1,.

generate the osculating space of the second order at the considered point. In this section we verify this statement like in the Section 2. In order to verify the geometrical interpretation of the second normal curvatures and vectors we apply analogous considerations like in the Section 2, where instead of the tangent vectors Y 1 , .. . ,Y n we consider the vectors y i , . . . ,Y n , N1 = Y n + 1 , . . . ,N k l = Yn+ kl its tangent vectors to the first osculating space and instead of the normal vectors N 1 , . . . ,Nk we consider the vectors N k l + l , ..., N k as normal vectors and denote also by Y ( n + k l + l ) , . . ', y ( m ) .

In order to consider the angles between the normal spaces, i.e. the space generated by the vectors N k l + l , . . ., N k at the considered point and the normal space of displacement dxz close to it, we considered the following ( k - k l ) x ( k - k l ) matrix

+ AY(r))

Msr = Y ( s ) . ( Y ( r )

(n

+ kl + 1 I

S, T

I m)

and should consider the eigenvalues of A = M M T . Analogously as in the Section 2, in this case we obtain n+k,

n

n

i=l u = l v = l

n+k,

n

n

i = l u= lv= l

We are interesting for the non coinciding subspaces of the normal spaces of dimension k - kl of two close points, i.e. for the eigenspaces for eigenvalues different of 1. Thus we are interesting for the eigenspaces for the non-zero eigenvalues of the matrix

cC C r:" r:,

n+kl

B$$)=

n

n

i=l u=lv=l

dx" dx" ds ds

--

and the sum of these eigenspaces together with the osculating space of the first order yield to the osculating space of the second order at the considered point. Note that B(') is a matrix depending on the choice of the direction .. . , in the tangent space. We choose any n orthogonal directions p l , . . . ,p n on the tangent space of M' and define matrix

g,

419

One can verify that the matrix B(2)does not depend on the choice of the orthogonal directions and it is given by n+kl

n

i=l j=1

Analogously as in Section 2, we obtain that the space of sum of the eigenspaces of Bbz) together with the first osculating space gives the second order osculating space at the chosen point. Next we prove the following lemma.

Lemma 3.1. The matrix P(') defined by (3.2) is equal to B ( 2 ) . Proof. Since m

u=l

we obtain

Theorem 3.1. The vectors

Y 1 ,

. . . ,Y n Nil), , . . . ,Nc ) ,N f ) , . . . ,N k(2)2,

generate the second order osculating space at the considered point. Analogous to (2.3), we consider a modified expression of (2.3) where we consider the projection of the partial derivatives on the osculating space of the first order, instead on the tangent space. Analogously t o the Theorem 2.2 we have the following theorem, which gives the geometrical meaning of the eigenvalues and eigenvectors (eigenspaces) of P ( 2 ) .

Theorem 3.2. The eigenvalues of the operator P ( z ) ,i.e. B(2)are the values En a. for the corresponding eigenvectors qa N,, k 1 + 1 5 a 5 k. 3=1 a p , Moreover, the eigenvectors (eigenspaces) of P(') have such directions where C3n.= 1 6N.m a p j a p j has critical value (specially achieves its maximal and minimal value). Now we introduce the second principal values and second principal directions, i.e. the principal values and principal directions of second degree. Namely, they are defined as eigenvalues and eigenvectors of the following matrix

i=l p=n+kl+l

i=l

p=n+kl+l

420

Hence we obtain that t r a c e C ( 2 ) = t r a c e B ( 2 ) ,which means that the sum of the second principal values (i.e. the trace of C(2))is equal to the sum of the second normal curvatures (i.e. the trace of B(2)).The geometrical interpretations of the second principal directions and curvatures are given by the following theorem. We omit the proof, because it is analogous to the proof of the Theorem 2.3.

Theorem 3.3. The directions o n the tangent space are such directions where the expression (%)2 takes its critical value (especially it takes its minimal or maximal value), where Acp is the angle between two osculating spaces of the first order of two close points, or equivalently Acp is the angle between the normal spaces (generated by N k l + l , . - ., N m ) of the two close points. The second principal curvature in a chosen principal direction is just the value ( $&2 in that direction. 4. Canonical normal vectors and normal curvatures of the higher order Continuing this procedure, the normal curvatures and normal vectors of higher degree are introduced [ 21. This procedure is finite since the number m is finite. We give only the inductive steps for the matrices P('+'), B('+') and C (l+l). Namely, P(l+') is ( k - kl - . . . - kl) x ( k - k l - . . . - kl) matrix given by

and r a n k P('+') 5 n . r a n k P@). Further for n kl ... kl 1 5 s , r 5 m it is

+ + + +

i=l

j=1

The geometrical interpretations are analogous like osculating spaces of the first and second order. The principal values and principal directions of higher order are defined analogously. The inductive step for the matrix c @+1)is given by

42 1

i=l

p=n+kl+...+kr+l

+

and the geometric interpretation of the ( I 1)-st principal values and the (1 1)-st principal directions is analogous like for the first and the second order. It is also true that traceC(') = traceB('1, which means that the sum of the 1-th principal values (i.e. the trace of C ( ' ) )is equal to the sum of the 2-th normal curvatures (i.e. the trace of &'I).

+

5. Curvature tensors of higher order

Now we are going to find the expression for the curvature tensor on M' with the given notations. Let us denote by V the covariant derivative in the Euclidean space R", and thus

R(X,Y) = [ vx,VY I - V[X,Y] = 0. The projection of V on the base manifold induces a connection V' given by

V&Z = vXz -

k

n

a= 1

T=l

C(N,-vxZ)N, = C(Y,. V~Z)Y,

for vector fields X and Z on the base manifold. The corresponding Riemannian curvature tensor yields to n

R'(X,YZ

=

([Vfc,VLl - v ~ x , Y , ) z = ~ I I R ( X , Y ) Z ] . Y r } Y r r=l n

k

r = l a=l n k r = l a=l

where Yi is an arbitrary basis of the tangent space. Using R ( X ,Y) Z = 0, we obtain n

k

r=l a=1

r=l a=1

422

and hence

n

k

r=l a=l n

k

n

r = l a=l

r=l

where

Note that the components of the curvature tensor depend only on the symbols for n 1 I :a 5 m and 1 5 i , j 5 n. These components also determine all principal directions, values and normal curvatures. Note also that the torsion tensor on M’ is automatically zero tensor which follows from the fact that the distribution determined by the vectors Y 1 , . . . ,Yn is integrable. Finally we determine the curvature tensors of higher orders. Namely, the first curvature tensor is just the above tensor field, i.e.

+

m

~$1” =

C

(rijrLi - rii rLj),

(1 5 T , S,i, j 5 n ) .

(5.3)

p=n+l

The second order curvature tensor is defined by (1 5 1 5 2 , j 5 n) ’(2)

R+

-

C

T, s

(rijrLi - riirij),

5 n

+ kl, (5.4)

p=n+kl+1

the third order curvature is defined by (1 5

T,s

5 n+kl+

k2,

1 5 i , j 5 n)

m

p=n+kl+kzfl

and so on. The geometrical interpretation of these curvatures is the following. Having in mind the geometrical interpretation of the curvature tensor (i.e. the first curvature tensor) by the parallel displacement of any vector along a closed circuit, the second curvature tensor is represented as the

423

curvature tensor applied to the first order osculating space. The third curvature tensor is represented as the curvature tensor applied to the second order osculating space and so on. Note that in the second curvature tensor the indices s and T take values from 1 to n Icl with the analogous geometrical interpretation. In the third curvature tensor the indices s and T take values from 1 to n Icl k2 with the analogous geometrical interpretation and so on. Namely, the indices are related to the principal vector bundles (the first curvature is related to the tangent space, the second curvature is related to first osculating space, the third curvature is related to the second osculating space and so), while the indices i and j always take values from 1 to n, because they are related to the coordinates on the submanifold M'. Thus the second curvature tensor can be defined also by

+

+ +

where X and Y are tangent vectors and Z is vector from the first osculating space. Then the left (right) side of this equality is also vector from the first osculating space. If we consider the third curvature tensor, then X and Y are tangent vectors and Z is a vector from the second osculating space and so on. Note that these curvature tensors are defined in the special choice of the orthonormal frame at the considered point as it was done in the previous section. If we change the basic frame then we should apply the tensor transformation of them such that these curvature tensor fields are well defined. References 1. I. B. Risteski, K. G. TrenEevski, Principal values and principal subspaces of two subspaces of vector space with inner product, Contribution to algebra and geometry 6, 289-300, (2001). 2. K. TrenEevski, N e w approach f o r the submanifolds of the Euclidean space, Balkan Journal of Geometry and Its Applications 2, 117-127, (1997). 3. K. TrenEevski, Principal directions f o r submanifolds imbedded in Euclidean spaces of arbitrary codimension, General Mathematics 5 , 385-392, (1997). 4. K. TrenEevski, O n the submanifolds of the Riemannian manifolds, Novi Sad J. Math. 29, 369-375, (1999). 5. K. TrenEevski, Geometric interpretation of the principal directions and principal curvatures of submanifolds, Differential Geometry - Dynamical Systems 2, 50-58, (2000).

