Differential geometry, Valencia 2001 : proceedings of the international conference held to honour the 60th birthday of A.M. Naveira, Valencia, July 8-14, 2001

A a + ip+ =

e127+

e347 + e567 + e135

_ e146 _ e236 _ e245_

<• >

The basis (e l ) is orthonormal for the metric determined by the inclusion Gi C -90(7), and allows us to consider *

^

'

The structure of a general G^ -manifold will not reduce to SU(3), and these descriptions are only valid pointwise or locally. The intrinsic torsion space

T* ® g£

= xx ® x2 e x3 e xA

has four components of respective dimensions 1,14,27,7, first described by Fernandez-Gray. 22 Various constructions 16 ' 6,1 ' 13 of metrics with holonomy equal to G2 are based on the significant fact that the holonomy reduction is characterized by the simultaneous closure of

T* = T

AV s 02 e T A3T* s R © T © SgT.

(14)

It follows that 02" — ^> a n ( ^ * n e s P a c e s ^i,X2,X^,Xi are isomorphic to R, 02, SQT, T respectively. The components of Vy? can be recovered from those

122

Table 2. G2 torsion

component

dim

G2-module

Xi

1 14 27 7

R

x2 x3 x4

SU (3)-modu\e

R T «u(3) R T Su(3) R T

02 2

2

50 T

[S '°J

T

of d

x2 + x4

(15)

d*ip, (*d*tp) A (*y>).

Xi

The G2-structure is called calibrated (respectively cocalibrated) if dp = 0 (respectively d *

wAa, 2

V»- A a,

V-

(16)

ip+ A a

UJ ,

u2 A a is a list of the exterior forms on M fixed by 577(3). With reference to the summands in (14), we may assert that 2.1 L e m m a

3tp+-4uAa£

2

ST

C AT,

ip+ A a e T 2

3V>_ A a - 2u £ S$T

c A 4 r.

123

Proof. Given (6), 7 = 3^+ - 4w A a satisfies 7 A (*ip) = 0 and is therefore orthogonal to -, it lies in S$T. The invariant 4-forms are obtained by observing that *V>- = i/>+ A a and *(u A a) = 3U 2 . D Remark. The existence of various SU(3) -invariant elements of A T gives rise to a choice of induced G-i -structures in the passage from 6 to 7 dimensions. For example, 3V>+ - 4w A a = 3 [w A ( - | a) + ip+] determines a G2 structure with reversed orientation on T and different scalings relative to (11). Other choices of coefficients will have the effect of modifying combinations in Theorem 3.1 below. The three components of T* 02" isomorphic to T can be detected from corresponding components of dip and d*/p. It is useful to record the following list for diagnostic purposes. 2.2 Lemma C = e 1347 + e 1567 - e 1236 - e 1245 £ T c T rj = e1347 + e1567 + e1236 + e1245 g j , Q gg T £= #=

e13456 + 2e 13456

e12357

_ e12467

_ e12357 + e12467

c A*!*, tfj.

c

^ 5

€ T c T e T

C 02

C

/\ |», cAV.

Proof. Each of these forms represents the element of T dual to e 1 in an appropriate guise. For example, £ = e 1 A

3

Product manifolds

We now suppose that M is a 7-manifold with an SU (3) -structure, so that the differential forms a, w, tp+,tp- of respective degrees 1,2,3,3 and constant norm are all defined globally. In this and the following sections, we shall investigate properties of the G2-structure defined with the convention of (12). In general, one may write da = a A f3 + 7 , where /3,7 are forms with values

124

in the subspace T* at each point. For example, the equation 7 = 0 is the integrability condition for the 6-dimensional distribution now determined by (11). We shall consider various special cases, the simplest of which is that in which M is the Riemannian product of M with an interval or circle, so that V a (and so da) vanishes. In the product situation, we choose to write a — e7 = dt, so that dip = dw Adt + dip+, d*

. Let Tl

= (W++Wr)

+ (Wf+Wz)

+ W3 + W4 + W5,

Ti = X\ + X2 + X3 + X4 denote the respective intrinsic torsion tensors, as denned in the previous sections. Since V

+, the tensor T-I is determined by T\. Any SU(3) -invariant component of Tv must be a linear combination of W^ and W{~, and any component isomorphic to T a linear combination of W4 and W$. The precise statement is 3.1 Theorem The four components of T2 are determined by the seven components of T\ as follows.

X2 <—> (Wf, 2W4+W5) X3 <—• (W+, W+, W3, W4+W5)

x4^(w4-w5, wn. Proof. The component X\ arises from ip A dip = ip+ A dw A dt + u A dt A dip+ =

2W^UJ3

A a.

Similarly, X4 is determined by the contraction

125

confirming that X4 = 0. In order to obtain £ instead of 1?, we need to take CLJ = i u A e 1 and d0_ = —^_ A e 1 , which corresponds to 2W4 + W5 = 0. The association of W2~~ with X2 and W2+, W3 with X 3 follows immediately from (15). The hypotheses do; = w A e 1 and dV>+ = -tp+ A e 1 are compatible with the constraint W4 + W$ = 0. This implies that dip = e 1347 + e 1567 - e 1236 - e 1245 = C, whence the T-component of X3 is proportional to W4 + W 5 .

D

Remark. The difference 49 — 42 = 7 of the dimensions of the spaces containing T\ and T2 is accounted for by the repetition of W± and a linear combination of W4, W5 in the above list. This redundancy is eliminated in the more complicated situations described in subsequent sections. The lack of repetiton between components of X\,X% is consistent with the result 9 that a connected G2 manifold with T2 S X\ © X2 has at least one of X\,X2 zero. 3.2 Corollary Suppose that M has an SU(Z)-structure. The G2-structure defined on M x R by (12) is cocalibrated if and only if T\ € W2 . Examples. 1. An almost-Hermitian 6-manifold is called nearly-Kahler 2 if V J belongs to the space W>i. Assuming that V J ^ 0, the structure reduces to SU(3) and we may suppose that T\ € W^ with V+ is proportional to dw. The product M x S1 then has a G 2 -structure with T2 G # 4 . Alternatively we may swap the roles of VM-> V>- to obtain T2 € X\ © X3. 2. A known example 11 of a calibrated nilmanifold can be interpreted as follows. Let g = E©rj be a 6-dimensional Lie algebra with structure determined by 0,

i = 1,2,4,5

25

de*= < e , D 24

t = 3, i = 6.

The definitions (5) furnish an associated nilmanifold M = T\G S,J7(3)-structure for which dw = 0,

dV+ = 0,

with an

# - = e 1234 ~ e 1256 ,

whence TJ € VUJ". It follows that M x S1 has both a calibrated G2 structure and (swapping ip+,if>-) a cocalibrated one with T2 € X$. The same Lie algebra g was incidentally used 15 in the construction of a closed non-parallel 4-form with stabilizer Sp(2)Sp(l).

126

4

Dynamic G 2 structures

Let M be a fixed 6-manifold. Suppose that (u>,ift+,ip-) is an SU(3) structure that depends on a real parameter t lying in some interval I C M., so that one may regard M = M x I as a warped product fibring over I. To avoid confusion, we denote exterior differentation on M by d in this section. Adopting a unit 1-form dt on I allows us to write d

^ t ) Adt + dif>+,

d*

) for which dtp+ = 0 and ui A dw = 0. Half-flatness is therefore characterized by the closure of %l>+ and LJ1 . It amounts to requiring that W\ + Wi has rank one and that both W4, W5 vanish. This eliminates 1 + 8 + 6 + 6 = 21 of the total 42 dimensions of T\ , which is constrained to lie in VV{" © W f © W3. If we now suppose that the Gi -structure on M has holonomy group contained in G2, we may conclude that M is half-flat for all t, and that the forms evolve according to the equations


d

«

(18)

Conversely, suppose we are given a half-flat SU(3)-structure at time t = to. The equations (18) may be regarded as a system constraining a closed 3-form V>+ and a closed 4-form J1. For, if (as in our situation) the stabilizer of ip+ is SX(3,C) then tp+ determines ip- via (3). In this way, Hitchin 12 proved that the compatibility equations (6) are conserved in time, and this leads to 4.2 Theorem Let M be an almost Hermitian 6-manifold which is half-flat. Then there exists a metric with holonomy contained in G% on M x I for some interval I. A key example of this construction is the following. Given (M, g), consider the conical metric g = t2g + dt2 on M x R + . Consistent with this, we set u) = t2u),

ip+ = t3ip+,

V-=*3^-,

127

where circumflex indicates a form independent of t. Then (18) becomes dui = 3^+,

#_ =

-2u2.

These equations determine the nearly-Kahler class for which T\ e W± . Further examples. 1. We now construct metrics with holonomy G 2 associated to each of the three nilmanifolds with b\ = 4 whose U(3) structures were classified by Abbena et al. 7 Starting with the Iwasawa manifold M, we modify the usual basis of 1-forms in order that t = 1,2,3,4, i — 5,

0, l

0 23

=14

de = ( -e

13

42

- e ,

(19)

i = 6.

The metric g

= t2 J2 e* ® e* + 1 ~ 2 £ e* ® e* + t 4 dt 2

is compatible with the natural fibration M —> T 4 , and the forms w = t 2 (e 1 2 + e 3 4 ) + t - 2 e 5 6 , ^ + + iip- = t(e* + ie2) A (e 3 + ie 4 ) A (e 5 + ie 6 ) determine a reduction to 51/(3) for which du = t~3tp+,

dip- = - 4 t e 1234 ,

whence TI e W f © WJ". Indeed, g is a metric with holonomy (?2 on M x R + . 2. The previous example was discovered by Gibbons et al 1 4 who exhibit it as arising from the standard complete metric with holonomy G2 by 'contracting' the isometry group SO(5). A more complicated 2-step example is associated to the Lie algebra with 0,

del=

* = 1,2,3,4,

I

i = 5, 0 24

i = 6.

Consider an orthonormal basis of 1-forms teL

t2e2,

teA

t2e\

t~2e\

t~le\

2tAdt,

128

and the SU(3) -structure for which u = t3(e12 + e 34 ) + *~ 3 e 56 , ip+ + ii/>- = (e 1 + ite2) A (e 3 + ite4) A (e 5 + ite*), by analogy to (5). This yields a G2 -structure with closed forms

Adt + 2* 5 (e 145 + e 136 + e 235 ) A dt + e 1256 + e 3456 + i 6 e 1 2 3 4 .

3. The nilpotent 3-step Lie algebra for which 0,

i = l,2,4,5 25

de* = I e , ei4

_

* = 3, e23;

{ =

6

gives rise to an example satisfying the hypotheses of Theorem 4.2. In fact, the forms (5) satisfy u A du = 0,

dV+ = 0,

# _ = -e1256.

An explicit determination of the metric requires an analysis of spaces of invariant exact forms. 12 5

Fibred G 2 manifolds

Let (M, g) be a Riemannian 6-manifold. In this final section, we consider the 7-dimensional total space of a circle fibration n : M —• M, endowed with a metric of the form g-a®a

+ n*g,

(20)

with da = n*p for some 2-form p on M. To begin with, suppose that M has an SU(3)-structure. The G2structure on M determined by (12) satisfies the following enhanced version of (17), in which we omit the pullback operator TT* : d

A2T* = {u) © [Aj'1] © [A2-0] p

= pQUJ +

pi

+

p2,

we obtain the following generalization of Theorem 3.1.

129

5.1 Theorem The four components of T2 are given by

X2 <—>• (W2", 2W4+W 5 -2p 2 ) Xz «—• (3W+-4po,

Wf+Pu

X4 *_> (W4-W5-P2,

W3,

Wi+W5+P2)

Wr).

Proof. Whilst Xi corresponds to

A a — 4i/>+) A dip. The various coefficients of p2 can be deduced from the observations: (i) d

= 0, and (ii) d *

d * tp = —ip_ A p = — -0_ A p2.

This situation applies when M is the torus T 6 = R 6 / Z 6 endowed with a constant SU(3) -structure. The set (1) of such structures compatible with g is isomorphic to the set of G2 -structures compatible with (20) (assuming that orientations are also preserved). We deduce that M is calibrated if and only if p = 0, and cocalibrated if and only if p2 = 0. Examples. 1. Let M be a circle bundle with curvature 2-form

over T 6 . Then there are no calibrated G2 structures on M compatible with g. The set C* of cocalibrated structures corresponds to SU(3) -structures relative to which p has type (1,1). By considering first the space CP 3 (regarded as the set SO(6)/U(3) of orthogonal almost complex structures 7 , 3 ), C* can be seen to be a disjoint union RP 1 U RP 5 . 2. We can apply the above theory to the real projective space RP by representing it as SO(5)/SU(2), which fibres over SO(5)/U(2) S C P 3 . It is well known that the latter has a homogeneous nearly-Kahler metric, so that there is a reduction to SU(3) with T\ € W^. Moreover, the curvature 2-form p of the above Hopf fibration is an element of [Aj'1] relative to the non-integrable complex structure. 25 Example 1 of Section 3 now implies that RP has a G2 -structure with r2 £ X\ © Xz and another with T2 € Xz © %4.

130

3. Let us return to the descriptions (1). The space of Gi -invariant differential forms on RP 7 is 1-dimensional, and it follows that RP 7 has a nearly-parallel G-i -structure, one with T2 € X\. More generally, we may regard (16) as a list of SO(6) -invariant differential forms on RP subject to the relations da = - | w and dip±=aArp^,

(22)

that also give rise to G2 -structures with T2 G X\ © X3. One can arrive at (22) by considering the canonical S1 -bundle B over a Kahler manifold M of real dimension 6 (see (4)). A local orthonormal basis {tp1, ip2} of sections of the bundle with fibre [A 3 ' 0 ] = R 2 gives rise to coordinates a1, a2 and radial parameter r = ^/(a 1 ) 2 + (a 2 ) 2 on B. The Kahler condition implies that dtp1 = a Aip2 for some 1-form a on M, and we use b1 = da1 - a2-K*a, b2 = da2 + a V c r to define global forms rdr = alb1 + a2b2, ip+ = a1*!)1 + a2xp2,

a = a1b2 — a2b1, ip- = a1^2 — a2^1.

A straightforward calculation 26 shows that rdr A ip+ + a A V- = ^{b1 A ip1 + b2 A ip2) — r2dip+, so that restricting to 1 (r = 1) we obtain (22). Similarly, da = ir*p, where p = da can be identified with the Ricci form. This leads to the following result of Baum et al, 2 9 which forms part of a more general theory of nearly-parallel Gi structures. 23 ' 10 5.2 Theorem If M is Kahler-Einstein with positive scalar curvature then B has a Gi -structure with TJ, G X\. We conclude this article by returning to the initial set-up of this section, together with the assumption that the holonomy of the metric (20) reduces to the subgroup G2 determined by the 3-form (12). This allows us to set Xt = 0 in Theorem 5.1. 5.3 Corollary If the holonomy group of the metric (20) reduces to G2 then the quotient M = M/S1 has an SU(3) -structure for which T\ G W ^ \

131

Observe that the resulting condition on T\ involves a change of sign from that in Corollary 3.2. Indeed, from (21), we obtain duj = 0 and dip- = 0 . It is convenient to regard SU(3) -structures with (dtp-)2'2 = 0 as 'self-dual', and those with (dip+)2'2 = 0 as 'anti-self-dual'. The latter type occurred naturally in Sections 3 and 4, and we focus attention on the (2,2) components so that the terminology is conformally invariant. In the present situation, we may therefore say that M has a self-dual symplectic structure with dip+ = u) A p,

where p = p\ is a closed effective (1, l)-form. Whilst the incomplete examples of Section 4 admit quotients of this type, a more realistic generalization of the condition T\ = 0 is obtained by dropping the assumption that the S1 orbits on M have constant size. In this situation, there exists a function f on M for which d(e2^u}) = 0. It follows that, applying the conformal transformation (10), M once again has a self-dual symplectic structure, though this time W$ = —df is non-zero and P = Pi + P2 = -e~3f{W+

+ 2dfj V-)

is closed. A study of this particular class of structures may be valuable in the construction of metrics with holonomy Qi.

Acknowledgments This paper was conceived for the conference 'Differential Geometry Valencia 2001', though the material was developed in the light of feedback from the Durham-LMS Symposium 'Special Structures in Differential Geometry' that took place a month later. The authors wish to thank the organizers of both conferences. References 1. G. W. Gibbons and D. N. Page and C. N. Pope, "Einstein metrics on S3, R3, and Ri bundles", Commun. Math. Phys. 127, 529-553 (1990). 2. A. Gray, "The structure of nearly Kahler manifolds" Math. Ann. 223, 233-248 (1976). 3. V. Apostolov and G. Grantcharov and S. Ivanov, "Orthogonal complex structures on certain Riemannian 6-manifolds" Diff. Geom. Appl. 11, 279-296 (1999).

132

4. M. Atiyah and E. Witten, "M-theory dynamics on a manifold of G
0 15. S. Salamon, "Almost parallel structures" Contemp. Math., Global Differential Geometry: The Mathematical Legacy of Alfred Gray 288, 162-181 (2001). 16. R.L. Bryant, "Metrics with exceptional holonomy" Annals of Math. 126, 525-576 (1987). 17. C. Bar, "Killing spinors and holonomy" Commun. Math. Phys. 154, 509-521 (1993). 18. F. Belgun and A. Morioanu, "Nearly-Kahler manifolds with reduced holonomy" preprint, (2000) 19. M. Berger, "Les varietes riemanniennes homogene normales simplement conexes a courbure strictemente positive" Ann. Sc. Norm. Sup. Pisa 15, 179-246 (1961). 20. M. Fernandez and L. Ugarte, "Dolbeault cohomology for Gi -structures" Geom. Dedicata 70, 57-86 (1998).