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ON COMMUTATIVITY OF THE LIE DERIVATIVE AND COVARIANT DERIVATIVE AT A NON-SYMMETRIC AFFINE CONNECTION SPACE *

LJUBICA

s. VELIMIROVIC, + SVETISLAV M. MINCIC + AND MICA s. STANKOVIC~

Faculty of Science and Mathematics, Viiegradska 33, 18000 N i i , Y U G O S L A V I A E-mails: [email protected]. ac. yu, [email protected]

At the present work are exposed some previous results, related to infinitesimal deformations in a space L N with non-symmetric affine connection L;k., and then two theorems are proved: the first one on the commutativity of Lie derivative and covariant derivative, and the second one-on commutativity of Lie and absolute differential.

1. Previous results

Deformations and infinitesimal deformations of geometric magnitudes have been studied by many authors. We refer to [ 11, [ 21, [ 31, [ 71, [ l o ] , [ 111, [ 121 for more details and references. Let us consider a space LN of non-symmetric affine connection Lik with the torsion tensor T j k = L;.k - L&, at local coordinates zi (i = 1,. . . ,N ) . At the beginning we are giving some basic facts according to [ 71, [ 91, [ l o ] , [111, [121.

-

Definition 1.1 A transformation f : LN L N : z = (d, . . . , zN)= (xi)+ 1= (&, . . . ,zN)= (z'),where 1=z

+

Z(Z)E,

(1.1)

* MSG2000: 53C25,53A45,53B05. Keywords :infinitesimal deformation, non-symmetric affine connection, Lie derivative.

t Work partially supported by the Serbian Ministry of Science, Technologies and Development under contract No MM1646.

425

426

or in local coordinates

zi= x i + Z Z ( l C j ) & ,

i,j =l,...,N,

(1.2)

where E is an infinitesimal, is called infinitesimal deformation of a space L N ,determined by the vector field z = (zi),which is called infinitesimal deformation vector field . For geometric object A(&) and corresponding deformed geometric object A(zi),we will have

+

d(zi)= A(zi) DA,

(1-3)

where

Dd

= &LC,A.

(1.4)

The Lie derivative of geometric object d ( z i )is defined by

L z A = lim

d(?) - d ( z i ) &

E'O

with respect t o the vector field z = z ( d ) . For a tensor of the kind ( u , ~we ) get

where we denoted .

.

,

)a(;

. . ti?...i? = tt?...tu 31...3w

31 . . . j q - l p j p + l . . . j w7

(1.7)

and for the connection coefficients we have

LzL"j k

= L" J ~ , z Pp

+ zi. , j k - z:pL;k

f

ZSLfk

f

.$Lip,

(1.8)

Because of non-symmetry of connection, a t LN we can consider two types of covariant derivatives for a vector and four types for general tensor. SO, denoting by I(0 = 1,. . . ,4) derivative of the type 0, we have ([ 41, [ 51, [ 61): e

As it is proved in [ 10, (eqs. 2.12 a-d)], the Lie derivative of a tensor of the type (u, u)in L N can be presented by virtue of covariant derivatives in the

427

form

2. Commutativity conditions of covariant and Lie derivative, absolute and Lie differential 2.1. At the beginning we shall examine the commutativity condition of covariant and Lie derivative of a tensor. Using the first kind derivative for . . a tensor based on (1.9) i.e. for (A, p , v) = (1,2,2) we have

ti:::;z,

By subtraction of these equations one obtains

428

and

the equation (2.3) becomes

On the base of (2.10') from [ 71

the previous equation gets the form

From here one concludes: if the Lie derivative of the connection is zero, the commutativity of Lie derivative and covariant derivative of a tensor is valid. Conversely, if the commutativity is valid, i.e. the left side at (2.4) is zero, the right side will be zero too,. and the Lie derivative of the connection . will be zero, because the tensor ti:;:;: is arbitrary. Remark that in place of I one can use 1, for 6 = 2,3,4 and obtain equations analogous to (2.4). 1

e

429

In this manner, we obtain

With respect t o exposed, we obtain

Theorem 2.1. Necessary and suficient condition for commutativity of the Lie and covariant derivative (of the kind 8, f o r 8 = 1,2,3,4) is the annulment of Lie derivative of the connection. 2.2. Starting from (2.4), we can get a commutativity condition of the Lie and absolute differential. Composing (2.4) with dxm and taking into consideration that L,(dx") = 0, we get

Noting with b the absolute differential obtained with regard to 1

1 , we have 1

Multiplying both sides with .E and noting a Lie differential with V ,from the previous equation one gets

430

and, based on (2.5), we obtain 2 3

2 3

4

4

Analogously to previous case, we conclude that is valid

Theorem 2.2. A necessary and suficient condition for commutativity of the Lie and absolute differential i s V L ; , = Lc,Lj, = 0 , i.e. annulment of the Lie differential (Lie derivative) of the connection. References 1. I. Ivanova- Karatopraklieva, I. Kh. Sabitov, Surface deformation, J. Math. Sci. 70, No 2, 1685-1716, (1994). 2. I. Ivanova-Karatopraklieva, I. Kh. Sabitov, Bending of surfaces II, J. Math. Sci. 74, No 3, 997-1043, (1995). 3. J. Mikes, Holomorphically projective mappings and their generalizations, J. Math. Sci. 89, No 3, 1334-1353, (1998). 4. S. M. MinEiC, Ricci identities in the space of non-symmetric a f i n e connexion, MatematiEki vesnik 10 (25), Sv. 2, 161-172, (1973). 5. S. M. MinEiC, New commutation formulas in the non-symmetric a f i n e connexion space, Publ. Inst. Math. 22 (36), 189-199, (1997). 6. S. M. MinEiC, Independent curvature tensors and pseudotensors of spaces with non-symmetric a f i n e connezion, Coll. math. SOC.J&nosBolyai 31, Dif. geom. , 445-460, (1979). 7. S. M. MinEiC, L. S. VelimiroviC and M. S. StankoviC, Infinitesimal Deformations of a Non-symmetric A f i n e Connection Space, Filomat NiS, 15, 175-182, (2001). 8. J. A . Schouten, Ricci calculus, Springer Verlag, Berlin- Gotingen- Heidelberg, 1954. 9. R. StojanoviC, Osnovi diferencijalne geometrije, Gradjevinska knjiga, Beograd, 1963. 10. L. S. Velimirovid, S. M. MinEid and M. S. StankoviC, Infinitesimal deformations and Lie derivative of a non-symmetric affine connection space, submitted. 11. K. Yano, Sur la theorie des deformations infinitesimales, Journal of Fac. of Sci. Univ. of Tokyo 6 , 1-75, (1949). 12. K. Yano, The theory of Lie derivatives and its applications, North-Holland Publ. Co., Amsterdam, 1957.

SPECIAL CLASSES OF THREE DIMENSIONAL AFFINE HYPERSPHERES CHARACTERIZED BY PROPERTIES OF THEIR CUBIC FORM *

LUCVRANCKEN~ LAMATH, ISTV2, Uniuersite de Valenciennes, Campus du Mont Houy, Cedex 9, 59313 Valenciennes, FRANCE E-mail: luc. [email protected]

It is well known that locally strongly convex affine hyperspheres can be determined as solutions of differential equations of Monge-Ampere type. The global properties of those solutions are well understood. However, due t o the nature of the MongeAmpere equation, not much is known about local solutions, particularly if the dimension of the hypersurface is greater then 2. By the fundamental theorem, affine hyperspheres are completely determined by their metric h and their difference tensor K which together build the symmetric cubic form C. Following an idea of Bryant [ l ] ,we want to investigate affine hyperspheres for which a t each point there exist equiaffine transformations of the tangent space preserving both the affine metric h and the cubic form. The first non-trivial case is the case that M is 3-dimensional which is also the case which is investigated further in this paper.