133

21. L. Falcitelli and A. Farinola and S. Salamon, "Almost-Hermitian Geometry" Differ. Geom. Appl. 4, 259-282 (1994). 22. M. Fernandez and A. Gray, "Riemannian manifolds with structure group G2" Ann. Mat. Pura Appl. 32, 19-45 (1982). 23. Th. Priedrich and I. Kath and A. Morioanu and U. Senunelmann, "On nearly parallel G2 structures" J. Geom. Phys. 27, 155-177 (1998). 24. A. Gray and L. Hervella, "The sixteen classes of almost Hermitian manifolds" Ann. Mat. Pura e Appl. 282, 1-21 (1980). 25. R. Reyes-Carrion, "A generalization of the notion of instanton" Differ. Geom. Appl. 8, 1-20 (,)1998. 26. S. Salamon, Riemannian geometry and holonomy groups, Pitman Research Notes in Mathematics 201, Longman, 1989. 27. S.M. Salamon, "Complex structures on nilpotent Lie algebras" J. Pure Applied Algebra 157, 311-333 (2001). 28. A.F. Swann, "Weakening holonomy", Quaternionic structures in mathematics and physics (Rome, 1999), World Sci. Publishing, River Edge, NJ, 405-415, (2001). 29. H. Baum and Th. Priedrich and R. Grunmewald and I. Kath, Twistor and Killing spinors on Riemannian manifolds, Teubner-Verlag, StuttgartLeipzig, 1991.

tf-HYPERSURFACES

W I T H FINITE TOTAL CURVATURE

MANFREDO P. DO CARMO Instituto Nacional de Matemdtica Pura e Aplicada Estrada Dona Castorina 110 - Jardim Botanico 22460-320, Rio de Janeiro-RJ - Brasil E-mail: manfredoQimpa.br To Antonio Naveira, on his 60 th birthday 1

Introduction: historical perspective

Let x: Mn —> M (c) be a complete hypersurface with constant mean curvature H (to be called an iJ-hypersurface) of a complete, simply-connected Riemannian manifold with constant sectional curvature c (c = 0,1, —1). Recently a notion of finite total curvature was introduced for this situation which, as in the case of minimal surfaces, has some topological and geometrical consequences. We want to describe some of these results here. Let us put some perspective on the subject. We can start in 1957 when A. Huber 16 proved the following conjecture of H. Hopf. Let M2 be a 2-dimensional complete Riemannian manifold and assume that JM\K~\dM < oo, where K~(p) = mm{0,K(p)}, p e M, and K is the Gaussian curvature of M2. Is it true that JM K+dM < oo and that M is homeomorphic to a compact manifold M minus finitely many points {pi,...,pr}7 Huber's proof of the above result is rather intrincate and a simple proof can be found in White 23 . As far as I know, no n-dimensional generalization of Huber's intrinsic result, for arbitrary n, has been found. In 1964, Osserman 18 observed that for minimal surfaces x: M2 —> M.3 with finite total curvature, i.e., with JM2 \K\ dM < oo, the above equivalence is actualy conformal and, more importantly, that the Gauss map g: M2 —» Si C i 3 , which is (anti) holomorphic in this case, extends to the punctures as a (anti) holomorphic map. This yields a compactification of M by the Gauss map and allows to bring results of compact Riemann surfaces into the theory of complete minimal surfaces with finite total curvature. Osserman's result was extended by Chern-Osserman 10 in 1967 to the case of complete minimal surfaces x: M2 —> RN in arbitrary euclidean spaces. So far, the results are esentially 2-dimensional. This remained so until 1985 when M. Anderson 3 proved a beautiful result for which we need some

134

135

notation. Let x: Mn —> M n + 1 be a complete, oriented, minimal hypersurface (actually, Anderson considered minimal immersion into euclidean spaces with arbitrary codimension but, for simplicity, we will restrict to hypersurfaces). Let A: TPM -> TPM be given by (AX,Y) = (VXN,Y), where V is the covariant derivature in the ambient space and N is the unit normal vector in the orientation of M. Assume that x has finite total curvature in the sense that JM \A\ndM < oo, where |A| = (^"=1 *f) 1 / 2 a n d the fcj's are the eigenvalues (the principal curvatures) of A. This is a good generalization of the notion of finite total curvature for minimal surfaces, since \A\2 = fe? + fc| = -2fc!fe2 =

-IK.

With this notation, Anderson proved: i) x is proper ii) Mn is diffeomorphic to a compact manifold M minus finitely many points iii) the Gauss map extends C°° to the punctures. A crucial point in the proof of Anderson's theorem is the following result: ( M a i n L e m m a ) . Assume JM \A\ndM < oo and fix po G M. e > 0, there exists Ro > 0 such that, for R > RQ, we have sup

Then given

\A\(p) < e,

pEM-BPQ(R)

where Bpo(R)

is the geodesic ball in M with center po and radius R.

Roughly saying: finite total curvature => limij^oo |>1| = 0 (actually, Anderson proved and needed the sharper result lim^-xx, i?2|^4|2 = 0, but the above statement suffices for our purposes). Anderson's result was partially extended by Oliveira17 in 1993 for minimal hypersurfaces of the hyperbolic space H n + 1 ( - 1 ) . (Again, we are considering hypersurfaces for simplicity, since Oliveira's results are for arbitrary codimensions). The Main Lemma holds true and the consequence is that Mn is homeomorphic to the interior of a compact M with boundary. Although there is no information on the Gauss map, it is proved that the immersion extends to a continuous map x: M —> H (here H denotes the canonical compactification of the hyperbolic space by the "sphere at infinity".) So far, we have restricted ourselves to minimal hypersurfaces. Can we define a notion of finite total curvature for if-hypersurfaces which give interesting results?

136

2

Finite total curvature for complete .ff-hypersurfaces. Topological structure

Consider a complete, oriented, iJ-hypersurface x: Mn —> M (c), H ^ 0. Let A° — A - HI. It is easily seen that Trace A0 = 0, so we call A0 the traceless second fundamental form of the hypersurface. Also, if we denote by fii the eigenvalues of ^4°, we obtain

1

l
It follows that \A°\ = 0 if and only if all principal curvatures are equal so that A0 measures how much M deviates from being totally umbilical. It has been observed in [2] that many theorems on minimal hypersurfaces involving A are naturally extended to iJ-hypersurfaces if we replace A by A0. A number of papers following [2] (see [*], [5], [13], [19]) confirmed this general observation. It is then reasonable to define finite total curvature for il-hypersurfaces by requiring that /

\A°\ndM


JM

This was proposed in 1998 in a joint paper of Berard, do Carmo and Santos [5], and the fundamental result proved there is that the Main Lemma mentioned in the Introduction holds true for this situation. The proof uses Simons' equation for ff-hypersurfaces [7] and a Sobolev inequality to obtain a general inequality satisfied by u = |A°| which can be used to apply the de Giorgi-Nash-Moser interation process. The precise statement of the result proved in [5] is as follows. Theorem 1 (Berard, do Carmo, Santos). Let x: Mn —> M (c) be a complete H-hypersurface with finite total curvature and let u = |A°|. Fix p0 6 M and denote by Br(po) the geodesic ball in M wth centerpo and radius R. Then there exist constants CM, C'M depending only on n, H and c and a radius RM determined by the condition C'm JER undM < 1, ER = M-BR(PQ), such that sup u{p)
( \JER

undM)

,

for all

R>RM-

J

This theorem has a large number of consequences. The first surprising one is that, in many cases, finite total curvature implies compactness. Let me

137

describe a special case of this situation where, assuming the above theorem, the proof is quite easy. Theorem 2 [ 5 ]. Letx: M2 —> R3 be a complete, oriented, H-surface, H ^ 0. Assume that JM \A°\2dM < oo. then M2 is compact. Proof: An easy calculation shows that \A°\2 = 2(H2 - K) > 0. Since \A°\ tends to zero at infinity, we have that K approaches H2 at infinity. It follows that outside a compact set, K is positive. Thus the negative part K~ of K has compact support, i.e., JM \K~\dM < oo. Prom Huber's theorem, this implies that JM K+dM < oo. But, outside a compact set Cl, K+ > \H2. Therefore, oo > / JM-a

K+dM

>\

I 2

H2dM = ^2

JM-U

f JM-

dM. Omega

Now if M is complete, noncompact and H ^ 0, vol(M) = oo [14]. This is a contradiction, and shows that M is compact. • The same proof works for iJ-surfaces immersed in hyperbolic space EI3(—1) provided that H2 > 1. Actually, the theorem holds in a more general situation, namely Theorem 3 (do Carmo, Cheung, Santos [ n ]). Letx: Mn -> M " + (C), C < 0, be a complete H-hypersurf ace with H2 > \c\ and finite total curvature. Then M is compact. The idea of the proof is to show that given a compact set Q C M, there exist a number S > 0, such that RiceM-QM

> 5,

and then adapt the proof of Bonnet-Myers to M — CI. For the case c > 0, the theorem also holds, and this was shown by Berard and Santos in [8]. To complete our knowledge of the topology of finite total curvature Hhypersurfaces of M (c), it remains to consider the case where c < 0 and H2 < \c\. The case H2 < \c\ was settled by Castillon. Theorem 4 (P. Castillon [9]). Let x: Mn -> W1*1^) be a complete Hhypersurface with finite total curvature and H2 < \c\. Then M is diffeomorphic to intM, where M is a compact manifold with boundary, and the

138

immersion x extends continuously to x: M —> H (c), where H (c) is the compactification o/H" + 1 (c) mentioned in the Introduction. Notice that the above theorem generalizes Oliveira's result in the Introduction. Thus, only the case c < 0 and H2 = \c\ remains to be settled. As for as I know this is still an open question. I believe that the following conjecture is true. Conjecture: Let x: Mn —> H n + 1 (c), c > 0, be a complete H-hypersurface with H2 = \c\ and finite total curvature. Then M is homeomorphic to a compact M minus finitely many points. Notice that for n — 2, and c = — 1, we easily compute that \A°\2 = 2 2{H + c — K) = —IK. So, by Huber's theorem, the conjecture holds in this case. The table below summarizes the topology of finite total curvature complete ff-hypersurfaces of M (c).

x: Mn

3

c > 0 => M compact 'H2 > \c\ =>• M compact ^ M™+1(c)< H2 = c =>? c<0< 2 H < \c\ =^> M ~ int M , M a compact manifold with boundary

Complete iJ-hypersurfaces with finite total curvature. Further results.

In this section, we will describe a few geometric results which follow from the basic Theorem 1 and are not related to the topological structure described in the previous section. We will need some preliminary definitions. Let us consider the case where x: Mn —> H " + 1 ( - l ) is a complete orientable iJ-hypersurface. The stability operator of M is defined by L = A + \A\2 -n = A + \A°\2 - n(l - H2). Here A is the (negative definite) Laplacian operator, i.e., A satisfies f fAfdM JK

= - f

\Vf\2dM,

JK

where K c M is a compact domain with smooth boundary dK, and / is smooth function that vanishes on dK.

139

The index Ind^M of M in the compact domain K is the number of negative eigenvalues of L counted with multiplicity. The index of M is defined by Ind M = sup Ind^M. KCM

We say that K C M is stable if Ind/cM = 0 and that M is stable if Ind M = 0. If we consider the operator L restricted to smooth functions with compact support that satisfies JM fdM — 0, we obtain the notion of weak stability. In constrast, the above notion is called strong stability. T h e o r e m 5 [ 6 ], [ 5 ]. Let x: Mn —> H n + 1 ( - 1 ) a complete H-hypersurf ace withH2 < 1. Then J \A°\n dM < oo => Ind M < oo. JM

The proof involves the essential spectrum of L, and to give a sketch of it, we will need a few more definitions. Because the hypersurface has finite total curvature, \A°\2 is bounded in M, hence the quadratic form - f


JM

n(l-H2)
JM

defined in the space of compactly supported smooth functions cp on M is bounded below. It follows that (see [6], and also [15]) the operator L is essentially self-adjoint and has a unique self-adjoint extension (also denoted by L) to the Hilbert space L2(M), and that the spectrum a(V) of L is a(L) = {A G K; L — XI has no continuous inverse}. An element A £ a(L) is an eigenvalue of L if Kern (L - XI) ^ 0. Also, A G o-(L) is non-essential if both A is isolated and the multiplicity of A (i.e., the dimension of Kern (L - A7) is finite). Finally, the essencial spectrum o~ess{L) is defined by aess(L)

= a — {non-essential elements}.

In the lemma below, we collected two basic facts about the essential spectrum. L e m m a . Let K c M be a compact domain with smooth boundary. Then 1) cress(LM)

2) infaess(LM)

=

oess{LM-K)

= infa {- fM ipLtpdM; fM
140

We can now present the Sketch of proof of Theorem 5 By finite total curvature, |^4°| is small outside a compact set (Theorem 1). Thus, since Maets(LM)=w.fv\-

I


JM

ip2dM = 11 )

= inf„./M v,=1 ( / M ( I W | 2 - l^°l V + n(l - ff 2)v>2) dM. and H2 < 1, there exists by (1) of the Lemma an a > 0 such that inf cress{LM) = at. On the other hand, again by Theorem 1, \A°\2 is bounded in M. It follows that — JM ipLipdM is bounded from below. Since there are no elements of <7ess (LM) below zero, there are only a finite number of negative eigenvalues of L, each with finite multiplicity. Thus, Ind(M) < oo. • Remark. Theorem 5 was first proved in [6] with a longer proof then the one given here. The proof given here comes from [5]; its apparent simplicity is due to the fact that the main burden of the proof has been displaced to Theorem 1. Remark. The converse of Theorem 5 is false. The example comes already in dimension two and can be found in [22] p. 635. It should be mentioned that for n = 2 and H = 1, the converse holds and can be found in [12]. In the following result, when we say that a complete iJ-hypersurface {H may be zero) x: M™ —> R n + 1 is stable we mean strong stability when H = 0, and weak stability when H ^ 0. Theorem 6. Let x: Mn —> R n + 1 be a complete H-hypersurface with finite total curvature. Assume that M is stable. Then x(M) is a hyperplane or a sphere. Proof: If H ^ 0, M is compact by Theorem 3. It has been shown by Barbosa and do Carmo in ([4], Theorem 1.2) that a stable compact if-hypersurface is a sphere. If H = 0, it has been proved by Yi-Bing Shen and Xiao-Hua Zhu 20 that a complete stable minimal hypersurface of K n + 1 with finite total curvature is a hyperplane. • We will close this brief survey with the following question. Let C be the class of orientable, complete .H-hypersurfaces x: Mn —> M (c), H ^ 0,

141

with finite total curvature. Is there a constant C, depending only on n, H and c with the following property? Let x e C with/ M \A°\n dM < C. Then the total curvature of a; is actually zero (i.e., x is a totally umbilic hypersurface). Assuming the existence of a C satisfying the above property, it is natural to ask whether C m a x = sup{all C with the above property} xec is realized as the total curvature of some x € C. For the case where c > 0 (hence M is compact), the first question has been solved in Shiohama and Xu ([ 21 ], Theorem A). There, they consider n+p

submanifolds of M (c) of arbitrary codimension p. For consistency with the rest of this survey, we have taken p = 1. For the case where c < 0, to the best of my knowledge, the question is still open. Also, as far as I know, the answer to the second question in the case of hypersurfaces, is unknown, even for the simple case of iJ-surfaces in R 3 . It is reasonable to expect that C m a x would then characterize (in a certain sense) the simplest Wente Torus. Acknowledgements. This a an expanded version of a lecture given in the conference to commemorate the 60 th birthday of Professor Antonio Naveira, held in Valencia, Spain. I want to thank the Organizing Committee for inviting me to participate in that happy occasion. I also want to thank Maria Fernanda Elbert for criticism and suggestions to the manuscript. References 1. Alencar, H. and do Carmo, M.: Hypersurfaces with constant mean curvature in spheres, Proc. Amer. Math. Soc. 120, 1223-1229 (1994). 2. Alencar, H. and do Carmo, M.: Hypersurfaces with constant mean curvature in space forms, An. Acad. Brasil. Cien. 66, 265-274 (1994). 3. Anderson, M.: The compactification of a minimal submanifold in Euclidean space by the Gauss map, Preprint I.H.E.S, (1985). 4. Barbosa, J.L. and do Carmo, M.: Stability of hypersurfaces of constant mean curvature, Math. Z. 185, 339-353 (1984). 5. Berard, P., do Carmo, M. and Santos, W.: Hypersurfaces with constant mean curvature and finite total curvature. Ann. Global Anal. Geom. 16, 273-290 (1998).

142

6. Berard, P., do Carmo, M. and Santos, W.: The index of constant mean curvature surfaces in hyperbolic 3-space. Math. Z. 224, 313-326 (1997). 7. Berard, P.: Simons' equation revisited. An. Acad. Brasil. Cien. 66, 397-403 (1994). 8. Berard, P., Santos W., Curvatures estimates and stability properties of CMC-submanifolds in space forms, Mat. Contemporanea, 17, 77-97 (1999). 9. Castillon, P.: Spectral properties of constant mean curvature submanifolds in hyperbolic spaces. Ann. Global Anal. Geom. 17, 563-580 (1999). 10. Chern, S.S. and Osserman, R.: Complete minimal surfaces in M.N. J. d'Analyse Mathematique, 19, 15-34 (1967). 11. Cheung, L.-F., do Carmo, M. and Santos, W.: On the compactness of CMC-hypersurfaces with finite total curvature, Arch. Math., 73, 216-222 (1999). 12. do Carmo, M., da Silveira, A.M., Index and total curvature of surfaces with constant mean curvature. Proc. A.M.S. 110, 1009-1015 (1990). 13. Fontenele, F.: Submanifolds with parallel mean curvature vector in pinched Riemannian manifolds. Pacific J. Math. 177, 47-70 (1997). 14. Frensel, K.: Stable complete surfaces with constant mean curvature. Bol. soc. Bras. Mat. 27, 129-144 (1996). 15. Glazman, I.M.: Direct methods of qualitative spectral analysis of singular differential operators. Israel Program of Scientific Translations, Jeruzalem, (1965). 16. Huber, A.: On subharmonic functions and differential geometry in the large. Content. Math. Helv. 32, 13-72 (1957). 17. de Oliveira, G.: Compactification of minimal submanifolds of hyperbolic space. Comm. Analysis and Geom. 1, 1-29 (1993). 18. Osserman, R.: Global properties of minimal surfaces in E 3 and E™. Ann. of Math. 80, 340-364 (1964). 19. Santos, W.: Submanifolds with parallel mean curvature vector in spheres. Tohoku Math. J. 46, 403-415 (1994). 20. Shen, Y.-B. and Zu, X-H.: On stable complete minimal hypersurfaces in R" + 1 . Amer. J. Math. 120, 103-116 (1998). 21. Shiohama, K. and Xu, H.: Rigidity and sphere theorems for submanifolds, Kyushu J. of Math. 48, 291-306 (1994). 22. da Silveira, A.: Stability of complete noncompact surfaces with constant mean curvature. Math. Ann. 277, 629-638 (1987). 23. White, B.: Complete surfaces of finite total curvature. J. Geom. Diff. 26, 315-326 (1987).

NULL HELICES A N D D E G E N E R A T E CURVES IN LORENTZIAN SPACES ANGEL FERRANDEZ, ANGEL GIMENEZ AND PASCUAL LUCAS Departamento de Matemdticas, Universidad de Murcia 30100 Espinardo, Murcia, Spain E-mails: [email protected], [email protected], [email protected]

Dedicated to A. M. Naveira In this paper we introduce a reference along a degenerate curve (null or non null) in an n-dimensional Lorentzian space with the minimun number of curvatures. That reference is called the Cartan frame of the curve. In the Lorentzian space forms case, we obtain a complete classification of helices (that is, curves with constant Cartan curvatures) in low dimensions. In all cases we present existence, uniqueness and congruence theorems.

1

Introduction

In a proper semi-Riemannian manifold there exist three families of curves depending on their causal characters. It is well-known1 that the study of timelike curves has many analogies and similarities with that of spacelike curves. However, the fact that the induced metric on a null curve is degenerate leads to a much more complicated study and also different from the non-degenerate case. Even more, a timelike or spacelike curve can have a null higher order derivative and then its study is also different from that of Riemannian case. In the geometry of null curves difficulties arise because the arc length vanishes, so that it is not possible to normalize the tangent vector in the usual way. A solution is to introduce the pseudo-arc parameter (already used by Vessiot2) which normalizes the derivative of the tangent vector (see the papers by W.B. Bonnor 3 and M. Castagnino 4 ). The importance of the study of null curves and its presence in the physic theories is clear 5,6 ' 7 ' 8 ' 9 . Recently, Nersessian and Ramos 10 show that there exists a geometrical particle model based enterely on the geometry of the null curves in Minkowskian 4-dimensional spacetime which under quantization yields the wave equations corresponding to massive spinning particles of arbitrary spin. The same authors 11 study the simplest geometrical particle model which is associated with null curves in 3-dimensional LorentzMinkowski space.