1. Introduction In this paper we study nondegenerate affine hypersurfaces M" into IWn+', equipped with its standard affine connection D. It is well known that on such a hypersurface there exists a canonical transversal vector field I ,which is called the affine normal. With respect to this transversal vector field one can decompose

DxY = VxY

+ h ( X ,Y ) c ,

(1.1)

thus introducing the affine metric h and the induced affine connection V. The Pick-Berwald theorem states that V coincides with the Levi Civita * MSC.2000: 53A15. Keywords :affine hyperspheres, afFine differential geometry. t Partially supported by a research fellowship of the Alezander von Humboldt Stzftung (Germany).

431

432

connection 6 of the affine metric h if and only if M is immersed as a nondegenerate quadric. The difference tensor K is introduced by

KxY

= VxY - GxY

(1.2)

It follows easily that h ( K ( X , Y ) , Z )is symmetric in X , Y and 2. The apolarity condition states that trace K X = 0 for every vector field X . The fundamental theorem of Dillen, Nomizu and the author, see [ 51 implies that an affme hypersurface is completely determined by the metric and the difference tensor K . Deriving the affine normal, we introduce the affine shape operator S by

DxJ = - S X .

(1-3)

Here, we will restrict ourselves to the case that the affine shape operator S is a multiple of the identity, i.e. S = H I . This means that all affine normals are parallel or pass through a fixed point. We will also assume that the metric is positive definite in which case one distinguishes the following classes of affine hyperspheres: (i) elliptic affine hyperspheres, i.e. all affine normals pass through a fixed point and H > 0, (ii) hyperbolic &ne hyperspheres, i.e. all affine normals pass through a fixed point and H < 0, (iii) parabolic afFine hyperspheres, i.e. all the affine normals are parallel ( H = 0). Due to the work of amongst others Calabi [ 21, Pogorelov 15, Cheng and Yau [ 41, Sasaki [ 171 and Li [ 111, positive definite affine hyperspheres which are complete with respect to the affine metric h are now well understood. In particular, the only complete elliptic or parabolic positive definite affine hyperspheres are respectively the ellipsoid and the paraboloid. However, in the local case, one is far from obtaining a classification. The reason for this is that affine hyperspheres reduce to the study of the MongeAmpkre equations. In this paper we want to make a distinction between different affine hyperspheres based on special properties of their difference tensor K . In order to do so, we apply an idea of Bryant [ 11 to the case of &ne hyperspheres. Namely we want to make a distinction between different affine hyperspheres based on the number of orientation preserving isometries of the tangent space preserving the difference tensor K at a given point.

433

In case that the dimension is two, it is easy to verify that either K vanishes identically at a point or K is preserved by rotations with angle 2 ~ 1 3 .So the first non trivial case is the case that the dimension of M is 3. This is the case that we will consider in this paper. At every point p of M , we can introduce a symmetric polynomial f p by

f P ( x , y , z )= h ( K ( x e l + Y e2

+ z e 3 , x e l + Y e2 + ~ e 3 )x,e l + Y e2 + z e 3 ) ,

where {el,e2,e3} is an orthonormal basis at the point p. The apolarity condition implies that the trace of this polynomial with respect to the metric vanishes. As far as such symmetric polynomials with vanishing trace on a 3-dimensional real vector space are concerned, we quote the following result by Bryant:

Theorem 1.1. Let p E M and assume that there exist an orientation preserving isometry which preserves fp. Then there exists an orthonormal basis of TpM such that either (i) f p = 0, in this case f p is preserved by every isometry, (ii) f p = X (2 x3 - 3 x y 2 - 3 x z 2 ) , f o r some positive number X in which case f p i s preserved by a 1-parameter group of rotations, (iii) f p = 6 X x y z f o r some positive number A, in which case f p is preserved by the discrete group A4 of order 12, (iv) f p = X ( x 3- 3 2 y 2 ) for some positive number A, in which case fp is preserved by the discrete group 5’3 of order 6 (v) f p = X ( 2 x 3 - 3 x y 2 - 3 x z 2 ) 6 p x y z , for some X,p > 0 , with X # p, in which case f p is preserved by the group Z2 of order 2, (vi) f p = ~ ( 2 - 32 x ~ y 2 - 3 x z 2 ) + p ( y 3 - 3 x y 2 ) f o r some X,p > 0, with p # fix, in which case f p is preserved by the group Z3.

+

In this paper, we will assume that one of the special cases of the above theorem is satisfied at every point of the hypersphere. The paper is organized as follows. In Section 2, we will deal with the case that at each point p , either Theorem 1.1(i) or (iii) is satisfied. The case that M is an affine hypersphere satisfying either Theorem 1.1 (ii) or Theorem 1.1(vi) is studied in Section 3, whereas the case that M satisfies Theorem 1.1 (iv), which also corresponds to Chen’s equality studied in [ 181, [ 71 and [ 91, is considered in Section 4. Finally, the case that M satisfies Theorem 1.1 (v) at every point is investigated in Section 5 . We will call M an a f i n e hypersphere of Type k i f and only i f Theorem 1.1 (k) is satisfied at each point p of M . To conclude this introduction, we remark that the basic integrability conditions for an

434

affine hypersphere state that

E ( X , Y ) Z = H ( h ( Y , Z ) X- h ( X , Z ) Y )- [ K x , K y ] Z ,

(1.4)

(?XK)(Y, Z ) = ( Q Y K ) ( X ,2).

(1.5)

Remark that by applying an affine transformation, we may always assume that H = E , where E E { - l , O , 1).

2. Affine hyperspheres of Type (i) or (iii) First, we remark that if M is an affine hypersphere of Type (i) or Type (iii), there exists a local orthonormal basis { e l ,e2, e3) such that

K ( e 1 , ~=) Xe3,

K(e27e3) = X e l ,

K(e3,el) = X e 2 ,

K(e1,ed = 0 ,

K(e2,e 2 ) = 0 ,

K(e3,e 3 ) = 0 .

Substituting this in (1.4) implies that

E ( X ,Y ) Z = ( H

+ X 2 ) (h(Y,Z ) X - h ( X ,Z ) Y ) .

Consequently, Schur’s lemma implies that X is a constant and that M has constant sectional curvature. Thus, from the classification of positive definite affine spheres with constant sectional curvature, see [ 121, [ 81 or [ 191, it follows that M is affine congruent with a positive definite quadric, if X = 0, or with the affine hypersphere described by X I . . . xn+l = 1, if X # 0. Summarizing the above we have that

Theorem 2.1. Let M be an afine hypersphere of Type (i). Then M is afine congruent with a positive definite quadric. Theorem 2.2. Let M be an afine hypersphere of Type (iii). Then M is affine congruent with the flat afine hypersphere described by X I . . . xn+l = 1. 3. Affine hyperspheres of Type (ii) or (vi)

In this section, we assume that M is an affine hypersphere of either Type (ii) or Type (vi). This means that at each point p there exists an orthonormal basis { e l , e2, e3) such that

K ( e l , e l )= 2 X e 1 , K(e2,e 2 ) = -A el

K(e1,e2) = - X e l ,

+ p e2,

K(e3,e l ) = -A e3,

K(e2,e3) = -p e3, K(e3,e3) = -A el - p e2,

where X is a positive number and p # f i x . If M is an affine hypersphere of Type (ii) then p = 0, whereas if M is an affine hypersphere of Type (vi) then p > 0.

435

Let { e l ,e2, e3) denote the orthonormal basis constructed before, and denote bY

1 G(Y, Z ) = - trace{X 2

++

R^(x,

Y)z),

the Ricci tensor associated with the affine metric h. Then, by a straightforward computation using (1.4) and the explicit expression for K it follows that:

Lemma 3.1. Let {el,e2, e 3 ) be the orthonormal basis defined previously. Then it follows that h

Ric(el, e l ) = E h

Ric(e2, e2) = E

+ 3 x2, +X2 +p2,

h

h

Ric(el, e2) = 0,

Ric(e3, e l ) = 0,

h

h

Ric(e2,e3) = 0 , Ric(e3, e3) = E

+ X 2 + p2.

As p 2 # 2 X 2 , we see as a consequence that the 1-1 symmetric tensor field P associated with the Ricci tensor has at each point two different eigenvalues, one with multiplicity 1, the other with multiplicity 2. We then have

Lemma 3.2. Let M be an affine hypersphere of Type (ii) or (vi) and let p E M . Then there exist orthonormal vector fields { E l ,E2, E3) defined on a neighborhood of the point p and diflerentiable functions X and p , with p 2 # 2 X 2 and X > 0 such that K(E17 E l ) =

El

K(E37 E l )

E37

=

7

K(E2,E3) = -P E37

K(El7E2) zz -XE17 K(E27E2)= K(E3,E3)

El

-A El

+ pE2, -

/J E2.

Moreover, if M is an afine hypersphere of Type (ii), then p = 0 , whereas if M is an afine hypersphere of Type (vi), then p > 0. Proof. From Lemma 3.1 it follows that the eigenvalues of the differentiable operator P have constant multiplicities. A standard result then implies that the eigendistributions are differentiable. We now take El a local vector field spanning the 1-dimensional distribution and for Ez and E3 local orthonormal vector fields spanning the second distribution. As El is uniquely determined, it follows that there exists a positive function X and differentiable functions p and Y such that

K(E1,El) = 2 X E l , K(E37 E l ) = -xE37 K(E2,E3) = --Y E2 - p E3,

K(E17E2) = -XEl, K(E2,Ez)= -XEl +pE2 - v E ~ , K(E3,E3)= -XEl - p E z + ~ E 3 .

436

+

If M is an affine hypersphere of Type (ii) then p 2 u2 = 0 and thus the proof is completed. Hence, we may assume that M is an affine hypersphere of Type (vi) implying that p 2 u2 # 0. Then, if we define

+

F2

+ sin 0 E3 , OE2 + cos OE3,

= cos 0 E2

F3 = -sin

we see that the basis {El1F2,F3} satisfies the conditions of the lemma provided that cos3Bp- sin3Ou > 0, cos 3 Bv

+ sin 3 Bp = 0.

+

As p2 v2 # 0 , it is clear that a differentiable function 8 satisfying the above conditions exists. 0 In the remainder of this section, we will always work with the orthonormal basis constructed in the previous lemma. We then introduce local functions a 1 , . . . , c 3 by Q E ~El = a1 EZ

+

A

a2 E3,

h

V E E3 ~ = -a2 El

- a3 E2,

+ b3 E3, Q E ~El = c1 E2 + c2 E3,

Q E ~= E -bi ~ El

+

V E ~= E -a1 ~ El a3 E3, QE~ El = bl E2 b2 E3,

+

Q,y2E3= -b2 El

- b3 E2,

Q E ~= E -ci ~ El

+

~3

E3,

A

V E , E ~= -CZ

El

- ~3 E2,

We now will use the equations of Codazzi (1.5) and Gauss (1.4), in order to obtain more information about the above defined functions.

Lemma 3.3. Assume that M is an a f i n e hypersphere op Type (ii) OT Type (vi). Denote by { E l ,E2, E3} the corresponding orthonormal basis. Then, we have a2 = b2 = c1 = a1 = 0, pa3 = 0 , b2 = c1, and the functions A, p and bl satisfy the following system of differential equations:

E1(X) = -4 bl A,

E l h ) = -P b l ,

El(b1) = -6

E2(X)

= 0,

E 2 ( p ) = -3pC3,

-J33(X)

= 0,

E3(P) = 3Pb3,

E2(bl) = 0, E3(bl) = 0 .

-

2

b1

- 3X2 ,

437

Proff. for a and on the other hand, we get that

(9EzK)(E1,E1)= 9E2(2%) - 2K(9E2E1,El) = 2 &(A) El f 4 bl A E2 -k 4 b2 AE3 Comparing both sides it follows that

&(A) = -a1 p - 4 b l A, &(A) = 2a1 A, Uzp

= 4b2A.

(3.1) (3.2) (3.3)

Similarly, using the other Codazzi equations, it follows that A(C1

- b2) = 0 ,

(3.4)

p(b2 + e l ) = -2Aa2 = -2p(3a3 - bz),

(3.5)

p (bl - c2) = 2a1 A,

(3.6)

4A(C2 - bl)

= 2pU1,

(3.7)

E3(A) = 2Aa2,

(3.8)

E2(4 = -p

(c2 - bl),

E3(4 = - p

(b2

(3-9) (3.10)

+ Cl),

- pc2, E2(p) = - A ( h - c2) - 3pc3,

(3.12)

E3(p) = -A

(3.13)

= -Xu1

(C1

- 3bz)

+ 3 p b3.

(3.11)

Proff. for a

438

El(b3) - E2(a3) = b

l ~ 2 a1 b2 - a1 a3 - bl

b3

+ a3 ~3 - ~3 b 2 ,

+ 2 p 2 + bz + + a3 b2- a3 c1 + bl c2 - b2 c1.

E3(b3) - E 2 ( ~ 3= ) E - X2

(3.21)

C:

(3.22)

As A # 0, it follows first from (3.4) that c1 = b2. We now consider 2 cases. First, we assume that p = 0, then it follows from (3.6) and (3.5) that al == a2 = 0. Combining (3.13) and (3.4), we get that c1 = b2 = 0. The fact that bl = c2 now follows from (3.7). In the case that p # 0, we proceed as follows. As c1 = b2 and p # 0, we deduce from (3.5) that a3 = 0. As Xp # 0, it follows from combining (3.6) and (3.7) that bl = c2 and a1 = 0. Similarly, it follows from combining (3.5) and (3.3) that a2 = b2 = 0. The differential equations for X and p now follow immediately from the remaining Codazai equations. In order t o obtain the differential equations for bl = c2, we use the Gauss equations (3.14), (3.18) and (3.19). These reduce to 2 El(b1) = --E - 3 X2 - b1,

E2(bl) = 0 ,

E3(bl) = 0 .

This completes the proof.

0

Remark that, as X > 0, the vector field El is globally defined on M . As a consequence of the previous lemma, we see that bl is independent of the choice of E2 and E3 and is therefore globally defined on M . It also follows immediately from the previous lemma that the distributions TI = span{E2,E3) and T2 = span(E1) are integrable and orthogonal with respect to the afine metric h. We also get that T2 is autoparallel and TI is spherical with mean curvature normal -bl El. Therefore according to [ 101 we have that ( M ,h) admits a warped product structure M = R x,f N 2 with f : R R satisfying

-

af = b l , at

dbl _ at - --E

- bf - 3 X 2 ,

dX

- = -4 bl A, at

where f , bl and X only depend on the variable t , with a / d t = El and the curvature of N 2 is given by

K ( N 2 )= e2f ( 6 - X 2

+ 2p2+ bf).

which we verify by a straightforward computation is indeed independent of t. Remark that it follows straightforward from the above differential

439

equations that d

-( E

at

+ b?) = -2 b l ( E - A2 + by), - X2 + b: vanishes identically on M - X2

implying that either E or it vanishes nowhere on M . In the latter case, we may, by translating f,i.e. by replacing N 2 with a homothetic copy of itself, assume that e2f ( c - X 2 + b : ) = E, where C = f l . It is also clear that U1 = e f E 2 and U2 = efE3 form an orthonormal basis on N 2 . Next, we introduce a positive function by the differential equation:

P, depending only on the variable t ,

a

at P = P (61 + A) It then follows that

DE1(EF +t)= 0 , DEz(fF

+ 6) = 0

DE3 (EF -k

(3.23) (3.24)

7

6) = 0,

DEI (P(-(bl

(3.25)

+ A) El -k 6)) = p (bl f A) (-(h f A) El f t ) )

(3.26)

-2p(bi+X)XE1+P(E+b?+3X2)Ei

- p ( b l + X ) ~ + 4 b l X p E l - € P E l= o , (3.27) DEZ (P (-(h A) El + 6)) = -P ( 6 b: - X2) E2, (3.28) D E Q ( P(-(hf A) El 5)) = -P ( E b: - X2) .&j.

+

+ +

+

+

Now we consider different cases. First, we assume that E b: - X2 does not vanish identically on M , in which case we have seen that the function vanishes nowhere on M . We then have the following theorem:

--

Theorem 3.1. Let $ : N 2 R3 be a proper, positive definite afine hypersphere and let y : I R2 be a curve. Let E = f l denote the mean curvature of the 2-dimensional afine hypersphere and define F : I x N 2 --+ W3 : ( t , u , U ) ++ ( y l ( t ) $ ( u , ~ ) , y 2 ( t ) ) . (i)

If Y = (71,721 satisfies (a) (7; 7 2 - 7; 71) 7 2 # 0 (b) Z Y 2 Y i ($7; - %7;) < 0 2 12 ( 4 (717; - 7; %I5 = - f 1% 7 2 (7Y 7; - Y2”74), then M is a 3-dimensional positive definite proper afine sphere,

(ii) If y = (y1,yz) satisfies

440

( 4 727; # 0 (b) c 7 2 7: (7; 7; - 7; 7 ; ) > 0 (c) 7:”; = - f 17h2(7Y-Y; - 7; 7 0 7 then M is a 3 dimensional positive definite improper a f i n e sphere. Conversely every afine hypersphere of Type (ii) or (vi) satisfying E bf 3 X 2 # 0 can be obtained in this way.