143

144

Motivated by the growing importance of null curves in mathematical physics, A. Bejancu 12 initiated an ambitious program for the general study of the differential geometry of null curves in Lorentzian manifolds and, more generally, in semi-Riemannian manifolds (see also his book 13 ). This paper is organized as follows. In Section 2 we introduce the Cartan frame and the Cartan curvatures of a null curve and compute the ordinary differential equation that helices satisfy. In the next section we obtain the existence and uniqueness theorems relative to Cartan curves (see Theorems 2 and 3) and determine the families of null helices in Rf, Sf and H 4 (Theorems 4, 5 and 6). In Section 4 we introduce the s-degenerate curves as an extension of the null curves (that are 1-degenerate curves) and, in Section 5, we present existence and congruence theorems for that kind of curves. Finally, in the last section we classify 2-degenerate Cartan helices in 4-dimensional Lorentzian space forms. The results of this paper can be extended to degenerate curves in pseudoEuclidean spaces of index two 14 and following the same ideas they can be also stated in §2 and ffif. With some extra effort they can be extended to pseudo-Riemannian space forms of higher indices.

2

The Cartan frame of a null curve

Let M " be an orientable Lorentzian manifold and consider C a null curve locally parametrized by 7 : I C R —* M™. Assume that { 7 ' , 7 " , . . . , 7 ^ } is a linearly independent family and define Ei = span{7',7",. • • ,7^}> i = l , . . . , n . Let L £ E\, so that 7' = k\L, for a certain function k±. Since E2 = E\ © span{7"} we can choose a unit spacelike vector W\ satisfying Ei = span{L, Wi}. Now, since E3 = E2 © s p a n ^ 3 ' } we obtain that £3 is a Lorentzian subspace of En, then there exists only one null vector AT such that (L,N) = s = ± 1 , (WUN) = 0 and E3 = span{L, WUN}. In general, for i = 2 , . . . , n - 3, we can find orthonormal spacelike vectors {W\,..., Wi} such that Ei+2 = span{L, Wi, N,... . W J and the basis {7', 7 " , . . . , 7 ( i + 2 ) } and {L,W\,N,... ,Wi} have the same orientation. Finally, the vector Wm, m = n — 2, is chosen in order that the basis {L, W\, N,... ,Wm} is positively oriented. An easy computation shows that there exist functions (ki,... ,km+3} such that the following equations hold (compare with Ref. 12)

145

L' = ek2L + hWi, W[ = ehL - sk3N, N' = -ek2N

- hWi + k5W2,

(1)

Wi = -ek5L + k6W3, Wl = -ki+zWi-!

+ ki+4Wi+1,

i=

3,...,m-l

W^ = - f c m + 3 W m _ i . Without loss of generality we may assume that 7 is parametrized by the pseudo-arc parameter, that is, (7", 7") = 1. Now choose L = 7' and Wi — 7", so that fci — 1, k2 — 0 and £3 = 1. Let us take N given by N = — £7^) — £ 5 ( 7 ^ , 7 ^ ) 7 ' , then the forth curvature is given by k~4 = —{N',Wi) = £ ^ ( 7 ^ , 7 ^ ) - After a direct computation, where the curvature functions are renamed (fci = £4, k2 = k$ and so on), we can show the following theorem. T h e o r e m 1 ( 15 ) Let 7 : I —* M™, n = m + 2, be a null curve parametrized by the pseudo-arc such that {l'(t),^"(t),... ,^n\t)} is a basis of T^M™ for all t. Then there exists only one Frenet frame satisfying the equations L' = Wlt W[ =ekxLeN, N' = -kxWx + k2W2, W'2 = -ek2L + k3W3, W[ = -kiWi-i + ki+iWi+i W' — -k W 1

(

'

i = 2,,... , m - 1,

and verifying i) For 1 < i < m - 1, {7', 7 " , . . . , 7 W } and {L,WUN,... the same orientation. ii) {L, W\,N,... , Wm} is positively oriented.

, W^2} have

Observe that when m > 1 then e = - 1 andfcj> 0 for i > 2; however, when m = 1 then e = — l o r e = l according to {7', 7", 7'"} is positively or negatively oriented, respectively. Definition 1 A null curve in M " satisfying the conditions of the above theorem is called a Cartan curve. The above frame {L, W\,N,... ,Wm} and

146

curvatures {fci, /C2,. • • , km} are called the Cartan frame and the Cartan curvatures, respectively, of the curve 7. It is not difficult to show that the Cartan curvatures of a curve C in Mf are invariant under Lorentzian transformations. Definition 2 A curve is said to be a helix if it has constant Cartan curvatures. A long and messy computation shows that if 7 is a helix then it satisfies the following differential equation 7(

n+1

) = ai7' + a27(3) + • • • + a s 7 ( n _ 1 ) ,

if n is even, n = 2s,

and 7 ("+!)

=

0l7

" + a 2 7 ( 4 ) + • • • + asj(n-1],

if n is odd, n = 2s + l,

where the coefficients can be easily computed (see Ref. 15). 3

Null curves in MJ(c)

In this section M"(c) stands for K™, S"(c) or H"(c), according to c = 0, c > 0 or c < 0, respectively. Our first goal is to prove the following theorem. Theorem 2 Let fci, /C2, • • • ,fcTO: [—<5, <5] -+ M be differentiable functions with hi < 0 and fcj > 0 for i = 3 , . . . , m — 1. Let p be a point in M™, n = m + 2, and consider {L°, W®, N°,... , W^} a positively oriented pseudo-orthonormal basis of TPM™. Then there exists a unique Cartan curve 7 in M™, with 7(0) = p, whose Cartan frame {L, Wi,N,... , Wm} satisfies L(0) = L°, N(0) = N°, Wi(0) =W°,

i = 1 , . . . , m.

Proof. Let us suppose M™ = M™ (the remaining cases are similar). According to the general theory of differential equations, there exists a unique solution {L, Wi,N,... , Wm} of (2), defined on an interval [-8,5], and satisfying the initial conditions of the theorem. A straightforward computation, bearing in mind (2), leads to j

—

/

m

eLiN3 + eLjNi + J2 WaiWaj

\

= 0,

i,je{l,...,n}.

147

Since {L(0),Wi(0),iV(0),... , W m (0)} is a pseudo-orthonormal basis, then the above equation jointly with Lemma 6 of Ref. 15 implies m

.

eLi(t)Nj(t)

+ eLj(t)Ni(t)

+ ] T Wai(t)Waj(t)

= VlJ,

t e [-6,6}.

a=l

Then by using again the same lemma we deduce that {L, Wi, TV,... , Wm} is a pseudo-orthonormal basis for all t, and this concludes the proof. The following result shows that the Cartan curvatures determine curves satisfying the nondegeneracy conditions stated in Theorem 1. T h e o r e m 3 If two Cartan curves C and C in M™ have Cartan curvatures {k\,... ,kn}, where hi : [—#,#] —> K are differentiable functions, then there exists a Lorentzian transformation of M™ which maps C into C. 3.1

Null helices in Rf

The goal of this section is to classify the family of null helices in Rf. Before to do that, we present some examples. From the general equation, we know that a null helix in Rf satisfies the following differential equation (6) 7

+ (2fci + fc32)7(4) - (k2 - 2fe1fe32)7" = 0,

which will help us to find the examples. Example 1 Let u, a and h be three non-zero constants such that -~7 < h2 < -\ and let 7 : M —> Mf be the curve defined by 7(i) =

\

ht, —asmuit, —acosuit, — bsmat, —bcosat w w a a

where a = y/h2a2 — 1/Vc 2 — u2 and b = y/1 — h2u>2/Va2 — w2. Then it is easy to see that 7 is a helix with curvatures kx = \{a2 + w 2 (l - a2h2)), k\ = -w2a2 (u2h2 - 1) (a2h2 - 1) and k2 = w2a2h2. Example 2 Let u>, a and h be three non-zero constants such that 0 < h2uj2 < 1 and let 7 : R —> Rj be the curve defined by / N /"l , * , !, • 1, 7(i) = — asmhuit, — acoshwt, — bsmat, — bcosat, ht \w

UJ

a

a

where a = \fl + h2a2/y/u2 + a2 and b = \ / l — h2oj2/yJuj2 + a2. Then 7 is a helix with curvatures fci = \{a2-u2{l+a2h2)), k\ = -w2a2(w2h2-l){a2h2 + 2 2 2 1) and kl = ui a h .

148

Example 3 Let a and h be two non-zero constants such that 0 < 2h2 < 1 and let 7 : R —> Rf be the curve defined by 2 3 2 2 3 ~t(t) , w = ( -at 2 + 6-h t + t , —=t , 2-at - 6-h t + t ,bsmcrt,bcosat)

V2

V

/

,

where a = (1 - 2h2)/(a2h2) and 6 = \/\ - 2h2/a2. Then 7 is a helix with curvatures h = \a2{\ - 2/i 2 ), fcf = 2aih2{\ - 2ft2) and fcf = 2er2/i2. Theorem 4 ( 15 ) A null curve fuIJy immersed in Rf is a helix if and only if it is congruent to a helix of the families described in Examples 1-3. 3.2

Null helices in S\

In this section we are going to classify the null helices in the 4-dimensional De Sitter space. A null curve 7 in Sf C Rf is a helix if and only if it satisfies the differential equation 7 ^ ' + 2fci7^3^ — (1 + ^2)7' = 0, whose general solution is 7(t) = A1 sinh wt+A2 cosh cut+A3 sin at+A4 cos at+A5, where A\, A2, A3, A4 and A5 are constant vectors in R^. Example 4 Let 7 : R -> Sf C Rf be the null curve defined by 7(i) =

, VU

1 2

-2

+ (J

/ 1 . , 1 1 . 1 ,~ — smhiot, — coshort, — sinai, — coscri, \ V W

W

CT

(T

V

W2

^

\-

where 0 < w2 1. A direct computation shows that 7 is a null helix with curvatures ki = \{p2 — w2) and k\ = uj2a2 — 1. Theorem 5 ( 15 ) A nuil curve fully immersed in S\ is a helix if and only if it is congruent to one of the family described in Example 4. 3.3

Null helices in Ef

Let 7 be a null curve in Hf, then it is a helix if and only if it verifies the ordinary differential equation 7 ^ + 2/ci7^3^ + (1 — ^2)7' = 0- Before we state the main result of this section we present some examples of helices in the 4-dimensional anti De Sitter space.

149

Example 5 Let 0 < J1 < 1 and let 7 be the curve in Mf defined by ^

'2*

coshwi, —^ I coshwt ur \

*

2

uitsmhtot I , —^ ( sinhwi / wz \

2

u)tcoshut I /

4

• v. * V 7 ! 3 ^

— sinhwi,

~— w2

2u>

Then 7 is a helix with curvatures k\ — —u>2 and k\ = 1 — w4. Example 6 Let 0 < a2 < 1 and let 7 be the curve in H 4 defined by If. 1 \ ± ( I . \ t 7(i) = I —^ I sincrt — -at cos at I , —^ ( coscrt + -at sin at I , — — coscrt, 4

_t B m. < r t , -yT^ _ _ j^. +

Then 7 is a helix with curvatures fci = a2 and fc2 = 1 — a4. Example 7 Let LO2 = 1 and let 7 be the null curve in H 4 defined by

^m -(,-*. 7 W

^ 3 + ^ ) ^ "(* 3 -*) *!\

V 24' 2 ^ '24' 2^3 ' 2 ; Then 7 is a helix with curvatures fci — 0 and k\ — 1. 7 will be called the null quartic in Hf. Example 8 Let 0 < LJ2 < a2 and uj2a2 < 1, and let 7 be the curve in H 4 defined by "fit)

—

.

=

Vo-2 -

w

'l . . 1 , 1 . . 1 . / 1 + g;4 — sin u>t, —— Osin cosO ^
l + o4"^ ^— . , a"'

Then 7 is a helix with curvatures k\ = \{u2 + a2) and k\ = 1 — uj2a2. Example 9 Let 0 < a2 < w2 and J2a2 defined by 7(t) =

< 1, and let 7 be the curve in Hf

1 /l , 1 , 1 , 1 /' l + cr4 — sinhwt, — coshut, — sinhat, — coshwf, \ = 2= Vw - a \ w a a u> V cH 2

Then 7 is a helix with curvatures k\ = — \{w2 + a2) and k\ = 1 — a>2cr2.

1 + a;4 w2

150

Example 10 Let a ^ 0 and let 7 be the curve in Mf defined by '2 + 2a4-a2t2 t t2 1 . 1 , , —, —, , —^ sin at, —K cos a 2o-2VTTa1 'or'2VTTc^
1 / 1 + ui4 1+a4 1 , 1 1 . 1 \ . 2 =2 \/ r2 1 -—, — sinhwr, — coshwt, — sincrf, — cosat 1 Vw +
Then 7 is a helix with curvatures fci = \{a2 — w2) and k2 — 1 + uj2a2. Example 13 Let w2 + a2 < 1 and let 7 be the curve in Hf defined by y(t) =

r—=

^- (2UJO cosh ut sin at + (OJ2 — a2) sinhwi cos at,

2

v

2uja (w + 0*)

'

—2wa cosh Lot cos at + (w2 — cr2) sinh wt sin 2 + cr2) sinh wi cos cri, (w2 + cr2) sinhw£sincr£,2wcrv/l — (w2 + cr2) I . Then 7 is a helix with curvatures fci = —uJ2 + a2 and k\ = \ — (w2 + cr 2 ) 2 . Theorem 6 ( 15 ) A nuIZ curve fully immersed in Hf is a helix if and only if it is congruent to one helix of the families described in Examples 5-13. 4

The Cartan frame for s-degenerate curves

Let 7 be a differentiable curve in M™, write Ei(t) = span {7'(£), 7" (*),... , 7 ^ ( * ) } a n d let d be the number defined by d — max{i : dim Ei(t) = i for all i). The curve 7 is said to be an sdegenerate (or s-lightlike) curve if for all 1 < i < d dimRad(.Ej(t)) is constant for all t, and there exists s, 0 < s < d, such that Rad(£ s ) ^ {0} and Rad(Ej) = {0} for all j < s.

151

Note that 1-degenerate curves are precisely the null curves studied in preceding sections. In this section we will focus on s-degenerate curves (s > 1) in Lorentzian spaces. Notice that they must be spacelike curves. To find the Frenet frames, we will distinguish four cases: Case 1: d = n and s < d. One can show that there exist a set T = {W\,... , Ws^i,L, Wg,N, W3+i,... , Wm} such that the following equations hold

W[ = k2W2, W^-kiWi-i+ki+iWi+i, W's_x = -k^Ws-2 + eksL, L' = eks+1L + ks+2Ws, W'a = eks+3L - efcs+2-/V, N' = -ksWs^ Ws+l = -£ks+4L W'j = -kj+aWj-!

2
- eks+1N - ks+zWs + + fcs+5Ws+2, + kj+4Wj+1,

s+

ks+4Ws+1, 2<j<m-l,

for certain functions {fci,..., km+3} called the curvature functions of 7 with respect to T. Case 2: d < n and s = d. If M™ is a Lorentzian space form, then 7 lies in a d-dimensional totally geodesic lightlike submanifold. This can be proved by adapting the proofs of Theorems 5 and 9 of Chapter 7 in Ref. 17. This case will be treated in a forthcoming paper 16 . Case 3. d < n and s = d — 1. This case can not occur. Case 4: d < n and s < d — 1. Working as in the non-degenerate case (see, for example, the book of Spivak17 ) this case reduces to Case 1. Note that the type s does not depend on the parameter of the curve. Also, this kind of curves are invariant under Lorentzian transformations, in the sense that the type s does not change under a Lorentzian transformation. Now we are going to find a Frenet frame with the minimal number of curvatures and such that they are invariant under Lorentzian transformations. We will restrict to Case 1. Without loss of generality, let us assume that 7 is arc-length parametrized, so that W\ — 7' and k\ = 1. By taking ks = 1, a straightforward computation shows that

152

7'

= Wi,

w{ = hw2, W! =-ki^Wi-i

+ kiWi+1,

2
L' = ks^Ws, W's = eksL - eks-xN,

( }

N' = - e W 8 _ i - ksWs + ks+1Ws+1, Ws+1 = -eks+1L + ks+2Ws+2, W^ = -kjWj.i

+ kj+1Wj+1,

s + 2 < j < m - 1,

W'm = -fc m W m _i, for certain functions {fci,..., km}. We can easily deduce the following result. Observe that when m > s then e = — 1 and ki > 0 for i ^ s; however, when m = s then e — - 1 or e = 1 according to {7', 7 " , . . . , 7 ^ } is positively or negatively oriented, respectively. Theorem 7 ( 18 ) Let 7 : 1 -» Mf, n = m + 2 , be an s-degenerate (s > 1) unit spacelike curve and suppose that (7'(t),7"(t),. • • , 7 ^ ( * ) } spans T^^M™ for all t. Then there exists a unique Frenet frame satisfying the equations (3). In this case 7 is said to be an s-degenerate Cartan curve, the reference and curvature functions given by (3) are called the Cartan frame and Cartan curvatures of 7, respectively. 5

s-degenerate curves in Lorentzian space forms

Let 7 : I —> M"(c) be an s-degenerate (s > 1) Cartan curve, where M"(c) stands for R", §™ o H", according t o c = 0 , c = l o r c = - l , respectively. If {Wi,... , Wa-i, L, WS,N, Ws+\,... , Wm} is the Cartan frame then in equations (3) we must write W{ = fciW^ — cy. The following results can be obtained in a similar way as in the null case (for the proofs see Ref. 18). Theorem 8 Let k\,... ,km : [—6,6] —> R be differentiable functions with ki > 0 for % ^ s,m. Let p be a point in M", n = m + 2, and let { Wf,... , W°_!, L°, W°, N°, W ° + ! , . . . , W°,} be a positively oriented pseudoorthonormal basis of T p M"(c). Then there exists a unique s-degenerate, s > 1,

153

Cartan curve 7 in M"(c), with 7(0) = p, whose Cartan frame satisfies: L(0)=L°,N(0)

= N°,Wi(0)=W9,

ie{l

m}.

Theorem 9 (Congruence Theorem) If two s-degenerate Cartan curves C and C in M"(c) have Cartan curvatures {fci,... , km}, where ki : [-6, S) —> R are differentiable functions, then there exists a Lorentzian transformation of M"(c) which maps bijectively C into C. 6

s-degenerate helices in Mf (c)

This section is devoted to classify s-degenerate Cartan helices in Lorentzian space forms Mf(c), c = —1,0,1. If we assume that k\ and fe are constant, then 7 satisfies the differential equation 7 ^ ' - (2ek\k2 — c)j^ (k\ + 2eck\k2) 7' = 0. Without loss of generality, we can assume that 7 is positively oriented, that is, e = — 1. In what follows, we will present examples of 2-degenerate Cartan helices in M|(c) and show the corresponding characterization theorems. 6.1

2-degenerate helices in R^

Example 14 Let 7 be the curve in Rf defined by 7ft) =

. = ( — cosh uit, — sinh wt, — sin at, — cos at) , ua > 0, \/u)2 + u 2 Vw u a a ) -2 — Then 7 is a helix with curvatures k\ = ua and £2 = (o u2)/{2ua). Theorem 10 ( 18 ) An s-degenerate spacelike Cartan curve Rf is a helix if and only if it is congruent to a helix of Example 14. 6.2

2-degenerate helices in Sf

Example 15 Let 0 < a2 < 1 < u2 and let 7 be the curve in Sf defined by

'r

lit) = [

-l)(l-a ui2a2

2

1 > 1 1 1, ) —a sirlut, —a coscut, —bsmat, —b cos at U a u J 5

2 where a = \ / l — a2/y/u2 -- a-2 and b = Vw^ - 1/^U2 - 0• . Then 7 is 2 2 -l)(l-a a helix with curvatures fci ) and /c2 == * ( « a + a2

1)/V(^-1)(1-^ 2 ).