+ +

Proof. As (3.28) that

E

+ bf - X2 vanishes nowhere on M , it follows from (3.26) t o

4 = P (-(h+ A ) El + 51, defines a map from N 2 into R4.Moreover, it follows that

+ b: x ~ ) D E ~ E ~ + b: - A 2 ) ( - ( h + A) El + p + = p4*(E2) + b3 4*(E3)+ b? - X2) 4 - Ee2f4. = p & ( E 2 ) + b3

D E ~ ~ * (=E-P~ )( E

(3.29)

-

- -P ( E

E2

b3

E3

+ 5)

(E

Similarly, we obtain that DEz d* (J%) = --b3

$* (E2)- p $* (J%)

DE3 d* (’%) = (c3 - p ) h (&) D E ~ ~ * (=E(-3~ ) - p )

7

- ce2fd.

The above implies that 4 defines an immersion of N 2 as an equiaffine sphere in a linear subspace R3 of R4.The affine metric introduced by this immersion corresponds with the metric on N 2 . Next, we remark that’if we put

it follows that

D E 6~= D E 6~= 0. Hence 6 depends only on the variable t. Now, we consider two cases. First, we assume that E # 0. In this case, we may by applying a translation assume that ( = - E F . Solving then (3.29) and (3.30) for F , we find that

F ( t ,u,v ) = b ( t ) -

P(t)(E

1 b! - X 2 )

+

4(% v),

441

where u and v denote local coordinates on the surface N2. As

D,gl(E1 - (bl - X)F) = 2XE1+ E - (--E- b: = (3X - bl)El

- 3X2 f

4blX)F - (bl - X)E1

+ (b: + 3X2 - 4blA)F

= (3X - bl)(El - (bl - X)F),

it follows that 6’ and 6 are proportional. As F is linearly full this implies that 6 and 4 lie in mutually transversal subspaces. Consequently there exists a curve 7 = ( 7 1 , ~in ~ R2 ) such that after an affine transformation F can be written as

F(G u, v> = (7l(t>,72(t)4(U, v)),

(3.31)

where 4 is a positive definite proper affine hypersphere. It follows that

Ft = (7;,7;4>,

Fu

= (0,724zL),

Fv = ( 0 , 7 2 4 u ) .

As F is an immersion and F itself is a transversal vector field it follows that 7 2 # 0 and (7; 7 2 - 7; 71) # 0. We also have that

where g denotes the affine metric on the 2-dimensional f i n e hypersphere. Hence taking into account that 4 is an equiaffine sphere with mean curvature <, it follows that F defines an equiaffine hypersphere if and only if

4 b17;

2

-

12

7: Y Z ) = ~ flT~ y1 (7; 7;

- 7; T;),

whereas the condition that the induced metric is positive definite implies that (7: 7; - 7; 7 ; )2 7 2 7;

< 0.

442

This completes the proof in this case. In the case that as follows. We first remark that (3.30) reduces to

b(t)=

1

b: - X2 1 -bq - X2 ~

'-

1

bl - X

E

= 0, we proceed

El + F

(6 - (bi + A) El) + F

=

1

P (b? - X2)

4+F.

Hence, we still have that

F ( t ,U , W) = b ( t ) -

1

P

4@: X2)

As

and by an affine transformation we may assume that 6 = (1,0, 0,O). It still follows that if necessary after applying a translation we may assume that there exists a curve y = (yl,y2) in R2 such that

where that

4 :N2

-

F(t7U1w)

= (71(t)772(t)

R3 is a positive

Ft=($iy;4),

4(u7w))7

(3.32)

definite aEne hypersphere. It follows

FU=(o,%4U),

FW

=(O,%4tJ)*

As (1,0,0,0) is a transversal vector field it follows that 727; # 0. We also have that

Ftt Ft,

= (y;, 7;

4) = . . .Ft + (7; 4 - 7;

= ( 0 , ~&) ; =

7;

7;

- Fu, 72

7:)

[,

443

Hence taking into account that is an equiaffine sphere with mean curvature 2, it follows that F defines an improper equiaffine hypersphere with affine normal (1,0,0,0) if and only if

Ti5?;

= *1T;2(7?Yi

-7h;),

whereas the condition that the induced metric is positive definite and non degenerate implies that

which completes the proof of the theorem. Next we consider that E = &1 and E We then have the following theorem:

+ b:

0

- X2 vanishes identically on M .

-

Theorem 3.2. Let : N2 R3 : (u,w) ++ ( u , v ,f(u,w)) be a n i m proper positive definite a f i n e hypersphere with a f i n e normal (0, 0 , l ) and let y :I R be a curve satisfying

-

$J

-

where c i s a non-zero constant. Then,

F :I x N 2

R3 : ( 4u,w)

++

( Y I ~ , Y I Wyif(u, ,

w)

+72,711

defines a proper a f i n e hypersphere, which is positive definite provided (7: 7 2 - 71

4)# 0,

( Y X -7h&)+Y1 > 0 .

Conversely every proper a f i n e hypersphere of Type (ii) or (vi) satisfying E b: - X2 = 0 can be obtained in this way.

+

Proof. From (3.25) to (3.28) it follows that we can introduce constant vectors C1 and C2 such that

E F + <= C1,

P (-(hf A) El + E ) = c 2 .

Hence it follows that

or equivalently (3.33)

444

Using that

E

+ b!

-

A2 = 0, we can rewrite the above as

a

+

(3.34) F - E C1 (bi - A) C2 E (bi - A) p-' . at We now fix an initial value to. As mentioned before, we know that the distribution TI is integrable. Let NO denote the integral manifold through to. AS - F = ( b l - A)

+

+

DE~J% = - ( b l + A) El pE2 b3 E3 + (, D E ~= E -b3 ~ E2 - P E 3 , DE~= E (c3 ~ - p) E3, D E ~= E (-c3 ~ - p ) E2 - ( b l + A) El E.

+

and DE2(-(bl -k A) El

+ E ) = 0 = D&(-(h + A) El + r)

it follows that this integral manifold is contained as an improper affine sphere in a 3-dimensional affine subspace of R4with affine normal a multiple of C2. Moreover, choosing the initial conditions for p appropriately, we may assume that the affine normal actually is C2. Hence by applying a translation and an affine transformation, we get that

W t o , u , v ) = (u, 21, f(u, v), O ) ,

(3.35)

where ( u , v ) ++ (u,w,f(u,v) defines an improper affine sphere with afFme normal (O,O, 1). As the immersion F itself is nondegenerate it follows from (3.34) and (3.35) that the vector C1 is transversal to the space spanned by F(t0,u,v). Therefore by applying an affine transformation, it follows that C1 = (0, O,O, 1). We then get immediately from (3.34) and (3.35) that there exists a curve y = (yl,y2) such that

F ( t , u , v )= (ylu,ylv,ylf(.u,v) + Y 2 , Y 1 ) . It then follows that

Ft = ( 7 ; u > 7 ; W ; f ( ~ ? J )+7;,79, W F = (O,Yl,YlfW,O)

Fu = ( Y l , O , Y l f U , O ) ,

and

Fu, = 7; - Fu, 71

(3.36)

445

As

we see that the hypersurfaces defined by (3.36) defines an affine sphere provided that there exist a constant c such that

where, in order for the immersion to be positive definite and nondegenerate, 7 satisfies moreover:

(7; 7 2

-

71 74) # 0 ,

(7; 7; - 7:’ 74)7; 71

>0.

0

Finally we deal with the case that M is an improper affine hypersphere such that b? - X2 = 0. Those hyperspheres are obtained as in the next theorem:

-

Theorem 3.3. Let ?c, : N 2 R3 : ( u , v ) H (u,v, f ( u , v ) )be a positive definite improper a f i n e sphere with a f i n e normal (0,0 , l ) . T h e n F ( t , U , v ) = ( U t , v t , t ,f ( U , V ) t $ C t 4 ) ,

+

F(t,U,v) = ( u ,v , f ( ~v ), c t 3 , t 4 ) , where t > 0 and c i s a positive constant, define improper afine hyperspheres of Type (ii) or (vi). Conversely every improper a f i n e hypersphere of Type (ii) o r (vi) satisfying b? - X2 = 0 can be locally obtained in this way.