V(^

154

Example 16 Let a2 > 1 and let 7 be the curve in Sf defined by y(t) = ( — acoshut, —asmhut, —bsmat, — b cos at, — \ / ( L O 2 4- l)(a2 - 1) I , \u) cv a a wa J where u) ^ 0, a — \ A T 2 — l/\/ui2 + a2 a n d b = \fuj2 + \jsjup- + a2. T h e n 7 is a helix with curvatures k\ = \/{a2 — l ) ( w 2 + 1) a n d k,2 = (a2 — w 2 —

1)/V<%2 - 1)(^2 + 1). Example 17 Let a2 > 1 and let 7 be the curve in S | defined by ,

/1 A / ^ ^ T

N

2

/(J2 - 1

^^T^I

v7^7! 2 1

1

Then 7 is a helix with curvatures k\ = \Jo2 — 1 and &2 = ^Vo"2 — 1. Theorem 11 ( 18 ) An s-degenerate spacelike Cartan curve in Sf is a helix if and only if it is congruent to one of the families described in Examples 15-17. 6.3

Helices in H?

Example 18 Let 0 < a2 < 1 < J1 and let 7 be the curve in Mf defined by l(t) — I —acoshwi, — bcoshat, — asmhwt, —bsinhat, \/{w2 - 1)(1 - a2) I \u) a u> a wa J where a = \ / l — a2/^co2 — a2 and b — y/u2 — l/\/u2 — a2. Then 7 is a helix with curvatures ki = y/(to2 — 1)(1 — a2) and &2 = — (to2 + a2 —

l)/V4(W2-l)(l-0Example 19 Let u>2 > 1 and let 7 be the curve in Mj defined by (u>2-l)(a2 + l) 1 , 1 .u 1 . A/ o~o J — acoshwi, — asinhwi, — bsmat, \ V bi^a* to u> a 2 2 2 2 where a ^ 0, a = V<7 + 1/ya; + a and & = Vaj - 1/Vw2 7 is a helix with curvatures k\ = i/(o; 2 — l)(a2 + 1) and £2 l)/^(u2-l)(a2 + l). .. 7(f)

=

/

1 \ — bcosat \, a I 2 + a . Then = (CT2 — UJ2 +

Example 20 Let w2 > 1 and let 7 be the curve in Wf defined by 7(f) =

5

^ o—22 * , - ^ c o s h w i , —2 2sinhwt, 2 w +l 'w2 'w 'V

w2

'2(l+a;

Then 7 is a helix with curvaturesfci= y/u>2 — 1 and &2 = — A\/w2 — 1.

155

Theorem 12 ( 18 ) An s-degenerate spacelike Cartan curve in Hf is a helix if and only if it is congruent to one of the families described in Examples 18-20. Acknowledgments This research has been partially supported by DGICYT grant BFM20012871-C04-02. The second author is supported by a FPPI Grant, Program PG, M.E.C. References 1. B. O'Neill. Semi-Riemannian Geometry. Academic Press, New York London, 1983. 2. E. Vessiot. Sur les curbes minima. Comptes Rendus, 140:1381-1384, 1905. 3. W.B. Bonnor. Null curves in a Minkowski space-time. Tensor, N.S., 20:229-242, 1969. 4. M. Castagnino. Sulle formule di Frenet-Serre per le curve nulle di una V4 riemanniana a metrica iperbolica normale. Rendiconti di matematica, Roma, Ser. 5, 23:438-461, 1964. 5. L.P. Hughston and W.T. Shaw. Classical strings in ten dimensions. Proc. Roy. Soc. London Ser. A, 414:423-431, 1987. 6. L.P. Hughston and W.T. Shaw. Constraint-free analysis of relativistic strings. Classical Quatum Gravity, 5:69-72, 1988. 7. L.P. Hughston and W.T. Shaw. Spinor parametrizations of minimal surfaces. In The Mathematics of Surfaces, III (Oxford, 1989), pages 359372. Oxford Univ. Press, New York, 1989. 8. W.T. Shaw. Twistors and strings. In Mathematics and General Relativity (Santa Cruz, CA, 1986), pages 337-363. Amer. Math. Soc, PJ, 1988. 9. H. Urbantke. On Pinl's representation of null curves in n dimensions. In Relativity Today (Budapest, 1987), pages 34-36. World Sci. Publ., Teaneck, New York, 1988. 10. A. Nersessian and E. Ramos. Massive spinning particles and the geometry of null curves. Phys. Lett. B, 445:123-128, 1998. 11. A. Nersessian and E. Ramos. A geometrical particle model for anyons. Modern Phys. Lett. A, 14(29):2033-2037, 1999. 12. A. Bejancu. Lightlike curves in Lorentz manifolds. Publ. Math. Debrecen, 44:145-155, 1994. 13. K. L. Duggal and A. Bejancu. Lightlike Submanifolds of SemiRiemannian Manifolds and Applications, volume 364 of Mathematics and

156

14.

15. 16. 17. 18.

its Aplications. Kluwer Academic Publishers Group, ,Dordrecht, The Netherlands, 1996. A. Ferrandez, A. Gimenez, and P. Lucas. Degenerate curves in pseudoEuclidean spaces of index two. To appear in Proceedings of Geometry, Integrability and Quantization, Varna. Ed. I. Mladenov, 2002. A. Ferrandez, A. Gimenez, and P. Lucas. Null helices in Lorentzian space forms. International Journal of Modern Physics A 16 (2001), 4845-4863. A. Ferrandez, A. Gimenez, and P. Lucas. Spacelike curves in lightlike submanifolds of a Lorentzian space form. In preparation, 2001. M. Spivak. A Comprehensive Introduction to Differential Geometry, 3rd Ed, Vol. IV. Publish or Perish, Houston, Texas, 1999. A. Ferrandez, A. Gimenez, and P. Lucas. s-degenerate curves in Lorentzian space forms. Preprint, 2001.

MINIMAL DISCS B O U N D E D B Y S T R A I G H T LINES LEONOR FERRER AND FRANCISCO MARTi'N Departamento de Geometria y Topologia, Universidad de Granada 18071 Granada, Spain E-mail: [email protected]; [email protected] Dedicated to the 60th birthday of A. M. Naveira Minimal surfaces containing straight lines have special properties. Among them, we emphasize Schwarz's reflection principle (see, for instance, [7]). Many authors studied the problem of determining minimal surfaces bounded by straight lines. In particular, Schwarz, Weierstrass and Riemann obtained existence results for minimal surfaces with boundary a given polygon whose sides could be of finite or infinite length. We should mentioned also a classical result by Jenkins and Serrin about the existence and uniqueness of minimal graphs bounded by straight lines ([2]). In [2] these authors gave necessary and sufficient conditions to solve the Dirichlet problem in a compact convex domain bounded by a polygon assuming values +oo, —oo and continuous data on different straight segments in the boundary. We deal with the existence and uniqueness of properly embedded minimal discs whose boundary Tg d consists of the following configuration of straight lines: Fix Q £ [0,7r] and d > 0, and consider two half-lines rf and rj" in R , meeting at an angle of 8. Let qf and gf be two points in r^ and r± , respectively, such that they are symmetric with respect the inner bisector of this half-lines. We choose qf and q{ in such a way that either q\ = q{ or the half-lines l\ and l~{ on r+ and r~{ starting at q+ and gj", respectively, do not intersect. Write d = dist{qf ,q^[). Let 7Ti be the plane determined by i\ and l\ and let -n^ denote a plane parallel and distinct to -K\. Let t^ and l^ be the orthogonal projections to 7r2 of l\ and £±, respectively. Denote q£ (resp. q^) as the orthogonal projection to 7r2 of qf (resp. gf), and label £ j (resp. IQ ) as the segment [qi ,q%] (resp. [Qi iQ^})- We also denote by {Tthe vector with origin q^ and end q%• Finally, we define

»"(y«)u(y«r). 157

Figure 1. The curve T ^ .

With this notation, we consider the following generalized Plateau's problem: Problem (P) Determine a properly immersed minimal surface X : M —> K

satisfying:

(1) M is homeomorphic to the closed unit disc D minus two boundary points E\ and E2, that we call the ends of M. (2) X{d{M))

= Ted.

(3) If d > 0, X is an embedding. (4) In the limit case l£ = £Q (i.e., d = 0), the maps X\M—y+ and X\M_^~ are injective, where j + and j ~ are the two connected components of d(M). (5) X(M)

lies in S, where S is the horizontal slab determined by iri and -K2-

Let us recall briefly some previous results on ( P ) : • For 9 = 0 and 0 < d < \\v\\, it is known from the result by Jenkins and Serrin mentioned before that there exists a unique solution of ( P ) which is a graph over the rectangle {x2 = 0} n £ ( r 0 d ) , where E(T0d) denotes the convex hull of the boundary (see Fig. 2).

159

Figure 2 A Jenkms-Seirin graph

In case d — IT and d = 0 we have the example illustrated in Fig. 3 that had appeared the first time in a paper by F.J. Lopez, M. Ritore and F. Wei (see [5]). Later F J. Lopez and F. Wei proved in [6] that this example is the unique solution of (P) with this boundary.

Figure 3

160

Concerning the general case, F.J. Lopez and the second author studied the existence and uniqueness of solutions of the aforementioned problem with a more restrictive condition (5) in the papers [3] and [4]. Particularly, they assume: (5')

X(M)c£(T0d)

instead of our condition (5). We denote by ( P ' ) the problem with this stronger condition. Their results about existence and uniqueness of solutions of (P ? ) can be sumarized in the following theorem; Theorem

For 0 < 9 < w, there exists dg with 0 < dg such that:

(i) If d> dg there are no solutions of the problem ( P ' ) . (ii) If d = 0 or d = dg, ( P ' ) has a unique solution. (Hi) Ifde

}0,dg[, ( P ' ) has two solutions.

aw*-:;..-

Figure 4. The two solutions in case 0 = w/3, d = 0.56 ||v||. The first figure corresponds to the stable example and the second figure corresponds to the unstable one.

161

Figure 5. The unique solution in case 9 = w/3 and d = 0.

Now, we recall the Weierstrass representation of Lopez-Martin's examples. The following representation is obtained after applying a slight change to the original one: Consider the following Riemann surface M = {(u,v) eC*xC\v2

= (u2-

eit0)(u2 -

e~it0)}

where to G ]0, ir[, and define in the ti-plane the set of curves illustrated in Fig. 6. ^

s:'0

it0/2

SJ

sf

e'

4 Figure 6.

•it0/2

162

Yo

Y7

4

n

/ 0+

Jo •

^

'2

a)

h)

Figure 7. a) The domain «(M). 6J The surface M.

Then, label C cN

as the connected component of

U-l(C\({J(StUs7))) containing the point Po = ( l , + \ / 2 ( l — cos(io))) - Define M =C , where C means the closure of C in Af. Finally, we consider the following meromorphic data on M 5 =«" ,

where

n=j™

Je[0,f

(*) #3 = B # ,

where

J5 > 0.

In order to prove the existence part of the theorem, F.J. Lopez and F. Martin made a detailed study of the distance function d :]0,7r[—> R, to —• d(*o), for a fixed 6. They proved that d is an analytic function and that there exists t'6 € ]0,TT[ satisfying: ® d is positive in ]t^,7r[, ® d(ta) = lim d(to) = 0,

163

« d has only one critical point tg € ]te,ir[ which is a maximum. Moreover,

d(t£) = do. One can prove that for 9 e [7r,27r], these Weierstrass representations give examples of minimal surfaces satisfying conditions (l)-(5) of our problem ( P ) . These new examples are not only in S, but all of them are also contained in

(S\6(T0d))urm. We have then two families of solutions of our problem: Lopez-Martin examples, that lie in the convex hull of their boundary, and another family of solutions that lie in the exterior of the convex hull. We prove that these are all the solutions. To be precise we obtain the following result: P r o p o s i t i o n Let X : M —• M be a proper conformal minimal satisfying conditions (l)-(5) of problem ( P ) . (%) IfO<0<

v, then X{M)

immersion

lies either in £(T0d) or in (S \ £ ( r M ) ) U Fgd.

(ii) If 8 = IT, then X(M) lies in one of the half-slabs determinated by the plane that contains £(r„-<j).

Figure 8. Two solutions for 6 = ir/3, one in £{Tg,i) and the other in (S \ £ ( 1 ^ ) ) U T$4.

164

Figure 9, Two solutions for 6 = 0, one in £(Tod) and the other in (S \ £(Tod)) U T,

Finally, we obtain the following result: Mala Theorem ® Case 0<e<w. (i) Ifd>de

There exist de and d'e with 0 < d'e < de such that: there are no solutions of the problem ( P ) .

(ii) Ifd = de, ( P ) has a unique solution. (Hi) Ifde

]d'e,de[, ( P ) has two solutions,

(iv) Ifd = d'e> ( P ) has three solutions. (v) Ifde

]0, dg[, ( P ) has four solutions,

(vi) If d = 0, ( P ) has two solutions. 9 Case 6 = ir. There exist dv with 0 < d s such that: (i) Ifd>dw

there are no solutions of the problem ( P ) .

(ii) lfd = 0 ord = dw, ( P ) has a unique solution. (Hi) Ifd€

]0,4-[, ( P ) has two solutions.

® Cose 0 = 0. There exist d'0 with 0 < d'Q < 1 such that: (i) Ifd>\

there are no solutions of the problem ( P ) .

(ii) lfd = 0orde

]d'0,l[, ( P ) has a unique solution.

(Hi) If d = d'0, ( P ) has two solutions, (iv) Ifde

}0,d'0[, ( P ) has three solutions.

!65

Figure 10. The solution in case 0 = 0 and d = 0.

For the proof of this theorem it is sufficient to study the existence and uniqueness of the family of examples that lie in the exterior of the convex hull of their boundary. The existence part can be obtained similarly to Lopez-Martin existence results. However, the uniqueness part presents some additional difficulties that can be solved with more elaborated arguments. We enumerate the crucial points in this proof and refer the reader to [1] for details. Sketch of the proof of uniqueness: 1. Conformal type of M: It can be proved that M is conformally equivalent to D\{Ei,E2}Moreover, the Weierstrass data extend meromorphically to the ends. 2. Horizontal symmetry: Using Schoen's ideas based on Alexandrov method (see [8]), it is proved that the solution is symmetric respect to the plane {x3 = 0}. 3. Determine a model of the complex structure and Weierstrass representation: Taking into account the previous steps it is possible to see that any solution of ( P ) is, up a horizontal translation, one of the surfaces described by (*).

166

Acknowledgments Research partially supported by DGICYT grant number BFM2001-3489. References [1] L. Ferrer and F. Martin, Properly embedded minimal disks bounded by non compact polygonal lines, preprint. [2] H. Jenkins and J. Serrin, Variational problems of minimal surface type.II. Boundary value problems for the minimal surface equation. Arch. Rat. Mech. Anal. 21, 321-342 (1966). [3] F.J. Lopez and F. Martin, Minimal surfaces in a wedge of a slab. Commun. Anal. Geom., (to appear). [4] F.J. Lopez and F. Martin, A uniqueness theorem for properly embedded minimal surfaces bounded by straight lines. J. Aust. Math. Soc. (Series A) 69, 362-402 (2000). [5] F.J. Lopez, M. Ritore and F. Wei, A characterization of Riemann's minimal surfaces. J. Differ. Geom. 47, 376-397 (1997). [6] F.J. Lopez and F. Wei, Properly immersed minimal disks bounded by straight lines. Math. Ann. 318, 667-706 (2000). [7] R. Osserman, A survey of minimal surfaces. Dover Publications, New York, second edition, 1986. [8] R. Schoen, Uniqueness, symmetry and embeddedness of minimal surfaces. J. Differ. Geom. 18, 791-809 (1983).

V O L U M E A N D E N E R G Y OF V E C T O R FIELDS O N S P H E R E S . A SURVEY OLGA GIL-MEDRANO Departamento de Geometria y Topologia, Universitat de Valencia Burjassot (Valencia) Spain E-mail: [email protected] Dedicated to A. M. Naveira on the occasion of his 60"1 birthday The contents of this paper correspond roughly to the talk given at the Conference. The paper is divided into three parts. In the first one, we describe the functionals volume and energy of vector fields and the conditions that characterize their critical points in a general Riemannian manifold. Second part is devoted to the study of some critical unit vector fields on round spheres. In particular we will describe those that have been relevant in the different results concerning the problem of estimating the infimum of the functionals and that of the existence and regularity of minimizers. Last section contains a survey of these results. 1

Critical vector fields

The energy of a smooth map ip : (M, g) —> (N, h) from a Riemannian manifold to another is defined as

E(f) = \ I tr(Lv)dv9, 1

JM

where Lv is the (1,1) tensor field completely determined by (
tr(Lv) =

^2(ho(p)(
The volume of an immersion ip : M —* (N, h) is the volume of the Riemannian manifold (M,ip*h), that is Vol(
167

dv^h.

168

If we choose a metric g on M then Vol(p) = / JM

JdetiL^dVg.

V

It is well known that the Euler-Lagrange equations, of the corresponding variational problems, give rise to the definition of tension of a map and of mean curvature of an immersion. Both of them are vector fields along the map and their vanishing defines harmonic maps and minimal immersions, respectively. In a g-orthonormal frame as above, the tension is expressed in terms of the Levi-Civita connections V 9 and Vh as n

rg(
f*(ygEiEi)).

i=l

The mean curvature vector field coincides with the tension of the map

(N,h). If we consider the tangent bundle ir : TM —> M and a metric go on M, we can construct a natural metric on TM as follows: at each point v € TM, we consider on the vertical subspace of Tv (TM) the inner product g0 ( up to the usual identification with TVM, where p = TT(V)), we take the horizontal subspace determined by the Levi-Civita connection as a suplementary of the vertical and we declare them to be orthogonal; finally, we define the inner product of horizontal vectors as the product of their projections, with the metric go- The so constructed metric g$ is sometimes referred as the Sasaki metric. The geometry of (TM, g$) is well known, a good description can be found in [ * ]and we have used it to compute Tg(V), the tension of a vector field V in M considered as a map V : (M,g) —> (TM,go). We will represent by V the Levi-Civita connection of go, and by R the (1,3) curvature tensor given by R(X, Y, Z) = - V x V y Z + Vy VXZ + V[XX]Z. P r o p o s i t i o n 1 Let V be a vector field in M, then ~

\ hor

( Ei^((VV)(£ ),^^)+r (Id)) ( (VV)(r (Id)) + E (Vs (VV))( B_ ))\ ver 4

9

fl

i

i

J

i

where for a vector field X we have represented by Xver its vertical lift and by Xhor its horizontal lift. {Et} is any g-orthonormal frame and Ts(Id) is the tension of Id : (M,g) —> (M,g0).