Proof. As b? - X2 = 0 and A > 0, we have two cases to consider. First, we assume that bl = X > 0. In this case, the differential equation for ,B states that

from which we deduce that we can take p = 1/&. Proceeding as in the previous theorem, we find that there exist constant vectors C1 and C2, where C2 is a multiple of (O,O, 1 , O ) and C1 is a multiple of (O,O, 0 , l ) such that (3.37)

446

and such that the integral surface through the point to is given by F(to7

). = ( u 7

w7

f(.7

v)7

O),

(3.38)

where (u,u)H (u, w,f(u, w)) describes an improper affine sphere with affine normal (O,O, 1). Combining (3.37) and (3.38) it is then clear that there exists a curve y = ( ~ 1 ~ 7 such 2 ) that

F ( t ,u,w) = (u, 21, f(u,v ) +

YZ(~)).

%(t)7

(3.39)

As

Ft = (o,o,Y:,y~),

Fu

Fv = (0717fv70)7

=(1,o,fu,o),

and

Fu,

= f ~ ~ ( 0 17, 0 O )7= . . . Ft - f u u

Fuv

= fUv(0, O 7 l 70) =

Fvv

. . . Ft

7; 7(07070,117 71

ra

- fuv - (0707 071),

7;

7;

= fvv(0,O , l , O ) = . . . Ft - f v v - (0707 07 1 ) 7

r’l

Fut = 0,

Fvt = 0,

Fit = .. . Ft

+ (7; ri - rY rb) (o, 0, 0, l), ri

we see that F defines an improper affine sphere with affine normal a constant multiple of (O,O, 0 , l ) if and only if

where C is a constant. It then follows by reparameterizing such that n ( t )= t , if necessary after applying a translation, that yz(t) = c t % . Next, we consider the case that bl = -A. By applying an affine transformation, we may assume that E = (O,O, 0,l). Proceeding as before, we obtain that

W o , u7 w) = (u7 0, 0, f(u7 w)), where (u,w ) H (u, u,f(u, w)) defines an improper affine sphere with affine normal (0, 0 , l ) . As the function X is deterermined by

a

- X = 4X2

at

> 0,

(3.40)

447

it follows that after a translation of the t-variable, we may assume that ~ = 2 X El 5, we get that F is determined by the X(t) = -1/4t. As D E El differential equation:

+

a2

1 8 (3.41) - F +<, 2 t at with initial condition F(to,u,v) = (u,w,O,f(u,v)). As El - (61 - A) F is constant along and transversal t o the integral submanifold we may, if necessary after applying an affine transformation, take as second initial condition that

-F = -at2

&(to, u,v)

+ 2 X(to) F(to, a ,v) = (O,O, 1,O).

As it follows from (3.40) that the map t H A is a diffeomorphism, we can take X as a variable. Then the differential equations reduce to (3.42) with initial conditions at the point Xo given by v,o, f(%v)), F(Xo, 21, ). = (u,

Solving the above differential equation (3.42), we find that

F(X,U , W) = C ~ ( U W),X-4

1 + C ~ ( UW) ,+ 4%

<.

Comparing now with the initial conditions it follows that 1 3 C z ( u , v ) = O,O,-,-Xx, 2 x 0 48

(

Hence applying an affine transformation and a change of variables it is then easy to check that F is congruent with an open part of

F ( t , u , v )= ( u t , v t , t , f ( U , v ) t + d t 4 )

0

Remark 3.1. It is actually quite surprising that a 3-dimensional affine hypersphere of any type (elliptic, hyperbolic or improper) can be constructed starting from a 2-dimensional affine hypersphere of arbitrary type. In particular, a 3-dimensional elliptic affine hypersphere can be constructed starting from a 2-dimensional hyperbolic affine hypersphere. This is not the case

448

when studying minimal Lagrangian submanifolds of the complex projective space, see [ 161. There, it is shown that in order to obtain a minimal Lagrangian submanifold for which the second fundamental form has a similar form as the difference tensor in Type (ii) or (vi) one has to start from a minimal Lagrangian submanifold in @ P 2 ( 4 ) .

Remark 3.2. If in the previous theorem, we assume that the starting hypersphere is hyperbolic, we see that a special curve y is given by

In this case, the resulting solution is the well known Calabi product of hyperbolic affine hyperspheres, see also [ 61 and [ 131.

Remark 3.3. It is well known that a 2-dimensional positive definite improper affine hypersphere can be locally written as 1 1 ( z - G ), - ( z z - G G) 2 8 where G is a holomorphic function. Combining this with the previous theorems yields many explicit examples of 3-dimensional positive definite affine hyperspheres.

(-

+

4. Affine hyperspheres of Type (iv)

In this section, we assume that M is an affine hypersphere of Type (iv). This means that at each point p there exists an orthonormal basis { e l , e 2 , e 3 } such that

K(e1,ed = Xe1, K(e2,ez)= - X e l ,

K(e1,e2) = -Xe2, K(ez,e3)= O ,

K(e3,e l ) = 0, K(e3,es)= 0,

where X is a positive number. From the above expression, assuming that E = A l , it is clear that M realizes at every point the equality in the following inequality, which was derived in [ 181 and which states 3R(p) -

sup

r;a(rI) 2 2 E .

n€Gz(TzM)

Here k and K denote respectively the normalized scalar curvature and sectional curvature. Furthermore, G2(TzM)denotes the Grassmannian of 2-dimensional subspaces of T z M . This inequality was motivated by the work of Chen for submanifolds of real space forms ([ 31). It was shown in [ 181 that a 3-dimensional proper affine hypersphere realizes at every point

449

the equality if and only if M is a proper affine hypersphere of Type (iv). As 3-dimensional elliptic and hyperbolic affine hyperspheres realizing the equality were respectively classified in 191 and [ 71, we will restrict ourselves in the remainder of this section to the case that M is an improper affine sphere, i.e. E = 0. First, we remark that a straightforward computation shows the following

Lemma 4.1. Let {el, e2, e3) be the orthonormal basis defined previously. Then it follows that h

h

h

Ric(e1,el) = A2,

Ric(e1,ez) = 0,

Ric(e3,el) = 0,

h

h

h

Ric(e2, e2) = A2,

Ric(e2, e3) = 0,

Ric(e3,es) = 0 .

We again see as a consequence that the 1-1 symmetric tensor field P associated with the Ricci tensor has at each point two different eigenvalues, one with multiplicity 1,the other with multiplicity 2. As in the previous section we then can show that there exist orthonormal vector fields {El,E2, E3) defined on a neighborhood of the point p and a non vanishing differentiable functions A, such that K(E1,El)=XE1,

K(E17E2)

K(E2, E2) z=

K(E27 E3) = 0 ,

El,

K(E3,El) = O , K(E3,E3) = 0.

=-XE2,

Remark that the choice of El and E2 is not unique. However it is possible to define two distributions To (the distribution determined by E3) and the distribution TI (spanned by El and E2), which will play a crucial role in the classification. Computing now all components of the Gauss equation, it then follows by a long but straightforward computation that the functions A, a1, . . . ,c3 have to satisfy the following system of partial differential equations:

+ + bf + b2 - b2 a3 + a2 b3 - a3 ~ 1 , E3(a2) - E l ( C 2 ) + cg + b2 c1 f b2 a3 - a1 c3 + a3 c1, Ez(a2) - El(b2) = bl a3 + bl b2 + a1 a2 - a1 b3 + c2 b2 c2 a3, E3(al) - El(c1) = a2 c3 - a3 c2 + a1 a2 + bl c1 + c1 c2 + bl a37 E ~ ( c z-)E3(b2) = blc3 - b3 a2 b2 + a2 ~1 - b2 b3 E2(~1) - El(b1) = 2X2

~1

U:

z=

-

~2 ~ 3 ,

~1 -

E2(~1)- E3(bl) = b3 ~2

-

~3 b2

E3(a3) - Ei(~3) = ai c2 - a2 ci

+ a1 ~1 - a1 b2 - ~

+

a2 a3

+

a3 b3

El(b3) - &(a3) = b l a 2 - a1 b2 - a1 a3 - bl

b3

b3, 3

+ b3 + c2 c3, ~1

+ a3 ~3 - ~3 b2,

bi + C; + a3 b2 - a3 CI + b i c2

E3(b3) - E Z ( C=~ )

1-~ bl

-

b2

~ 1 .

(4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9)

450

The number of unknowns in the above equations can be reduced using that K is a Codazzi tensor with respect to the affine metric. In particular, we can show the following:

Lemma 4.2. W e have (21)

c2

= c3 = 0 ,

(i3) a3 = 4

2 ,

1 3

(22)

~1

= - b2,

(24)

b3

= a2.