169

In the particular case g = go, the expression above simplifies due to the vanishing of r 3 (Id) and in fact, the vertical part of the tension is just the rough Laplacian of V. This result goes back to Ishihara [ 2 1 ]. For a compact manifold M, the vector field (Vy)(r 9 (Id)) + ^ ( V ^ ( V ^ ) ) ( ^ ) i

vanishes if and only if W = 0, as can be seen combining Corollary 15 and Proposition 17 of [ 12 ]. Then it is more interesting to consider the problem of determining which unit vector fields, with respect to the metric go, give harmonic maps from (M,g) to the unit tangent bundle {TlM,gQ). It is not difficult to see that Theorem 1 ([ 12 ]) Given a unit vector field V in a Riemannian manifold (M,go) and a metric g on M, the map V : {M,g) —> {TlM,g$) is harmonic if and only if • (a) 'EiR((yV)(Ei),V,Ei)+Tg(Id)=0

and

• (6) (VV)(T 9 (Id)) + E i ( V j 3 . ( W ) ) ( £ i ) is collinear to V. Apart from the particular case g — g0, mentioned above and for which condition (6) is that of V being an eigenvector of the rough Laplacian, the other special case involving only one metric on M is g = V*go- In that case the tension computed in Proposition 1 is the mean curvature vector field of the immersion V : M - • {TlM,g$). In [ 12 ] we have shown that if g = V*g$', condition (b) implies condition (a) and then Proposition 2 Given a unit vector field V in a Riemannian manifold (M,go), the immersion V : M —> {T^M^Q) is minimal if and only if the vector field

(W)(7v.floS(Id)) + E ( V E , ( W ) ) ( ^ ) , i

is collinear to V. A different question is when a unit vector field is a critical point of the energy restricted to unit vector fields. To answer this, we only have to take into account that the tension Tg(V) of a unit vector field is an element of the tangent space at V, to the space of smooth maps from M to TXM. In fact Tg(V) € Tv(C°°(M, TXM)) can be seen as the gradient at V of the energy functional (for details see[ 12 ]). Since we are now concerned with a variational problem with constraints, the condition will be the vanishing of the projection of the tension onto the

170

subspace Tv{T°°{M,TlM)) c Tv(C°°(M,TlM)), of those elements that are tangent to the submanifold T°°(M, TlM), consisting in all smooth sections of the unit tangent bundle. It is easy to see that if 77 is a vector field on TlM along V, then it is tangent to the manifold of unit sections if and only if the vector r](x) e Tv(x){TlM) is vertical, for all x € M. As a consequence, Corollary 1 A unit vector field is a critical point of the energy restricted to unit vector fields if and only if satifies condition (b). It is a critical point of the volume restricted to unit vector fields if and only if it is a minimal immersion. Let us say now that this is not the usual approach to the problem. Instead, we can go back to the definition of energy and volume of a map and, using that for a vector field V we have V*g0s(X,Y) = g0(X,Y) + g0{VxV,VYV) and consequently Ly = Id + (VV)*(VV), we can see that the energy of the map V : (M, go) —> [TlM, g^), that is known as the energy of the vector field, is given by E{V)

=

\

+

\\M

H W H 2dv °

and the volume of the vector field is

F(V) = f

y/det(Lv)dv0.

JM

The relevant part of the energy, B(V) = JM || W | | 2 d v o is sometimes called the total bending of the vector field. The condition for a unit vector field to be a critical point has been obtained by direct computation of the Euler-Lagrange equation of these variational problems. The second order differential operators involved are the rough Laplacian V*V and the operator V*D where D, similar to V, is a first order differential operator from the space of vector fields to the space of (1, l)-tensor fields given as DV = ^Jdet(Lv){W)Lvl. Let us recall that V* represents the formal adjoint of V that can be expressed in an orthonormal frame as V*(AT) = ^(V^-K")!^. Moreover the 1-form associated by the metric to this vector field is given by X X V E ^ ' ) ' T h e o r e m 2 . Given a unit vector field V in a Riemannian manifold (M,go) then 1- ([29 ])V is a critical point of the energy if and only if it is an eigenvector of the rough Laplacian. %• ([li ])V is a critical point of the volume of vector fields if and only if it is an eigenvector of the second order differential operator V*D. Moreover, it is critical if and only if it is a minimal immersion.

171

Remark. In [ 12 ]we show, by a direct argument, that condition in part 2 above and that of Proposition 2 are equivalent. Let wy and wy be the 1-forms associated by the metric to V*VV and V*DV, respectively. The covariant version of Theorem 2, as it appears in [ 14 ], has been very useful for the study of particular examples and also to compute the second variation of the volume in [ 15 ] . The second variation of the energy was previously computed by a different method in ([ 2 9 ]). Proposition 3 Given a unit vector field V in a Riemannian manifold (M, go) then V is a critical point of the energy if and only if cjy{X) = 0 for all X e V1- and V is a critical point of the volume if and only if u>v(X) = 0 for

allXeVL.

2

Critical unit vector fields on round spheres

Using the characterizations of Proposition 6, several authors haved recently developed many examples of unit vector fields that are minimal immersions, harmonic maps or critical for the energy of unit vector fields, sometimes called also harmonic sections. In this survey we will concentrate on examples defined (globally or not) on a round sphere, or on those closely related to them. For more examples de reader may consult [ 2 ] , [ 5 ] , [ 13 ] , [ 17 ] , [ 18 ] and [ 19 ]. It is well known that Hopf fibration 7r : S2m+1 —> (DPm determines a foliation of S2m+1 by great circles and that a unit vector field can be choosen as a generator of this distribution. It is given by V = JN where N represents the unit normal to the sphere and J the usual complex structure on jR 2 m + 2 . V is the standard Hopf vector field, but we will call a Hopf vector field any vector field in 5 2 m + 1 obtained as JN for J a complex structure on M2m+2, that is J € End(M2m+2) such that J* o J = Id, J 2 = - I d . Hopf vector fields are exactly unit Killing vector fields of 5 2 m + 1 , as can be seen in [ 30 ]. In [ 22 ] it was shown that for any variation of a Hopf vector field by unit vector fields, the first derivative of the volume vanishes. In view of Theorem 2, this means that Hopf vector fields are minimal immersions. In fact we have Proposition 4 • ( [ 29 ]) Unit Killing vector fields on an Einstein manifold are critical points of the energy with energy E(V) = ^ ^ V o ^ M ) , where X is the Einstein constant. • (115 D Unit Killing vector fields on a manifold of constant curvature k are minimal immersions with volume F{V) = (1 + fc):V"Vol(M). Moreover, it is easy to see from Theorem 1 that Hopf vector fields are harmonic maps. An interesting result is the following:

172

Proposition 5 ([ 20 J) The only unit vector field on S3 that are harmonic maps are Hopf vector fields. The proof is rather technical and can not be extended to higher dimensional spheres. Concerning these problems, 3-dimensional manifolds have a very particular behaviour and we will see this phenomena to appear frequently in the sequel. For instance, in [ 15 Jit is shown that a unit Killing vector field V on a 3-dimensional manifold is minimal if and only if R(X, Y, V) = 0, for all X, Y that are orthogonal to V, but that this is not longer true if the dimension of the manifold is greater than 3. Hopf vector fields are the characteristic vector fields of the usual Sasakian structure of odd-dimensional spheres (see [ 1 ], for the definitions). So, that they are critical points of the volume and of the energy can be seen also as a particular case of the following general result Proposition 6 ([ 29 /, [15]) The characteristic vector field of any Sasakian manifold is a minimal immersion and a critical point of the energy. The construction of Hopf vector fields on spheres as JN, described above, can be generalized to any real hypersurface of a Kahler manifold. In [ 27 ] and [ 28 Jthe reader can find many interesting results concerning the problem of determining for which hypersurfaces the corresponding Hopf vector field is a minimal immersion, a critical point of the energy or provides a harmonic maps. Next example is that of unit radial vector fields that are only defined on the complementary of two antipodal points. Let p b e a point of Sn and let T be the unit vector field tangent to radial geodesies from p. The field T is defined on Sn — { — p, p}. It is clear that these vector fields have totally geodesic flow and the complementary distribution T1 is integrable and has umbilical leaves. Thanks to these properties it is not difficult to show, using Theorem 1 and Proposition 3, Proposition 7 Unit radial vector fields on Sn - {-p,p} are critical points of the energy, minimal immersions but they are not harmonic maps. For any point p of a Riemannian manifold M and any normal neighbourhood U, we can define a unit radial vector field on U — {p}. In [ 3 ] and [ 4 ] the authors have studied these unit vector fields and they have shown the following results Proposition 8 In a two-point homogeneous space, the radial unit vector fields are minimal immersions. Theorem 3 A Riemannian manifold is harmonic if and only if all radial unit vector fields are critical points of the energy. They are harmonic maps only

173

if the manifold is flat. Finally, let us consider vector fields W defined on Sn — {—p} by parallel transport of a given w e TpSn, along the great circles of Sn passing through p. We will call such a W, a parallel translation unit vector field. In [ 26 ], it is shown that the generalized Pontryagin cycle is minimal at each smooth point as a submanifold of the corresponding Stiefel manifold. Since W(Sn — {—p}) can be seen as the set of smooth points of one of these cycles, then W is a minimal unit vector field. In [ 15 ], we use direct computation of W and VW to show that tiwiW-1) = {0}. The expression of VW can be used also to compute tow (the details can be seen in [ 24 ]) and then Proposition 9 Parallel translation unit vector fields defined on Sn — {—p} are minimal immersions into the unit tangent bundle but they are critical for the energy only if the dimension is 2.

3

The infimum of the functionals on spheres

It is clear that lower bounds for the energy (resp. the volume) of unit vector fields on a manifold M, with finite volume vol(M), are given by ^vol(M) and vol(M), respectively. Moreover they are realized only by parallel unit vector fields, when they exist. The problem of minimizing the energy and the volume among unit vector fields becomes then interesting when M does not admit a gobally defined unit parallel vector field. This is the case for odd-dimensional spheres. The first known result is the famous Gluck and Ziller theorem Theorem 4 ([ 16 J) The unit vector fields of minimum volume of S3 are the Hopf vector fields and no others. To obtain this result they find a smooth closed 3-form \i on TXS3 such that fi(uAv Aw) < vol(u Av Aw) with equality holding for the tangent plane of V(S3) where V is a Hopf vector field. From here, V(S3) are absolutely minimizers in their homology class and now the result is obtained from the fact that any unit vector field on 5 3 is in the same homology class. For the energy the first result was Proposition 10 ([ 30 J) Hopf vector fields on S3 are stable critical points of the energy with nullity 2. Here the author uses deeply the fact that S3 admits a global frame consisting on Hopf vector fields obtained from the quaternionic structure of 2R4. And finally, in [ 8 ], it is established that Hopf vector fields are the unique absolute minimum by showing the following Theorem 5 Let M be a closed 3-dimensional manifold, then for every unit

174

vector field E{V)>\

1

[ (3 +

p(V,V))dv

JM

and

F(V)> f (l + ZL(V,V))dv. JM

In both cases equality holds if and only if the flow determined by V is geodesic and the distribution V1- verifies 2o~2 = S where ai is the second symmetric function of curvature of the distribution and S is the square of the norm of the second fundamental form. To obtain these estimates, the author uses the integral equality 2 f CT2dv = / p(V, V)dv JM

JM

which is a rewriting of the well known formula (see [ 2 5 ]) f ||W||2)dv = / (p(V,V) + JM

JM

z

hcvgf-(divV)2dv,

because, as it is easy to see, ±\\Cvg\\2-(divV)2

= \\VV\\2-2
Corollary 2 Let M be a closed 3-dimensional manifold such that p(V, V) > X, for all unit vector field V. Then E(V) > | ( 3 + A)vol(M) and F(V) > (1 + ^A)vol(M). In particular if VQ has geodesic flow, the distribution V1verifies 2
and F(V) = 2 m Vol(5 2 m + 1 ).

On the other hand, if T is a unit radial vector field on S2m+1 E(T) = ((

m

_ +l)m+^)Vol(52m+1)

and F(T) = | ^ L 4 m Vol(5 2 m + 1 ).

175

For higher dimensional spheres the situation is very different. In [ 22 ] it is shown that Hopf vector fields on S5 are unstable for the volume. In [ 3 1 ] (see the corrected version in [ 32 ]) it is shown that they are energy unstable for S2m+1 for m > 1 and finally in [ 14 ] we have shown that they are volume unstable for m > 2. So, apart from the case of S3, Hopf vector fields can not be minimizers of neither the volume nor the energy of smooth unit vector field on the sphere. In fact to find smooth minimizers of such a functional is not very likely, in general, as can be deduced from the regularity result proved in [ 2 3 ] asserting that for any homology class of one-dimensional foliations in a compact oriented Riemannian manifold of dimension greater than 3, there is a volume minimizing representative with singular locus of codimension at least 3. Therefore, the problem will be for a given Riemannian manifold: to compute the infimum value of the functionals, to study if these values can be attained by some smooth unit vector fields and if not, to look for minimizers as regular as possible. This problems has been completely solved for the energy on spheres. In I11], using integral inequalities similar to those mentioned in Theorem 5, it has been shown Theorem 6 ([n ]) Let V be a unit vector field with a finite set of singularities in a compact Riemannian manifold M of dimension n > 3 then

E(V)>1 [ (n+-?—p(V,V))dv. z

JM n—A Ifn>4, equality holds if and only if V has geodesic flow and V1- is integrable and with umbilical leaves. In the particular case of the the sphere S2m+1, with m > 1 they obtain Corollary 3 (f11 ]) Let V be a unit vector field with a finite set of singularities in S2m+l with m > 2 then E(V) > E(T) and equality holds if and only

ifV = T. Finally, in [ 6 ] (see also [ 7 ]), we have constructed a family of smooth unit vector fields Te such that E{T€) converges to E(T) and then Theorem 7 E(T) is the infimum value of the energy of smooth unit vector fields on S2m+1 with m > 2. The same problem concerning the volume functional appears to be much more complicated. In the recent work [ 9 ], the authors have been able to show the following estimate Theorem 8 ([ 9 ]) Let V be a unit vector field in a compact Riemannian manifold M of dimension 1m + 1 > 3 and constant curvature c then

176

m

(m) \c\l

Ifm>2, equality holds if and only if V has geodesic flow and V is integrable and with umbilical leaves. As a consequence, Corollary 4 Let V be a unit vector field with in S2m+1 with m > 2 then F(V) > F(T) and equality holds if and only ifV = T. The proof is long and technical because in this case it has been necessary to use the integral equalities involving all the symmetric functions of curvature of V1- that where evaluated in [ 10 ]. It is not clear, up to now, if a sequence of smooth unit vector fields can be found such that their volumes converge to F(T). In fact in [ 26 ] the author conjectured that the infimum of the volume of unit smooth vector fields on 5 2 m + 1 is precisely the volume of the parallel translation vector fields W, the value of which F(W) = 7 4 ^ v o l ( 5 2 - + 1 ) \2m)

is grater than F(T). So, Theorem 8 is a first step towards the solution of the interesting open problem of determining the infimum of the volume of unit vector fields on an odd-dimensional spheres. Acknowledgments Work partially supported by a DGI Grant No. BFM2001-3548. References 1. D. E. Blair, Riemannian Geometry of Contact and Symplectic manifolds, Birkhauser, 2002. 2. E. Boecks and L. Vanhecke, "Harmonic and minimal vector fields on tangent and un it tangent bundles", Diff. Geom. and its Appl. 13, 77-93 (2000). 3. E. Boecks and L. Vanhecke, "Harmonic and minimal radial vector fields", Acta Math. Hung. 90, 317-331 (2001). 4. E. Boecks and L. Vanhecke, "Radial vector fields on harmonic manifolds", Bull. Soc. Sci. Math. Roumanie 93 , 181-185 (2000).

177

5. E. Boeckx and L. Vanhecke, "Isoparametric functions and harmonic and minimal unit vector fields", Global Diff. Geom.: The Mathematical Legacy of Alfred Gray (eds. M. Fernandez and J.A. Wolf) Contemp. Math. Ser., Amer. Math. Soc. 288, 20-31 (2001). 6. V. Borrelli, F. Brito and O. Gil-Medrano, "An energy minimizing family of unit vector fields on odd-dimensional spheres", Global Diff. Geom.: The Mathematical Legacy of Alfred Gray (eds. M. Fernandez and J.A. Wolf) Contemp. Math. Ser., Amer. Math. Soc. 288, 273-276 (2001). 7. V. Borrelli, F. Brito and O. Gil-Medrano, "The infimum of the energy of unit vector fields on odd-dimensional spheres" Preprint. 8. F. Brito , "Total Bending of flows with mean curvature correction", Diff. Geom. and its Appl. 12, 157-163 (2000). 9. F. Brito P. Chacon and A. M. Naveira, "On the volume of unit vector fields on spaces of constant sectional curvature", preprint. 10. F. Brito R. Langevin and H. Rosenberg, "Integrales de courbure sur des varietes feuilletees", J. Diff. Geom. 16, 19-50 (1981). 11. F. Brito and P. Walczak, "On the Energy of Unit vector Fields with Isolated Singularities", Ann. Math. Polon. 73, 269-274 (2000). 12. O.Gil-Medrano, "Relationship between volume and energy of vector fields", Diff. Geom. and its Appl. 15, 137-152 (2001). 13. O.Gil-Medrano, C. Gonzalez-Davila and L. Vanhecke, "Harmonic and minimal invariant unit vector fields on homogeneous Riemannian manifolds", Houston J. Math. 27 , 377-409 (2001). 14. O.Gil-Medrano and E. Llinares-Fuster, "Second variation of Volume and Energy of vector fields. Stability of Hopf vector fields ", Math. Ann. 320 , 531-545 (2001). 15. O.Gil-Medrano and E. Llinares-Fuster, "Minimal unit vector fields", Tohoku Math. J. 54 , 71-84 (2002). 16. H. Gluck and W. Ziller, "On the volume of a unit vector field on the three sphere", Comment. Math. Helv. 61 , 177-192 (1986). 17. C. Gonzalez-Davila and L. Vanhecke, "Examples of minimal unit vector fields", Ann. Global Anal. Geom. 18, 385-404(2000). 18. C. Gonzalez-Davila and L. Vanhecke, "Energy and volume of unit vector fields on three-dimensional Riemannian manifolds", Diff. Geom. and its Appl. , (to appear). 19. C. Gonzalez-Davila and L. Vanhecke, "Minimal and harmonic characteristic vector fields on three-dimensional contact metric manifolds", J. of Geom. , (to appear). 20. D.-S. Han and J.-W. Yim, "Unit vector fields on spheres which are harmonic maps", Math Z. 227,83-89 (1998).

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21. T. Isihara "Harmonic sections of tangent bundles", J. Math. Tokushima Univ. 13, 23-27 (1979). 22. D. L. Johnson, "Volume of flows", Proc. Amer. Math. Soc. 104, 923-932 (1988). 23. D. L. Johnson and P. Smith, "Regularity of Volume-Minimizing graphs", Indiana Univ. Math. J. 44, 45-85 (1995). 24. E. Llinares-Fuster, Energia y volumen de campos de vectores. Puntos criticos y estabilidad, Tesis Doctoral, U. Valencia, 2002. 25. W.A. Poor, Differential Geometric structures, McGraw-Hill, 1981. 26. S.L. Pedersen, "Volumes of vector fields on spheres", Trans. Amer. Math. Soc. 336, 69-78 (1993). 27. K. Tsukada and L. Vanhecke, "Minimal and Harmonic Unit Vector Fields in G2((Vm+2) and Its Dual Space", Monatsh. Math. 130, 143-154 (2000). 28. K. Tsukada and L. Vanhecke, "Minimality and harmonicity for Hopf vector fields", Illinois J. Math, to appear. 29. G. Wiegmink, "Total bending of vector fields on Riemannian manifolds", Math. Ann. 303, 325-344 (1995). 30. G. Wiegmink, "Total bending of vector fields on the Sphere S 3 ", Diff. Geom. and its Appl. 6, 219-236 (1996). 31. C.M. Wood, On the energy of a Unit Vector Field, Geometriae Dedicata, 64, 319-330, (1997). 32. C.M. Wood, The energy of Hopf Vector Field, Manuscripta mathematica, 101, 71-88, (2000).