Moreover, the function X satisfies the following system of differential equations:

Proof. From

h

it follows immediately that V E ~ = E0.~ Consequently

c2

we deduce that K ( E ~ , ~ E=~KE( E ~ ~) , ? E ~ EAs ~ ) X. that b2 = -a3,

b3

= c3 = 0.: As

# 0,

this implies

= a2

The other equations are obtained similarly.

As the connection is torsion free and thus

El(E2(X))- E2(El(X))= (

W

2

- %El) A,

we deduce from the differential equations for X that El(a1)

+ E2(bl) = --23 a2 b 2 .

(4.10)

Using the previous lemma, the differential equations given by (4.1) t o (4.9)

45 1

now reduce to

(4.12) (4.13) (4.14) (4.15) (4.16) &(as)

+ Ez(b2) = 0 .

(4.17)

Now, we want to solve the above system of differential equations explicitly. In order to do so, we remark that by dividing M into several parts (and ignoring a set of measure zero), we may assume that either (i) the distribution spanned by El and E2 is integrable on M , (ii) the distribution spanned by El and E2 is nowhere integrable. We start with the first case. As the distribution is integrable, it follows that b = 0. In this case, we have the following theorem:

Theorem 4.1.

-

Let q5 : M 2 R3 : ( u , v ) H (u,w, f ( u , v ) ) be an improper a f i n e sphere with a f i n e normal (0,0,1). Then,

F(U,v,t ) =

(ul v,tl f (u,

1

+ 5t 2 )

7

-

is an improper a f i n e sphere of Type (iv) f o r which the distribution

TI is integrable. Let q5 : M 2 R3 : ( u , v ) H ( u , v ,f ( u , v ) ) be a proper elliptic a f i n e sphere. Then,

F ( u ,21, t ) =

(t 44% v),51 t 2 )

7

is a n improper a f i n e sphere of Type (iv) f o r which the distribution

TI is integrable. Conversely, every 3-dimensional improper a f i n e hypersphere of Type (iv) f o r which the distribution TI is integrable can be locally obtained in this way.

452

Proof. As b2 = 0, we have that both the distributions TO and TI are integrable. Hence there exist coordinates t, u,w such that E3 = a, and 8, and a,, span the distribution T I . It follows from (4.12), (4.13) and (4.17) that the function a2 is determined by ata2

= a27 2

=0.

&a2 =

First, we assume that a2 vanishes identically. In this case it follows that F(t0, u,w ) defines an improper &ne sphere with affine normal 5. Moreover, we have that along this surface, E3 is a constant vector. Hence by applying an affme transformation, we may assume that F(t0, u,w) = (u, w ,f(u, w),0 ) , E = (O,O, 1 , O ) and &(to, u,w) = (O,O, 0 , l ) . Taking into account that

it follows that F is congruent with F ( t , u,w) = (u, w ,t, f(u, w)

+ 51t2) .

This completes the proof in this case. By the differential equation for a2, we may now assume that a2 is a nonvanishing function oft. If necessary by replacing E3 with -E3, we may assume that a2 is positive. Solving the differential equation explicitly, we may assume that after a translation in the t-coordinate we have that l/a2 = -t. It now follows that F(t0, u,w) is an elliptic proper affine sphere with affine normal E3 5/a2. Using again that

+

a2

-F = D

at2

E ~= E E, ~

it now follows easily that we can write F(t, 21, ). =

(t 44% w),51 t2)

I

where (u, w)H (u,w,4(u,w)) defines an elliptic proper affine sphere in R 3 . 0 Next, we focus on the case that the distribution TI is nowhere integrable, implying that the function b2 is a nowhere vanishing function. In order to solve the system of differential equations (4.10)-(4.17) we introduce functions f,a1 7 ( ~ 2Pi , 7 Pz on by

453

where a1 and a 2 are well defined up to multiplication by egni, due to the nonvanishing of b2. We then have that

Consequently, we deduce that

+

E3(p1) = E3(El(f)) = EI(E3(f)) (vE3E1 - vElE3)f

--- _ 2 b 2 P 2 3 Similarly, we deduce that 2 E3(P2)=

5 b2P1+

a2 P 2 ,

Ei(P2)- E2(P1)= 2 b2 - a1PI- bl P 2 ,

2 El(a2) E~(a1) = -a1 a1 bl a2. 3 A straightforward computation, using the above differential equations, then shows that the vector fields T , U and V defined by E3(a2)= - b2 01 - a2 a27

+

+

T = E3, u = a1El +a2E2 + (alp1+a2/32)E3, V = - 02El + a1 E2 + (-a2P1+ a102)E3, satisfy that [ T ,U ]= [ T , V ] = [ U, V ] = 0. Consequently, we have that there exist local coordinates t, u and v such that a/% = T = E3,a/au = U and a/& = V . Solving now first (4.12) and (4.15), which can be rewritten as

a

- (a2+ i b2)

+

= (a2 i b ~ ) ~ , at we find that there exists functions C1 and C depending on the variables u and v such that -t c1 c a2 = (4.18) (-t C1)2 c2 ’ b2 = - (-t C1)2 c 2 From the definition of P1 and P 2 it then follows that C1 is a constant and thus by a translation of the t-coordinate we may assume that C1 vanishes identically. Solving now the differential equation for A, we find that there exist a function D depending on u and v such that

+

+

+

+

+

(4.19)

454

Solving the differential equations (4.13) and (4.17) for that Q2 P1 = -

auc + a1 auc a;

+

a1

P2

7

=

and P2 we deduce

auc

-

Q2

+

auc

ff; ff;

ff;

Using then the differential equations for X t o define the functions a1 and b l , we find after a straightforward computation that the differential equations (4.14) and (4.16) become trivially satisfied. On the other hand, the differential equations (4.10) and (4.11) imply that the functions C and D satisfy the following system of differential equations:

Ac= - 2 C E - t D ,

(4.20)

A D = (3 + 6 E20)E -tD.

(4.21)

We now can formulate the following theorem:

Theorem 4.2. Let S be a open domain in R2 and let (C,D ) be a solution of the system (4.20)-(4.21) such that C is a nowhere vanishing function on S . Then S x R can be immersed as an improper a f i n e hypersphere of Type (iv). Conversely every improper afine hypersphere of Type (iv) for which the distribution TI is nowhere integrable can be characterized by giving a pair of nonzero functions satisfying the above system of elliptic equations. Proof. We define functions a2, b2, a1, a 2 , A, PI, P2, a1 and bl as described in the previous equations. We define vector fields El, E2 and E3 on S x R by

El =

Q2

01

@;+a;

a;

+ a;

a u - 01 at

1

E3 = a t . We introduce a metric h on S x I by the assumption that El, E2 and E3 form an orthonormal basis and a difference tensor K by

K(E11E2) = - X E 2 1 K(E1,El) = XEl, K ( E 2 , E 2 ) =-XEi PEE^, K(&,E3) = O ,

K(E37 El)

= O,

K(E3,E3) = 0 .

We also define a shape operator S = 0. It is then straightforward t o check that h, K and S satisfy all the conditions of the fundamental theorem of [ 5 1. Consequently S x I can be immersed as an improper affine hypersphere with affine metric h and difference tensor K . Clearly, it is an improper

455

affine hypersphere of Type (iv). It is straightforward to check that the distribution TI is nowhere integrable. The converse statement was proved I3 immediately before we stated the theorem. 5. Affine hyperspheres of Type (v) Finally, in this section, we assume that A4 is an affine hypersphere of Type (vi). It is easy to see that this implies that at each point p there exists an orthonormal basis {el, e2, e3) such that

K(e1,e l ) = A1 el, K(e2,e2) = A2 e l , where 0

< A1

K(e3,e l ) = A3 e3, K(e3,e3) = A3 e3,

K(e1,e2) = A2 e2, K ( e 2 , e3) = 0,

= -A2 - A3 and 0

# A2 # A3 # 0.

By a straightforward computation we obtain the following:

Lemma 5.1. Let { e l , e2, e3) be the orthonormal basis defined previously. Then it follows that

-

Ric(e1, e l ) = E

+ A; + A: + A2

h

Ric(e1, e2)

X3,

= 0,

Ric(e2, e2) = E

+ A,,

Ric(e3, e3) = E

+ A:.

h

-

Ric(e3, e l ) = 0, Ric(e2, e3) = 0,

2

Using the various conditions on A2 and X 3 , we see that the 1-1 symmetric tensor field P associated with the Ricci tensor has at each point three different eigenvalues, all with multiplicity 1. Hence there exist orthonormal vector fields {El,E2, E3) defined on a neighborhood of the point p and a non vanishing differentiable functions XZ, X 3 , with A2 A3 < 0 and A2 - A 3 # 0 such that

+

K(E1,El) = - ( A 2 + X 3 ) El, K(E3,El)= A3 E3,

K(E1,E2) = A2 E2, K(E2,E2) = A2 El, K(E3,E3) = A3 E l .