SPACELIKE J O R D A N O S S E R M A N A L G E B R A I C CURVATURE T E N S O R S I N T H E HIGHER S I G N A T U R E SETTING P E T E R B. GILKEY AND RAINA IVANOVA Mathematics Department, University of Oregon, Eugene Or 974-03 USA email: [email protected], [email protected] Dedicated t o Professor Navaira Abstract: Let R be an algebraic curvature tensor on a vector space of signature (p, q) defining a spacelike Jordan Osserman Jacobi operator JR. We show that the eigenvalues of JR are real and that JR is diagonalizable if p < q. Subject Classification: 53B20

1

Introduction

Let V b e a vector space that is equipped with an inner product (•, •) of signature (p, q). The inner product is said to be Riemannian if p — 0, Lorentzian if p = 1, and balanced (or neutral) if p = q. Let R be an algebraic curvature tensor on V; i.e. R e 4V* satisfies the curvature symmetries of the Riemann curvature tensor: R(x, y,z,w) = R(z, w,x,y) = —R(y, x,z,w), and R(x, y, z, w) + R(y, z, x, w) + R(z, x, y, w) = 0. The associated Jacobi operator JR is the self-adjoint linear map of V which is defined by the identity: (JR(X)V,Z)

=

R(y,x,x,z).

We say that an algebraic curvature tensor R is spacelike Osserman if the eigenvalues of JR are constant on the pseudo-sphere S+(V) of unit spacelike vectors. We say that R is spacelike Jordan Osserman if the Jordan normal form of JR is constant on S+(V). In the Riemannian context, these two notions are equivalent. However, in higher signature setting, the eigenvalue structure does not determine the conjugacy class, so we shall work with Jordan normal form rather than with the eigenvalues alone. The investigation of spacelike Jordan Osserman tensors is motivated by geometric considerations. Let (M,g) be a pseudo-Riemannian manifold of signature (p, q). If 9R is the Riemann curvature tensor of the Levi-Civita connection, then 9Rp is an algebraic curvature tensor on the tangent space

179

180

TpM for every P in M. On the other hand, every algebraic curvature tensor is geometrically realizable. We say that (M, g) is spacelike Jordan Osserman if the Jordan normal form of JOR is constant on the pseudo-sphere bundle S+(M,g) of unit spacelike tangent vectors. Let (M, g) be Riemannian. Then (M, g) is said to be a local 2 point homogeneous space if the local isometries of (M, g) act transitively on S+(M, g); such a manifold is either flat or locally isometric to a rank 1 symmetric space. If (M, g) is a local 2 point homogeneous space, then the Jacobi operator JR has constant Jordan normal form on S+(M,g) and hence (M,g) is spacelike Jordan Osserman. Osserman 13 wondered if the converse held. Chi 4 showed this was true if m = 4, if m = 1 mod 2, or if m = 2 mod 4; an important ingredient in Chi's work was the classification of the spacelike Jordan Osserman Riemannian algebraic curvature tensors in these dimensions. There are other partial results available 5 ' 11 , but the question is still open in the Riemannian context. It has been shown 1 ' 6 that any Lorentzian spacelike Jordan Osserman algebraic curvature tensor has constant sectional curvature. Thus classification is complete in this context. Finally, it is known 2 ' 3 ' 8 ' 7 that there exist balanced spacelike Jordan Osserman pseudo-Riemannian manifolds which are not even locally homogeneous. We shall work in the algebraic category henceforth. If W is an auxiliary vector space and if A is a linear map of V, then we define the stabilization A

®0:=(oo)

onV

®w-

The Jordan normal form of a spacelike Jordan Osserman algebraic curvature can be arbitrarily complicated in the balanced setting 9 ' 10 : T h e o r e m 1.1 Let J be a r x r real matrix and let q = 0 mod 2r. If V is a vector space of neutral signature (q, q), then there exists an algebraic curvature tensor R on V so that JR(X) is conjugate to J (BO for every x G S+(V), However, the situation is very different if p < q, i.e. if the spacelike directions in a certain sense dominate the timelike directions. We consider this case and show that the geometry defined by such a tensor is much more rigid. The main result of this paper is: T h e o r e m 1.2 Let R be a spacelike Jordan Osserman algebraic curvature tensor on a vector space V of signature (p,q), where p < q. Then JR{X) is diagonalizable for any x e S+(V). Here is a brief outline of the paper. In §2, we review certain results concerning self-adjoint maps in the indefinite setting. In §3, we establish

181

several technical results concerning vector bundles over projective spaces. In §4, we use the results of the previous sections to prove Theorem 1.2. 2

Results from Linear Algebra

Let 3?(A) and 3(A) be the real and imaginary parts of a complex number A. If J is a linear map of a real vector space V of dimension m, then let Jx be the real operator on V defined by: A

J J - A Id '~ \(J-A-Id)(J-A-Id)

if if

AeR, AeC-R.

, l

. >

We define the generalized eigenspaces by setting Ex = E( := ker{J A m }.

(2.2)

Lemma 2.1 Let V be a vector space of signature {p,q) and let J be a selfadjoint linear map of V. Then V can be decomposed as an orthogonal direct sum V = ©3(A)>o-E'A- Furthermore, the induced metrics on the generalized eigenspaces Ex are non-degenerate. Proof: Let A and \x be complex numbers with A ^ / J and A ^ p,. Since J™ is self-adjoint and vanishes on Ex, we have 0 = (J™x x , xM) = {x\, JxXn) for xx e Ex and xM e £ M . Since J commutes with J^, J preserves E^. Since the eigenvalues of J on E^ are JJ, and p,, the linear maps J — A • Id, J — A • Id, and hence J\ are isomorphisms of E^; thus J™(SM) = EM. It now follows that Ex ± E„ and ExnE/i

= {0}.

(2.3)

Let ^ c := V CS> C be the complexification of V. We extend J to V c to be complex linear and set E A := ker{( J - A) m }. A complex vector space may be decomposed as the direct sum of the generalized complex eigenspaces defined by a linear transformation. Consequently, Vc = ®XE^.

(2.4)

As Ef 8 Ef = Ex ® C, V = ®cS(X)>0Ex. By display (2.3), the direct sum given in equation (2.4) is orthogonal; thus, the induced metric on each Ex is non-degenerate. •

182

We recall a few notions from bundle theory which we will use in what follows. Let p : E —* M be a vector bundle over a smooth manifold M. The fibers Ep : = p~1(P) are real vector spaces which vary smoothly with the point P £ M. A non-degenerate fiber metric on E is a collection of non-degenerate inner products on each fiber which vary smoothly on M. A bundle morphism \p of E is a collection of smooth linear maps tpp of the fibers Ep which vary smoothly with P. We say ip is invertible if each ipp is invertible. We say \p is self-adjoint if each xpp is self-adjoint. Let V he a, vector space with a non-degenerate inner product. We can decompose V as a direct sum V+ ®V~ of complementary orthogonal subspaces, where V+ is a maximal spacelike subspace and V~ is a maximal timelike subspace. There is a similar decomposition possible in the vector bundle setting: Lemma 2.2 Let E be a vector bundle over a smooth manifold M which is equipped with a non-degenerate fiber metric. Then we can decompose E as a direct sum E+ © E~ of complementary orthogonal subbundles, where E+ is a maximal spacelike subbundle and E~ is a maximal timelike subbundle. Proof: As noted above, we can decompose each individual fiber as an orthogonal direct sum of a maximal spacelike and a maximal timelike subspace. The main technical difficulty is to make the decompositions vary smoothly with P e M. We can use a partition of unity to put a positive definite inner product (•, -) e on E. Define a bundle morphism if> of E by setting (v, w) = (tpv, w)e. Since each linear map tpp is self-adjoint with respect to the positive definite inner product (•, -) e on each fiber Ep, tpp is diagonalizable and has only real eigenvalues. As the original inner product (•, •) is non-degenerate, each ipP is invertable. Let E\(ipp) C Ep be the eigenspaces of xpp on Ep. We define: Ep := ®\
®x>0Ex{xpP).

By Lemma 2.1, the subspaces Ep and Ep are orthogonal and complementary. Since xpp is invertible, the fibers Ep and Ep have constant rank and define smooth subbundles of E. O We continue our preparation for the proof of Theorem 1.2 by studying vector bundles equipped with non-degenerate fiber metrics and self-adjoint bundle morphisms which have constant Jordan normal form: Lemma 2.3 Let E be a vector bundle over a smooth manifold M which is equipped with a non-degenerate fiber metric. Let J be a self-adjoint bundle morphism of E which has constant Jordan normal form. Let A be an eigenvalue of J. If J\ ^ 0 on E\, choose i > 1 maximal so J^{E\) ^ 0. Then J\E\ is a totally isotropic subbundle of E of non-zero rank.

183

Proof: Assume that E and J satisfy the hypothesis of the Lemma. Set: Ex,i := J{(EX). Since J has constant Jordan normal form, E\ti is a smooth vector bundle over M. Fix a point P of M and let v\ and v2 be vectors in the fiber E\yi{P). There exist vectors w\,wi € E\^{P) so vi = J^iui and U2 = J^u^- Note that 1% > i + 1, that JA is self-adjoint, and that J^ = 0 on E\. We demonstrate that £^A,t is totally isotropic and thereby complete the proof by computing: (vuv2) 3

= (J{wi, J\w2) = (Jliw1,w2)=0.

•

Bundles over projective space

Let V be a vector space of signature (p, <j). We decompose V — V+ ®V~ as an orthogonal direct sum, where V+ is a maximal spacelike subspace of dimension q and V~ is the complementary maximal timelike subspace of dimension p. Let P(V + ) be the projective space of lines in V+. We define trivial bundles over F(V+) by setting: V+ :=F{V+) x V+, V - :=F(V+) x ^ - , and V : = V + © V ~ =F(V+) xV.

(3.1)

These bundles inherit natural inner products from the given inner product on V. Let (x) := x-R be the line thru an element x e S(V+). The classifying line bundle 7 and the orthogonal complement 7 X over F(V+) are the subbundles of V + defined by: 7 := {((x),y) 7

X

e P(V+) x V+ : y e (x)}, and

:= {({^},y) € F(V+) xV+:y±.

(x)}.

(3.2)

Note that 7(x) = (a;), i.e. the fiber of 7 over an element (x) of P(V + ) is just the line (x) itself. For that reason 7 has also been called the tautological line bundle. This bundle plays an important role in the classification of real line bundles and in our further considerations. In the following Lemma, we compare non-trivial (i.e. positive rank) subbundles of 7 X and V~. Lemma 3.1 Let V = V+ © V" be a vector space of signature (p,q), where p < q. Let V - and -y1- be the bundles over F(V+) defined in equations (3.1) and (3.2), respectively. Then no non-trivial subbundle ofV~ is isomorphic to a non-trivial subbundle o / 7 x .

184

Proof: The Stiefel-Whitney classes12 are cohomological invariants of vector bundles. Let w\ :— ^1(7) be the first Stiefel-Whitney class of the classifying line bundle 7. The cohomology ring of the projective space F(V+) is the truncated polynomial ring 12 : JT(P(V+);Z2)=Z2[u;i]/K=0). To prove Lemma 3.1, we suppose the contrary, i.e. that there exist positive rank subbundles E\ of 7 X and E2 of V - so that E\ is isomorphic to E2, and argue for a contradiction. Let r = rank(E'i). Since E\ c 7 X , we may use Lemma 4.4.49 to see that wr{E\) — w\. Furthermore, as E\ is isomorphic to E2, we may conclude wr(-E2) = w[. As V - = E2 ®E^, we have the following factorization 1 = w{E2)w(E£)

= (1 + ... + wr!)w(Ei)

in

H*{¥(V+); Z 2 ).

Since r a n k £ 2 = P — r < Q, the truncation (w\ = 0) in iJ*(P(T/ + );Z 2 ) plays no role, so we have the factorization l = (\ + ...+w\)w(Ei)

in

Z 2 [tui],

which is impossible. • We use Lemma 3.1 to establish the following Lemma: Lemma 3.2 Let V — V+ @ V~ be a vector space of signature (p,q), where p < q. Let V~ and 7 X be the bundles over F(V+) defined in equations (3.1) and (3.2), respectively. There is no totally isotropic non-trivial subbundle of x

7 ev-.

Proof: We suppose, to the contrary, that there exists a totally isotropic nontrivial subbundle E of 7-1- © V - . Let 7r+ be orthogonal projection on 7 X and let 7r~ be orthogonal projection on V~. Set E+ := ir+(E) C 7 X

and

E~ := n~(E) C V".

Note that ker7r+ = V - and ker7r~ = 7 X . As E is totally isotropic, every vector in E is null. Thus EnY-

=Er\>y1

= {0}.

Consequently, ir+ and n~ define isomorphisms between E and E+ and between E and E~, respectively. Thus E+, which is a non-trivial subbundle of 7 + , is isomorphic to E~, which is a non-trivial subbundle of V~. This contradicts Lemma 3.1. •

185

4

P r o o f of T h e o r e m 1.2

Let V = V+ ®V~ be a vector space of signature (p, q), where p < q, and where V+ and V~ are orthogonal maximal spacelike and timelike subspaces of V, respectively. Let R be a spacelike Jordan Osserman algebraic curvature tensor on V. Let x € S+(V). Since JR{X) is self-adjoint and since JR(X)X — 0, JR(X) preserves the orthogonal complement x x ; we let JR(X) denote the restriction of JR(X) to x1-; this is often called the reduced Jacobi operator. The Jacobi operator can be represented in the form J

«W = (°o j°R(x))

on

x

®x±-

Thus to prove Theorem 1.2, it suffices to show that JR{X) is diagonalizable. If A e C, then let Jx and Ex be defined by JR using equations (2.1) and (2.2), respectively. We may then use Lemma 2.1 to decompose 7-L

© V" = e3(A)>o£A over P(V+),

where the induced metric on each eigenbundle Ex is non-degenerate. By Lemma 3.2, Ex does not contain a totally isotropic subbundle. Thus, by Lemma 2.3, Jx = 0 on Ex- Consequently JR is diagonalizable on Ex if A e R. To complete the proof, we must show that all the eigenvalues are real. Suppose, to the contrary, that there exists an eigenvalue A of JR SO that 9(A) / 0; we argue for a contradiction. By Lemma 2.2, Ex = E~£ © E^~ decomposes as the orthogonal direct sum of maximal spacelike and timelike subbundles. We define a bundle map X of Ex by setting: T

X

. _ jR-at(A)id —

9(A)

'

The definition of Jx given in (2.1) and the fact that Jx = 0 on Ex then imply that I2 = —id on Ex- Since I is self-adjoint, I is a para-isometry of Ex that interchanges the roles of spacelike and timelike vectors. Thus, 1 defines an isomorphism between E^ and E^. Let 7r+ and TT~ be orthogonal projections on •y1- and V - , respectively. Since E~£ contains no timelike vectors and since ker(7r+) = V - is timelike, ker7r+ n £ A + = {0} and 7r+ is an isomorphism from E~£ to ir+(Ex). Similarly, 7r~ is an isomorphism from E^~ to ir~(Ex)- Thus 7r+(i?,\), which is a non-trivial subbundle of 7-1-, is isomorphic to TT~(E^), that is a non-trivial subbundle of V - . This contradicts Lemma 3.1. •

186

Acknowledgments Research of both authors supported in part by the NSF (USA) and MPI (Leipzig). References 1. N. Blazic, N. Bokan and P. Gilkey, A Note on Osserman Lorentzian manifolds, Bull. London Math. Soc. 29 (1997), 227-230. 2. N. Blazic, N. Bokan, P. Gilkey and Z. Rakic,Pseudo-Riemannian Osserman manifolds, J. Balkan Soc. Geometers 12 (1997), 1-12. 3. A. Bonome, R. Castro, E. Garcia-Rio, L. Hervella, R. Vazquez-Lorenzo, Nonsymmetric Osserman indefinite Kahler manifolds, Proc. Amer. Math. Soc. 126 (1998), 2763-2769. 4. Q.-S. Chi, A curvature characterization of certain locally rank-one symmetric spaces, J. Differential Geom. 28 (1988), 187-202. 5. I. Dotti and M. Druetta, Negatively curved homogeneous Osserman spaces, Differential Geom. Appl. 11 (1999), 163-178. 6. E. Garcia-Rio, D. Kupeli and M. E. Vazquez-Abal, On a problem of Osserman in Lorentzian geometry, Differential Geom. Appl. 7 (1997), 85-100. / 7. E. Garcia-Rio, D. N. Kupeli, and R. Vazquez-Lorenzo Osserman manifolds in semi-Riemannian geometry, Lecture notes in Mathematics, Springer Verlag, to appear. 8. E. Garcia-Rio, M. E. Vazquez-Abal and R. Vazquez-Lorenzo, Nonsymmetric Osserman pseudo-Riemannian manifolds, Proc. Amer. Math. Soc. 126 (1998), 2771-2778. 9. P. Gilkey, Natural Operators Denned by the Riemann Curvature Tensor, World Scientific Press, ISBN 981-02-4752. 10. P. Gilkey and R. Ivanova, The Jordan normal form of Osserman algebraic curvature tensors, preprint. 11. P. Gilkey, A. Swann, and L. Vanhecke, Isoparametric geodesic spheres and a conjecture of Osserman regarding the Jacobi operator, Quart. J. Math. Oxford Ser. 46 (1995), 299-320. 12. J. Mil.nor and J. Stasheff, Characteristic Classes, Annals of Math. Studies, Princeton University Press (1974). 13. R. Osserman, Curvature in the eighties, Amer. Math. Monthly 97 (1990), 731-756.

STABILITY OF SURFACES W I T H C O N S T A N T M E A N CURVATURE I N THREE-DIMENSIONAL SPACE F O R M S MIYUKI KOISO Department of Mathematics, Kyoto University of Education, Fushimi-ku, Kyoto 612-8522, Japan E-mail: [email protected] We derive sufficient conditions for constant mean curvature surfaces in threedimensional space forms to be strongly stable. 1

Introduction

Barbosa-do Carmo 1 obtained the following sufficient condition under which a minimal surface in the three dimensional euclidean space R 3 is stable. Theorem A(Barbosa-do Carmo x) An oriented compact immersed minimal surface in R 3 is strictly stable if the area of its Gaussian image is less than 2K. As a corollary of Theorem A, one can get the following Corollary B (Barbosa-do Carmo 1 ) Let E be an oriented compact immersed minimal surface in R 3 . If the Gauss curvature K of T, satisfies the inequality J s |.Jsr|
3

Let M = M (c) be a three-dimensional complete Riemannian manifold with constant sectional curvature c. Let E be a two-dimensional orientable 3

compact connected C°° manifold with boundary 9E, and let X : E —> M (c) be an immersion. Denote by v the unit normal vector field along X, by K the Gauss curvature of X, and by dE the volume element of E induced by X. For any smooth variation X : (—e,e) x E —> M (c) of X (that is, X(0,u>) = X(w) for all w € E), the area function A(t) and the volume function V(t) are defined as follows. A{t) = f dZt, V(t) = [ X*dw,

187

188

where dE t is the volume element of E in the metric induced by immersion o

_

3

Xt : E —> M (c) (X t (w) := X(t,w)), do; is the volume element of M (c), and X*dU is the pull-back of cE7 by X. For any ffeR, define a function JH by JH{t) = A(t) + 2HV(t). Then the mean curvature of X is constant H (we will say that X is CMC-H) if and only if J'H(0) = 0 for all variations X of X that fix the boundary (i.e. Xt\dTi = X\g-£ (Vt)) (Barbosa - do Carmo - Eschenburg 3 ) . 3

Definition Let X : E —> M (c) be CMC-IT Then X is said to be strongly stable if J^(0) > 0 for all variations X of X that fix the boundary. Moreover, X is said to be strongly strict-stable if J#(0) > 0 for all non-trivial variations X of X that fix the boundary. (When only the case H = 0 is treated, X is said to be "stable" instead of "strongly stable", and it is said to be "strictly stable" instead of "strongly strict-stable".) We will prove the following sufficient condition for the stability. 3

Theorem 1.1. Let E be simply-connected, and let X : E —> M (c) be CMCH. Assume thatc+H2 > 0. If Js(2(c+H2)-K)dE < 2TT, then X is strongly strict-stable. In the case where c + H2 < 0, we will prove the following: 3

Theorem 1.2. Let E be simply-connected, and let X : E —> M (c) be CMCH. Assume jthat c +• H2 < 0. Set -3K -^9K2-8(C , + H1)KQ v Q K0 = rTK, Mo = _2(c + g 2 ) • Then, the following (i) and (ii) hold. (i) If KQ > 2(c + H2), then X is strongly strict-stable. (ii) IfK0<2(c+H2), then (a) 2/3 < Mo < 3 - V5 « 0.764. (b) If f^{/j,0(c + H2) — K}dY, < 2TT, then X is strongly strict-stable. The following Corollary is a direct consequence of Theorem 1.2. 3

Corollary 1.1. Let E be simply-connected, and let X : E —> M (c) be CMCH. Assume that c + H2 < 0. / / / E {(2/3)(c + H2) - K}dZ < 2ir, then X is strongly strict-stable. Theorem 1.1 was proved by Ruchert 7 for c = 0, by Barbosa-do Carmo 2 for H = 0, by Barbosa-Mori 4 for c > 0, and by da Silveira 9 for c = — 1 and H = 1. Corollary 1.1 was proved by Barbosa-Mori 4 for c 4- 4H2 < 0. Before giving comments on the sharpness of our results, we remark an isometric correspondence of CMC immersions given by Lawson ("Lawson correspondence") and the second variation formula of the function J # .

189

Denote by M0(c) the three-dimensional simply-connected complete Riemannian manifold with constant sectional curvature c. For any simplyconnected CMC-H immersion X : £ —> M0(c) and any constant ci < c + H2, there exists a one-parameter family of C M C V c + H2 - c\ immersions Xt : 3

E -> M 0 (ci) each of which is isometric to X (Lawson 5 ) . Therefore, Theorem 1.1 is exactly the case where each X is isometric to a CMC surface in R 3 . On the other hand, the following second variation formula is known. Fact (Second variation formula 3 ) Let X : £ -> ~M3{c) be CMC-H. For any variation X of X that fixes the boundary, J£(0) = - J [uAu + 2 {2(c + H2) - K) u2] d£, where u={(dXt/dt)\t=0,v) and A = div grad is the Laplacian on E induced by X. Therefore, the second variation J#(0) for a CMC-H immersion into 3

M (c) is the same as Jf{i{0) for corresponding CMC-.ffi immersion {Hi := \/c + H2 - c i ) into M 3 (ci). Now let us comment on the sharpness of our results. Theorem 1.1 is sharp. In fact, the part of Enneper's minimal surface in R 3 whose Gaussian image coincides with the hemisphere is not strongly strict-stable and it satisfies / E (2(c + H2) - K)dT, = 2-K (where c = H = 0). Also for the hemisphere in R 3 , the situation is the same. Moreover, we obtain corresponding examples in other space forms by using the "Lawson correspondence". On the other hand, the sharpness of Theorem 1.2 is hopeless. We cannot omit the assumption of simply-connectedness in Theorem 1.1. In fact, we have the following counterexamples. Each CMC surface 3

in M (c) (c < 0) obtained by the "Lawson correspondence" from a catenoid in R 3 is called a Catenoid Cousin. The whole of each Catenoid Cousin is not stable (da Silveira 9 ) . But we can construct a Catenoid Cousin for which the integral J s (2(c -I- H2) - K)dT, = J^(-K)dH > 0 is as small as we wish. The paper is organized as follows. In Section two, we will consider a new metric on E and estimate the Gauss curvature of E with respect to the new metric. In Section three, we will give the proofs of our main results.

190

2

Curvature Estimate 3

Let X : E —> M (c) be an immersion with constant mean curvature H. Denote by ds2 the metric on £ induced by X. For any real number JJL, we define a function / on E as follows. H2)-K.

f = fi(c + We would like to find conditions so that

ds2 = fds2 defines a new metric whose Gauss curvature can be estimated. By considering the isothermal coordinates of X, we regard E as a Riemann surface. Let £ = £* + i£ 2 be a local coordinate in £. Then the metric of E is represented as

ds2=w

((de)2+(d?)2).

The second fundamental form B of E is represented as B =

Vijpijdgdg.

The square of the norm of B is \\B\\2 = 2(c +

2H2-K).

Set ip = (Ai - HW) - i/312(= -(/%2 - HW) -

ip12).

Then

H2-K),

+ ( M - l ) ( c + tf2).

Therefore, Lemma 2.1. / / (/x - l)(c + H2) > 0, i/ien / > 0. Remark 2.1 ip = 0 if and only if X is totally umbilic. Assume that / > 0, and denote by K the Gauss curvature of E with respect to the metric ds2. The following lemma is proved essentially by the same way as the proof of Lemma 2.3 in Ruchert 7 . Lemma 2.2.

f(K-l)

-2(n-l)(c+H2) fW3

_ ¥>C

2WQip W

2

-{fx{c+H2)}2+{3fi-2)(c+H2)K.

191

Proof, ds is represented as ds2 = fds2 = W {(d^1)2 + (d£ 2 ) 2 ) ,

W = fW.

Therefore, [W2

\

f2K+-{kfr-ff W^^C t JHC,>

w,CC

W

By straight calculation, we see .12 /C

\W2)C

W2

W3

'

2W<; VC

c

w 2W^ 2

< p < - 2W W(
M2

From the above equalities we get the desired formula.

•

2

Lemma 2.3. Assume that c + H > 0. If fi = 2, then at each non-umbilic point, / > 0,

and

K < 1.

Proof. The positivity of / follows by Lemma 2.1. The curvature estimate is derived as follows. f2{K-l)

-2{c + H2) Vd fW*

2Wcip W

-2(c + H2)

2Wc

fW*

¥><

Lemma 2.4. Assume that c + H2 < 0. Set

2

~4(c +

H2)(c+H2-K)

-4(c+W^<0. D

192 Then at each point,

/ > 0, and K < 1. Before proving the above Lemma, we give a few remarks. If c + H2 < 0, then, in view of the formula \ 0. L e m m a 2.5. 2 - <

MO

< 1,

where the equality holds if and only if X is totally umbilic. Proof. Set -3K - V9K2 - 8AK = =2A ' where A is a negative constant. Then, ,„.

9{K)

-2Ag'(K) = - 3

9K-AA y/VK2 - &4/f

-2

Since c + fl — K > 0, it is enough to consider the case K
-K)>0,

and we can easily show that

g'(K) > o. Hence, g(K) is strictly increasing. Moreover, 9(A) = 1, g(K) =

* _ _ 2 3+ ^ M 3

as

K—»D

Proo/ o/ Lemma 2.4- Set

Then, by Lemma 2.2, f(K

- 1) <

-(MO)2^2

+ (3/i0 - 2)AK = i4(-A(/io) 2 + 3K Mo - 2/f).

(1)

193

The right hand side of (1) is not positive if and only if ^ -3K - y/9K2 - 8AK ^ < Z^4

°

r

^ -3K + ^9K2 - 8AK ^ =S4 •

• Remark 2.2 -3K + V9K 2 - %AK > 1. -2 A 3

Proof of Main Results

Set J E ( U ) = - [ [uAu + 2 {2(c + H2) - K) u2) d£. Denote by L2(E) the usual Hilbert space completion of C°°(£) with respect to the norm defined by the inner product (u,v)L2 = / We denote by HQ(E) the completion of defined by the inner product (u,v)Hi

uvcE. CQ°(E)

with respect to the norm

= / (uv + VuVt;)dE,

where VuVv is the inner product of the gradient Vit of u and the gradient Vu of v with respect to the Riemannian metric in E induced by X. Define a linear operator L : #o(E) —> L 2 (E) by Lu = Au + 2(2(c + H 2 ) - K)u, and consider the eigenvalue problem Lu=-\u,

ti|as=0,

«6ffo(S)-{0}.

(2)

Then, a CMC-H immersion X is strongly stable (resp. strongly strict-stable) if and only if the first eigenvalue of (2) is nonnegative (resp. positive). In order to prove Theorems 1.1 and 1.2, we consider the new metric ds2 = fds2 = {n(c + H2) -

K}ds2,

and the eigenvalue problem Au + Xu = 0,

u | a s = 0,

(3)

194

where Au is the Laplacian in £ in the metric ds2. Denote by Af (£) the first eigenvalue of the problem (3). For each domain !T2 in the sphere S 2 (l) with curvature 1, denote by Af (ft) the first eigenvalue of the Dirichlet eigenvalue problem for the Laplacian for the domain £1. Proof of Theorem 1.1. Let \x = 2. Then,

First, let us assume that c + H2 > 0. Then, / > 0, and, by Lemma 2.3, K < 1 in the new metric ds2. Let fii b e a geodesic disc in S 2 (l) with area f dS= JT

f {2(c + H2) - K}dY, < 2TT, JT

2

and let S . be a hemisphere of 5 2 (1). Then, in view of K < 1, we see Af(E)>Af(fi1)>Af(52)=2 (cf. Barbosa-do Carmo 2 ). Hence, for any u G Co°(E) — {0}, we see that 7 E (u) = - I (uAu + 2u2)dt

= I \Vu\2dt -2 [ u2dt JT.

JT, 2

> Af (S) / u dt - 2 f u2dt > 0, JT

JT

which implies that X is strongly strict-stable. Next let us assume that c+H2 = 0. f(() = 0 if and only if ( is an umbilic point of X. If X is totally umbilic, then, for any u e CQ°(T,) — {0}, it follows that J E (u) = / |Vu| 2 d£. JT

Therefore, X is strongly strict-stable. If X is not totally umbilic, then umbilic points are isolated. Hence, the same discussion as above can be applied (cf. Barbosa-do Carmo 2 ) and the proof is completed. • Proof of Theorem 1.2. C0°°(£)-{0},

If K0 > 2(c -I- H2), then we see that, for any u e

IT{U) = - f [uAu + 2 {2(c + H2) - K} u2} dE> JT

hence X is strongly strict-stable.

f \Wu\2dS > 0, JT

195 Next, set } = lx0{c +

H2)-K.

Then, by Lemma 2.4, / > 0 and

K<\.

Therefore, if / E {/x 0 (c + H2) - K}dY, < 2TT, then, by the same way as in the proof of Theorem 1.1, we see Af (S) > 2. Therefore, for any u € Co°(E) - {0}, we observe that 7 s (u) = - f [uAu + 2 {/io(c + H2) - K) u2] dE - 2(2-fi0)(c > -

+ H2) J u2dY,

(uAu + 2u2)dt

is >0, which implies that X is strongly strict-stable. For the function g in the proof of Lemma 2.5, we see g(2A) = 3 - VE. Hence, by the same way as the proof of Lemma 2.5, we see that if Ko < 2(c + H2), then

- < no < 3 - VE.

• References 1. J. L. Barbosa and M. do Carmo, On the size of a stable minimal surfaces in R 3 , Amer. J. Math. 98, 515-528 (1976). 2. J. L. Barbosa and M. do Carmo, Stability of minimal surfaces and eigenvalues of the Laplacian, Math. Z. 173, 13-28 (1980). 3. J. L. Barbosa, M. do Carmo and J. Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z. 197, 123-138 (1988).

196

4. J. L. M Barbosa and H. Mori, Stability of constant mean curvature surfaces in Riemannian 3-space form, Yokohama Math. J. 30, 73-79 (1982). 5. H. B. Lawson, Jr., Complete minimal surfaces in S3, Annals, of Math. 92, 335-374 (1970). 6. J. C. C. Nitsche, A new uniqueness theorem for minimal surfaces, Arch. Rat. Mech. Anal. 52, 319-329 (1973). 7. H. Ruchert, Ein Eindeutigkeitssatz fur Fldchen konstanter mittlerer Kriimmung, Arch. Math. 33, 91-104 (1979). 8. H. Ruchert, A uniqueness result for Enneper's minimal surface, Indiana Univ. Math. J. 30, 427-431 (1981). 9. A. M. da Silveira, Stability of complete noncompact surfaces with constant mean curvature, Math. Ann. 277, 629-638 (1987).

PSEUDO-PARALLEL SURFACES I N SPACE F O R M S GUILLERMO ANTONIO LOBOS DM, Universidade Federal de Sao Carlos, Rod. Washington Luiz Km 235 - C.P. 616, 13565-905, Sao Carlos, SP - Brazil E-mail: [email protected] Dedicated to Professor A. M. Naveira on occasion of his 60th birthday Pseudo-parallel immersions were defined in [ 4] as extrinsic analogues of semisymmetric spaces and as a direct generalization of semi-parallel immersions. In this note we describe the results obtained in [ 5] for pseudo-parallel immersions of surfaces into JV-dimensional real space forms, N > 3. We also present (see Theorem 2.6 below) a detailed proof of Theorem 4.4 of [ 5 ] which is only sketched there.

1

Introduction

Let Mn be an n-dimensional Riemannian manifold and let QN (c) be a simply connected, complete iV-dimensional manifold with constant sectional curvature c. If / : Mn —» QN (c) is an isometric immersion, we denote by V and V the Levi-Civita connections of M and QN (c), respectively, and by v (/) its normal bundle. Then, the second fundamental form a : TM ® TM —> v (/) is given by a (X, Y) = V ^ V - VxY. As usual, V 1 and R1 denote the normal connection and respective curvature tensor and A% : TM —> TM denotes the Weingarten operator in the direction of £ e ^(/)- The local geometry of / is described by the above data and the basic equations (R(X,Y)Z,W)

=c(XAY(Z),W)+(a{X,W),a(y,Z)) -{a(X,Z),a(Y,W)),

( V z « ) (X, Y) = (Vya) (X, Z),

(Gauss)

(Codazzi - Mainardi)

(R^ (X, Y) £, rj) = ([At, Av] X, Y),

(Ricci)

where X AY : TM —> TM is the endomorphism defined by X A Y (Z) = (Y, Z)X-

{X, Z) Y

and V « : TM ® TM TM -* v (/) is the covariant derivative of a defined by (Vza)

(X, Y) = V^ [a (Y, Z)\ - a (VZY, Z) - a (Y,

197

VZZ).

198

The second covariant derivative V a : TM <8> TM ® TM ® TM —• v (/) is defined by (VwVza)

(X,Y)

= V ^ [ ( V z a ) (X,F)] - ( V z a ) ( V W X , F ) - ( V z a ) (X, V ^ F ) - ( V V w 2 « ) (X, F ) .

We recall that an isometric immersion / : Mn —> QN (c) is parallel if V a = 0 and it is semi-parallel if R(X,Y) • a := ( V ^ V y a ) — ( V y V j a ) = 0, for all X, Y e TM. Clearly, parallelism implies semi-parallelism, but the converse is not true in general, as shown, for example, in the classification of semi-parallel surfaces in QN (c) given by Deprez for c — 0 in [ *] and by Mercuri for c ^ 0 in [ 2],[ 3 ] . An isometric immersion / : Mn —> QN (c) is pseudo-parallel if there exists a smooth function (f> : M —> .ffl such that 5(I,y)-a = («Ay.a,

(1)

for all X, V G TM, where the linear endomorphism X AY is extended to act on a as follows: [X A Y • a] (Z, W) := -a(X

AY (Z) ,W)-a(Z,X

AY

(W)),

for all Z,W £ TM. We warn the reader that the sign of <j> in (1) is the opposite of that given in [ 5 ] . It follows easily from the equations of Gauss and Ricci that the condition (1) of pseudo-parallelism can be rewritten as: Rx (X, Y) [a (Z, W)] =a(R {X, Y) Z,W) + a (Z, R {X, Y) W) -4> (p) (Y, Z) a (X, W)+4> (p) (X, Z) a (Y, W) -4 (p) (F, W) a (Z, X) +

(2)

Every semi-parallel immersion is a pseudo-parallel immersion with <j> = 0. The converse is false in general, see section 2 below and also [ 4],[ 5 ] . R e m a r k 1.1 Let / : Mn —> QN (c) be a (^-pseudo-parallel immersion. If g : QN (c) —> QN (c) is a totally umbical immersion, then gof : Mn —> QN (c) is a ^-pseudo-parallel immersion. 2

Pseudo-parallel surfaces

Consider now an isometric immersion / : M2 —> QN(c) of a 2-dimensional manifold into an ./V-dimensional space form. Let {ei,e2} be a local orthonormal tangent frame and set a ^ = a (ej, ej) and R1 = R1 (ei, e^). With these

199

notations the condition of pseudo-parallelism (2) becomes: i ? x a i i = -Rxa22

= 2{4>-K) a12, ) (3)

fl«i2

= {K-4>) ( a n - a 2 2 )

J

where K denotes the Gaussian curvature of M. A pseudo-parallel immersion of a 2-dimensional Riemannian manifold in a space form is said to be a pseudo-parallel surface. Remark 2.1 An immediate consequence of (3) is that if R1- = 0, then / is (^-pseudo-parallel with <j> = K (or any if the point is umbilic). We recall that a semi-parallel surface with i ? x = 0 is either umbilic or flat (see [ 1 ]). Therefore, any non umbilic, non flat surface with vanishing normal curvature is a pseudo-parallel surface which is not semi-parallel. We recall that an isometric immersion / : M —> QN(c) is X-isotropic, A : M -+ R, if Vp € M and VX € TPM, with ||X|| = 1, \\a(X,X)\\ = \{p) (see[ 6 ]). It is not difficult to show that any A-isotropic immersion with non vanishing normal curvature is ^-pseudo-parallel immersion for ,

4K - A2 - c

* =

§

(4)

•

Conversely, we have the following result. Theorem 2.2 Let f : M2 —> QN (c) be a — c, \\H\\2 = 2>K — 2(f> — c and K > . Proof: Consider U = V U int(M - V), where V = {p € M2 : R1- (p) ^ 0}. If U-y is a connected component of V, then using the equations of Ricci and Gauss and the fact that a.\2 and a.\\ — a.22 &re linearly independent we get : ||a 1 2 || 2 = K - > 0, | | a n - a 2 2|| 2 = 4(X -)> 0, ( o n , o22> = 2 # - <> / - c,

Haall2 - AK - 30 - c > 0,1 ( a n , a i 2 ) = 0, \\H\\2 =

(5)

3K-2~c>0,

where H = \ ( a n + a 2 2 ) is the mean curvature vector, and hence f\u in part (ii). _

is as

200

This theorem extends analogous theorems proven for semi-parallel surfaces by Deprez for c = 0 (see [ *]) and by Mercuri for c ^ 0 (see [ 2 ]). We recall that a minimal isotropic immersion is called superminimal (see [ 7 ]). Then we have: Corollary 2.3 Let f : M2 —> Q4(c) be an isometric immersion with -R-1 ^ 0. Then f is pseudo-parallel if and only f is superminimal. Moreover, if (j> is constant, then K = f > 0 and f{M) is a piece of a Veronese surface. Proof: First observe that an A-isotropic immersion in codimension two is minimal. If <j> is constant, then K is constant (see (5)) and the claim follows from a theorem of Kenmotsu (see [ 8 ]). B R e m a r k 2.4 By a theorem of Chern, every minimal immersion of a topological 2-sphere in <34(1) := S4(l) is superminimal (see [ 9 ]). So any such immersion is pseudo-parallel. Moreover, if the curvature is not constant, then the immersion is not semi-parallel. Finally we study a class of pseudo-parallel surfaces in 5-dimensional space forms. We start with the following example (see [ 10 ]): E x a m p l e 2.5 Consider the immersion T : M2 —> S5(c): T(x, y) = H|= (cos u cos v, cos u sin v, ^ cos2u, sin u cos v, sin u sin v, ^ sin2u) where u — ^J\x, v = Q^V- ^ direct calculation shows that T is a superminimal torus with A = ^J\ and non vanishing normal curvature (see [ 11 ]). In particular, from (4) we have that T is pseudo-parallel with 4> = — f • Theorem 2.6 Let f : M2 —> Q5(c) be a pseudo-parallel immersion of a connected, complete surface, with <j> non positive constant and c > . Then we have one of the following possibilities: 1. f is a totally umbilic immersion; 2. f is an immersion with R1 = 0 and K = (j>; 3. f{M2) is a Veronese surface in some totally umbilic S4(c) C Q5(c); 4- f{M2) C S5(c) is congruent to the torus of the preceding example. Proof: If the normal curvature is identically zero, then / is totally umbilic or K = (j> at the non umbilic points, by remark 2.1. But 4> constant implies K constant on all M. Suppose now there is a point x 6 M such that R^-{x) ^ 0. Let C be the connected component of the set of points where the normal curvature does not vanish, which contains x.