K(E21E3) = 0,

Computing now all components of the Gauss equation, it then follows by a long but straightforward computation that the functions X 2 , X 3 , a l , . . . ,c3 have to satisfy the following system of partial differential equations: E2(ai)

+ 2 A2 + Xz + US + b: + b2 - b2 +

- E i ( b i )= E

A3

~1

E3(a2) - El(c2) == 6

f 2X i f

a3

a2

b3 - a3 C I ,

A3 f a ;

+ + b2 c1 + b2 a3 - a1 c3 + a3 C;

(5.1)

~ 1 ,

(5.2)

456

The number of unknowns in the above equations can be reduced using that K is a Codazzi tensor with respect t o the affine metric. In particular, by a straightforward computation we obtain the following:

Lemma 5.2. There exists a local function c such that (1) b2 = c1 = a3 = 0 ,

(2) a2

=

(4)

(5) bl

=c

c2

= cA3,

Moreover, the functions equations:

A2

(1 -

x 2)

b3,

(3) a1 =

(x A2

-

X ~ .

and A3 satisfy the following system of differential

(x A2

Ei(A2) = - C x 2 ( 3 x 2 4- A 3 ) ,

E2(A2)

E3(x2)= b3 ( A 2 - A,),

El(X3) = -cx3 ( 3 x 3 +A,),

E2(A3)

= c3 ( A 2

-

1 ) c3,

A3)7

= 3 c3 A2

-

I),

E3(X3)=3b3X3

Using the previous lemma, the differential equations given by (5.3) t o (5.9) now imply that

If we now introduce a function cr by

457

we deduce from (5.1) and (5.2) that

1

E3(b3) =

- A,)

A2 A3 (A2

+ A; + (6

A3

(4A;

(-A2

(a - (c2 - 1) (A2

+ As) + +2

A3

(b: (A; - A);

.

A3))))

We now compute some integrability conditions. As 6 is torsion free, we know that for any function f , the following equations are satisfied:

0 =El(E2(f))

-

J92(El(f))

+ (f?EZEl)(fL

- (6E1E2)(f)

= El(E3(f))- E3(El(f))- (%iE3)(f)f (vi&E1)(.f), = E2(E3(f))

-

E3(E2(f))

- (%zE3)(f)

+ (vkiE2)(f).

It is straightforward to check that applying the above principle for the functions A2 and A3 does not yield any new equations. However, applying this principle for the functions c, c3 and b3 yields the following system of differential equations for the function a:

1 & ( a ) = -(c ( 4 4 A; A; A;

+ A;

(4 - 3 a (A2

(A2

- A3)2

+ A:

(4b:

- A3)2

(A2

+ A,) + 4 (c2- 1)(A; +

A2 A3

+ A:))))),

of which one can check that the integrability conditions are satisfied. We now consider several cases. By restricting to an open dense subset of M , we may assume that one of the following holds: (i) the functions b3 and c3 are non vanishing functions. In this case we introduce positive, nonvanishing functions p1, p 2 and p3 by p1 = (A2 A,)-$

,

_ _1

p 2 = c3

1

A T,

- _1

1

Ag ,

p3 = b3

- _1

A, *A;,

1

.

(ii) the function c3 vanishes identically on A4 and the function b3 is a non vanishing function. In this case we introduce positive, nonvanishing functions p1, p2 and p3 by 1

pi = (

~ 2 ~ 3 ) - + ,

p3

1

1

= b i z AT' A?,

458

and the function tem

is determined by the following integrable sys-

p2

E 2 ( ~ 2= ) 0,

E 1 ( ~ 2= ) cX2 ~ 2 ,

E3(p2)

=4 3 .

(iii) both the functions b3 and c 3 vanish identically in which case we introduce the function p1 by P1

=(XzX3)-4

and the functions p 2 and p 3 are determined by the following integrable system of differential equations:

E I ( P I= ) E2(p3)

~ 2 7

= 0,

El(P3) = -ex3 ~ 3 , E3(P3)

= 0,

E 2 ( ~ 2= ) 0, E3(P3)

=0.

In all three cases, it follows that

which implies that there exist local coordinates u, v and w such that

Replacing now the E l , E 2 and E 3 coordinates by derivatives with respect to u, v and w, we obtain a completely integrable system. Using then the existence and uniqueness theorem of affine immersions, we immediately obtain the following:

Theorem 5.1. Let M be a n a f i n e hypersphere of Type (v). Then, M is completely determined by giving initial conditions f o r the functions A 2 , As, b 3 , c3, c and a at a given point. Conversely, (a) Given initial values f o r X2 and X 3 at a point, we can construct a n a f i n e hypersphere of Type (v) by assuming that b3 = c3 = 0, Q = 2 (c2 - 1)( A 2 X 3 ) and E X 3 A 2 (c2 - 1) = 0 (ii) Given initial values f o r A 2 , X 3 , b3 # 0, c 3 # 0 , c and Q at a point, we can construct a n a f i n e hypersphere of Type ( v ) (iii) Given initial values f o r b3, X 2 , X 3 and c a t a point, we can construct a n a f i n e hypersphere of Type ( v ) by assuming that c3 = 0 and Q i s determined by

+

bi ( A 2 - A3)

+

+

A2 ( 6

+

(Q

- (c2- 1)( 2 A2

+

A3))).

459

References 1. R. L. Bryant, Second order families of special Lagrangian 3-folds, preprint. 2. E. Calabi, Complete a f i n e hyperspheres, I. Sympos. Math. 10, 19-38, (1972). 3. B.-Y. Chen, Some pinching and classification theorems f o r minimal submanifolds, Arch. Math. 60,568-578, (1993). 4. S. Y. Cheng and S. T. Yau, Complete a f i n e hypersurfaces. I: T h e completeness of a f i n e metrics, Commun. Pure Appl. Math. 39,839-866, (1986). 5. F. Dillen, K. Nomizu and L. Vrancken, Conjugate connections and radon's theorem in a f i n e differential geometry, Monatsh. Math. 109, 221-235, (1990). 6. F. Dillen and L. Vrancken, Calabi type composition of a f i n e spheres, Diff. Geom. Appl. 4,303-328, (1994). 7. M. Kriele, C. Scharlach and L. Vrancken, A n extremal class of 3-dimensional elliptic a f i n e spheres, Hokkaido Math. J. 30,1-23, (2001). 8. M. Kriele and L. Vrancken, Lorenttian a f i n e hyperspheres with constant sectional curvature, Trans. Amer. Math. SOC.352,1581-1599, (2000). 9. M. Kriele and L. Vrancken, A n extremal class of 3-damenszonal hyperbolic a f i n e spheres, Geom. Dedicata 77,239-252, (1999). 10. S. Hiepko, Eine innere Kennzeichung der verzerrten Produkte, Math. Ann. 241,209-215, (1979). 11. A. M. Li, Calabi conjecture o n hyperbolic a f i n e hyperspheres. II, Math. Ann. 293,485-493, (1992). 12. A. M. Li, U. Simon and L. Vrancken, A f i n e spheres with constant a f i n e sectional curvature, Math. Z. 206,651-658, (1991). 13. A. M. Li, U. Simon and G. Zhao, Global Affine Differential Geometry of Hypersurfaces, Vol. 11,of De Gruyter Expositions in Mathematics, Walter De Gruyter, Berlin-New York, 1993. 14. K. Nomizu and T. Sasaki, Affine Differential Geometry, Cambridge University press, Cambridge, 1994. 15. A. V. Pogorelov, O n the improper convex a f i n e hyperspheres, Geom. Dedicata 1, 33-46, (1972). 16. C. Rodriguez Montealegre and L. Vrancken, Lagrangian submanifolds of the three dimensional complex projective space, J. Math. SOC.Japan 53,603-631, (2001). 17. T. Sasaki, Hyperbolic a f i n e hyperspheres, Nagoya Math. J. 77,107-123, (1980). 18. C. Scharlach, U. Simon, L. Verstraelen, and L. Vrancken, A new intrinsic curvature invariant f o r centroafine hypersurfaces, Beitrage zur Algebra und Geometrie 38,437-458, (1997). 19. L. Vrancken. T h e Magid-Ryan conjecture f o r equiafine hyperspheres with constant sectional curvature, J. Diff. Geom. 54,99-138, (2000). 1

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