201

Let {ex, e 2 } be a local smooth orthonormal tangent frame in C. If H = 0, t h e n K is constant in x. Now, if H ^ 0, we want to show t h a t t h e function (p = 3K — 2 — c is constant. Taking a possible smaller open set we can assume

c/>, set k = (K — (f))1 and define: =

H_

_ (an

y/^'

- Qt-Ti)

=

Q=i2

fc

2fc

Let {U>A} be t h e dual frame of { e ^ } with respective connection forms {CJAB} , where A, S = 1,...,5. As usual capital Latin indices will run from 1 t o 5, small Latin indices from 1 to 2 and Greek indices from 3 to 5. Since w\ = 0 2

along M we have Wi\ = ^2 {atij,e\)wj,

and therefore, by our choise of t h e

e^'s we have: Wl3 = v ^ l '

W

14 = ^ 1 , (6)

^23 = V ^ Z , W24 =

-ku>2,

(see (5)). T h e structure equations give: 5 B=l da>i2 = — Ku>\ Aw2, 5 d^A\ = X) W ^ B AWBAB=l

(7)

Now set: dip = aWi + 6^2,

WAB = 0,ABU\ + bAB^2-

(8)

We want to show t h a t a = b = 0. For this we compute du)i\ using (6) and (7) as follows: dwi3 = d ( v ^ u / i ) = ( v ^ a i 2 C?Wi3 = Yl

W

_

27p)

w

i

A w

2,

(9)

1S A WB3 = ( y ^ a ^ - fc&34 + ^ 3 5 ) &! A a»2,

dwi4 = d (fewx) = (/cai2 - ^ ) wi A w 2 , d^!4 = XI W 1B A U>B4 = (~fc0l2 + -v/^34 + ^ 4 5 ) ^ 1 A a>2,

(

'

B

dui5 = d (kuj2) = (kbi2 + f ) w i A w 2 , dwi5 = £ O > I B A w B 5 = (—fc6i2 + -v/^35 + ^ 4 5 ) wi A w 2 ,

(n)

202

d^23 = d (y/ipu>2) = ( V ^ 1 2 + 2~^)

Wl A W 2 ,

U

du)23 = E 2 B A u>B3 = (^pb12 - ka34 - fc&35) wi A w2, dW24 = d ( - f c w 2 ) = (-fc&i2 - ^ ) w l A w 2 , C?W24 = ] £ W 2 B A W B 4 = (fcfri2 - v ^ 0 3 4 ~ ^ 4 5 ) Wl A 0»2,

dw25 = d (kui) = (ka12 - ^ ) wx A w2^ 2 5 = X ) W 2 B A U>B5 = (~kai2

- ^ a

+ /CCI45) W\ A UJ2-

3 5

(12)

(13)

(14)

Prom (9), ...,(14), we get: 035

& r ^34

•

-

2 a i 2 — Gt45

-

—

034 + ^35 -

* 2k^p>

fc~^34

= 6fe7;

-26l2 + #&35 + bi5

a

•\/v

26 12

-2fc7p>

034 — ^45 = — fifr, a

2 a i 2 + fc

6 '6fc2'

3 5 - «45

Algebraic manipulations of the above give: w

34 = 4 ^ (-awi + bw2), 35 = -ir^(bu>i+auj2), W45 = 2wi2 + j§p {-bwi + auj2) UJ

}

(15)

Differentiating ^34,^35, W45 and using the expression in (15), we obtain: du34 = j - ( y 3 +^ j a6wi A u 2 + wiAdo

(16)

- awi2 A w2 + d& A u>2 - bivu A Wi} ( 4 ^ ) , rfw

35 = { ( i f p f ) (a 2 - &2) w i A w 2 - * A wi

(17)

-6W12 Aw 2 -da/\u>2 +awi2 Awi} ( | A = | da;45 = ( - 2 X 36fc< (62 + a 2 ))wi Aw 2 + j2p- (-d& A w i - bw\2 Aw2 + da Au)2~ awn A wi).

(18)

The structure equations, together with (6) and (15) give: ^ 3 4 = (3fc7v) (2bUJl A W l 2

+ 2
2

A W

12 + W

w

l

A w

2) ,

(19)

203

^ 3 5 = \TiQ^) ( ~ 2 a w l

A

^12 + 2few2 A U)12 +

i2fc"

^1

A w

2 ),

du45 = - (2fc2 + ( i s p ^ ) (62 + a 2 )) wi A u 2 .

(20)

(21)

Comparing the expression (16) = (19), (17) = (20) and (18) = (21), we get: u>\ A da + db A W2 = ( g^

) a&wi A w2 + &wi A W12

(22)

+ ow2 AW12, u>\ A db — da A 0J2 — —auii A u>\2 + bu>2 A ui\2

WlAdfc

+daA

W 2

(23)

= ( ^ + (^f!)(a2+62))a;1AW2

(24)

+ bui\2 A u)2 — awi A LO12. Combining (22), (23), (24) with 0 = d (d^p) = —ui\ Ada + db Au>2 + aw\2 AW2 + bw\ A W12, we obtain: da = bivi2 ..

I2k2 ,

, f7ifi+6k2\

db = - a w 1 2 + [-^^)

(55y>+21fc 2 )a 2 -(l5yH-39fc 2 )fe 2

u

, ( l2k2S

,

-1 + ( W £ ) «**.

(55 V +21fc 2 )6 2 -(l5 V +39fe 2 )a 2 \

abu! + ^ - g - 2 + *

^ofe^

—J ^

Differentianting again the last two equations we get: 0=a

K

-

3(l0ip+23k2) (897k2-215V) 25«, + 2400fc^ 2

, , (a

,o\ +b)

(25) 3(l0ip+23k2)

(897fc 2 -215y) /

2

2\

js

As i f — 0, <> / < 0 and < c, (25) implies a = 6 = 0, i.e.

204

References 1. J. Deprez, Semi-parallel surfaces in Euclidean space, J. Geom. 25, 192200 (1985). 2. F. Mercuri, Parallel and semi-parallel immersions into space forms, Riv. Mat. Univ. Parma, IV. Ser. 17, 91-108 (1991). 3. A. C. Asperti and F. Mercuri, Semi-parallel immersions into space forms. Bolletino U. M. I. (7) 8-B, 883-895 (1994). 4. A. C. Asperti, G. A. Lobos and F. Mercuri, Pseudo-parallel immersions in space forms. Mat. Cont. 17, 59-70 (1999). 5. A. C. Asperti, G. A. Lobos and F. Mercuri, Pseudo-parallel submanifolds of a space form. Submitted. 6. B. O'Neill, Isotropic and Kaehler immersions. Can. J. Math. 17, 907-915 (1965). 7. R. L. Bryant, Conformal and minimal immersions of compact surfaces into the 4-sphere, J. Diff. Geom. 17, 455-473 (1982). 8. K. Kenmotsu, Minimal surfaces with constant curvature in 4-dimensional space forms, it Proc. Amer. Math. Soc. 89 No.l, 133-138 (1983). 9. S. S. Chern, On minimal spheres in the four-sphere. Studies and Essays, Math. Res. Center, Nat. Taiwan Univ., Taipei 137-150 (1970). 10. M. Barros and B.Y. Chen, Stationary 2-type surfaces in hypersphere. J. Math. Soc. Japan, 39, 627-648 (1987). 11. K. Sakamoto, Constant isotropic surfaces in 5-dimensional space forms. Geom. Dedicata, 29, 293-306 (1989).

ROTATIONAL TCHEBYCHEV SURFACES OF § 3 (1) TSASA LUSALA Technische Universitat Berlin, Institut fur Mathematik Sekr. MA 8-3, Strafie des 17. Juni 136 D-10623 Berlin, Germany E-mail: [email protected] We prove that a rotational Tchebychev surface of S 3 (l) with distinct principal curvatures Ai and A2 satisfying the condition -r-J = const, is rotational with an arc of an ellipse or an arc of a hyperbola as profile curve.

1

Introduction

Let x: M2 —> § 3 (1) C K4 be a non-degenerate surface immersion, i.e. the corresponding shape operator S has maximal rank. It is known that the three fundamental forms I, I and H are (semi-)Riemannian metrics on M2. We denote by V 1 , V 1 and Vm their Levi-Civita connections, respectively. The symmetric difference-tensor field C := ^ V 1 - V m ) has traceless part C given by C(u, v) := C(u, v)~-

(l(u, T)v + I(v, T)u + I(u, « ) T ) ,

where T is the so called Tchebychev vector field defined by n(u,T):=ltr(«.—>C(u,w)). The non-degenerate immersion x is called Tchebychev surface or is of Tchebychev type if and only if the covariant derivative V^C of the tensor field C with respect to V n is totally symmetric 6 , i.e. (V 1 , C) is a Codazzi pair according to ([ 3 ], [4]). Consider the following non-degenerate surface: x(u,v) = r(u) • (cos(u)Ci + sin(u)C 2 ) +acos£(u)C3

+ bsm£(u)C4,

(1.1)

4

where C\, C2, C3 and C4 are constant orthonormal vectors inR ,£ G {—1,1}, f cos if £ = 1 , . f sin if e = 1 cose := < , .c and sine '•= \ • , , n c [cosh if £ = —1 [sinn it e = — 1, 0 < a < l , 0 < 6 a r e two real constants such that b ^ 1 if e — 1 and 7 c E is a non-empty open interval such that the function r defined by r u

()

=

V1~

fl2 c o s

e(u)

205

—

b2 sin^(u)

206

is real and positive on I. Note that non-degenerate isoparametric surfaces of S 3 (1) with two distinct principal curvatures, i.e. parts of the Clifford tori S ^ n ) x S 1 ^ ) (0 < r i , r 2 and r i + r i — I)- correspond to the surfaces of revolution as in (1.1) with e = 1 and 0
Ai =

ab and

A

A2 =

a {u)

T—r,

a(u)

where
Ai

= -

2 ( 1 - a?){e - b ) 2 2 ab

const.

Moreover A2 - A i

..... eabr2{u)

i.e. there are no umbilical points on the surface x. Conversely, a rotational surface with nowhere-zero principal curvatures Ai and A2 satisfying the condition yj = const, is of (C = 0)-type. For non-rotational immersions of (C = 0)-type in S 3 (1), the two functions ^ and 4§- are not constant. They had been used in [7] as parameters to classify such surfaces. In this note we consider the situation when one of these functions is constant and the immersion is one of Tchebychev type, more precisely we prove the following result: Theorem 1.1. Let x: M2 —> S 3 (l) be a non-degenerate surface immersion of a connected and orientable C°°-manifold M2 into S 3 (l) without umbilical points. Assume that the corresponding principal curvatures \\ and A2 satisfy 4j = const. A

2

Then the immersion x is of Tchebychev-type if and only if there exist local coordinates (u, v) such that x locally can be described by the surface of revolution (1.1). 2

Integrability conditions

In this note we consider a non-degenerate surface immersion x: M2 —> S 3 (l) without umbilical points. Choose a I-orthonormal frame (ei, e 2 ) of principal vector

207

fields: I(ei,ej) = Sij

and

Se^ = A^e,,

i , j = l,2. 2

There exist two differentiable functions a,(3 e C°°(M ) such that V ^ e ! = ae 2

and

V ' 2 e 2 = /?ei.

Note that the I-orthonormal frame {e\, e 2 ) of principal vector fields is unique up to signs. Therefore the two functions a and p are uniquely determined. The GauB formula and the Codazzi equation (integrability conditions for immersions in unit Euclidean spheres, see details for example in [2]) i?1(u,t;)'u; = l(w,v)u — l(w,u)v + H.(w,v)Su — K(w,u)Sv, (ylus)v = (yls)u, where R1 is the Riemannian curvature tensor of the first fundamental form, read with respect to the frame (ei, e2) e1(/3) + e 2 (a) = a 2 + / J 2 + l + AiA2; ei(A 2 )=)9(A2-A 1 ), e2(A1) = a ( A 1 - A 2 ) .

(2.2) (2.3) (2.4)

From the integrability conditions (2.3) and (2.4), and the assumption & = const, we get the following equations: ei(Ai) =

3/?A 1 (A 2 -A 1 ) ^r A2

and

e2(A2) =

QA 2 (AI

^—

- A2) '-.

6\\

According to [*], [ 5 ], the Levi-Civita connection Vm of the third fundamental form is given by V*v = S-1(VluSv)

for all

u,v&TM2.

Thus the difference-tensor field C satisfies C(u,v) = ~S-1((V1uS)vy

(2.5)

Using (2.5), we get the following components for the difference tensor field C with respect to the frame (ei, e 2 ): 3/3(A2 - AQ 2A2 q(Ai - A2) °i2 — 2Ai ! /3(A 2 -AQ : a22 ~ 2A,

tfi = -

r2 c 12

rl — °22 ~

q(Ai - A2)

2A2 ' |9(A 2 -Ai) 2A2 ' q(Ax - A2) 6Ai

208

where C(e,, ej) = C^ek, for i,j, k e {1,2}. Moreover, one has:

Ti =n(ei)r) = \{c\1 + cl2) = T2 = K e 2 ; T ) = i ( C 2 2 + C 2 2 ) =

-

-fcM, ^

^

-

So the Tchebychev vector field T is given by l T T„ + lT ff(*2-Ai) T = — Tiei + —PT2e2 = r—:

A\

A2

e\

AiA2

n

Q(A: - A2) rr-r e2, 0A1A2

1

and the Levi-Civita connection V = V — C of the second fundamental form is given with respect to the frame (ei, e 2 ) by: „ ^61 = „n V 62 = n V Cl = 7 i „e e2 2 = V

3

3/?(A 2 -Ai) a(A 1 + A 2 )^ 6l + 62 2X2 2A2 ' a(A! + A2) , ^(A.-AQ 2A^ 6l + ^ A ^ 6 2 ' a(Ai-A2) /?(Ai + A2) Cl 2A! §A2~e2' /?(Ai + A2) a(\i-\a). -L-^ -ei -e 2 . Cl + - ^ 2Ai 6A2

Proof of Theorem 1.1

First define the following H-selfadjoint operator on M 2 : Lu := Su - 5 - x u - V^T

for all

u e TM2.

The non-degenerate surface immersion 1: M 2 —> § 3 (1) is of Tchebychev type if and only if the above operator L satisfies Lu = fx-u,

(3.6)

for some function // e C°°(M2) (see [6] for details). It follows from (3.6) that, with respect to the frame (ei, e2), the Tchebychev type is equivalent to the following equations: I(Lei,ei) = l(Le2,e2)

and

I(Lei,e 2 ) = 0 = I(Le 2 ,ei).

209

In order words we have Al

_ Ar 1 - I ( V ^ T , e i ) = A2 - A2-J - I(V* 2 T,e 2 ) I ( V j i r , e 2 ) = 0 = I(Vj i l T,e 1 ).

From the equations above, using the expressions for the Tchebychev vector field T, the Levi-Civita connection of the second fundamental form, and the fact that there are no umbilical points, we get ap{\i + 4A2) - A2ei (a) = 0,

(3.7)

a/ta 2 + 3Aie2(/?) = 0,

(3.8)

9Aiei(/J) + 3Aie 2 (a) + a 2 (Ai + 4A4) - 9Aj/3 - 9AiA2 = 0. The equations (3.7) and (3.8) give (4A 2 +Ai)a/3

ei(a) =

and

A2

(3.9)

X2aP e2(/?) = ———; 0A1

the derivatives e 2 (a) and e\(/3) are solutions of the linear system of equations (2.2) and (3.9) (with unknowns e 2 (a) and e\{(3))\ . . ea(a) =

(2A2 + 5A 1 )o 2 — ^ — '

ei(/?) =

-2{Xl + X2)a2 + ZXx{P2 + l + XlX2) :—3Ai •

If the function a vanishes identically on M2, then the immersion x is rotational and satisfies ex(Ai) = 3 ^ A l y 2 ~ A l ^; one uses a result from [7] to conclude that there are coordinates (u,v) so that the immersion x locally can be described by the parametrisation of the surface of revolution (1.1). Thus Theorem 1.1 will be completely proved if we prove that the function a vanishes identically. Suppose that there is a point p € M2 such that a(p) ^ 0. There is an open neigbourhood U C M2 ofp such that a ^ 0 everywhere on U. Using the expressions of the derivatives ei(a), e2(Q;), ei(/3) and e2(/J) above, from e i M / 3 ) ) - e 2 ( ei (/?)) - (V^ea - V ^ e O ^ ) = 0, one deduces that the following holds on U: 2 a

~

3Ai(3Ai-A2-A?A2+3AiA|) 2(10A1A2 + 7A? + 7Ai)

Calculating e 2 (a 2 ) and using e2(o;) = ^2 2 ^ " " , we have that Ai and A2 satisfy a polynomial equation with constant coefficients on U; therefore the immersion is isoparametric and then a = 0 on U, a contradiction.

210

Acknowledgments I would like to thank Professor Udo Simon of the Technische Universitat Berlin for fruitful discussions. This work is partially supported by the DFG Si-163/7-2. References 1. F. Brito, H.L. Liu, V. Oliker, U. Simon, C.P. Wang: Polar hypersurfaces in spheres, Geometry and Topology of Submanifolds, IX, F. Defever et al. (eds.) (1999), World Scientific, Singapore, 33 - 47. 2. M. P. do Carmo: Riemannian Geometry, Birkhauser Boston (1992). 3. H.L. Liu, U. Simon, C.P. Wang: Higher order Codazzi tensors on conformally flat spaces, Beitr. Algebra Georn. 39 (1998), 329 - 348. 4. H.L. Liu, U. Simon, C.P. Wang: Codazzi tensors and the topology of surfaces, Ann. Global Anal. Geom. 16 (1998), 189 - 202. 5. K. Nomizu, U. Simon: Notes on conjugate connections, Geometry and Topology of Submanifolds, IV, F. Dillen et al. (eds.), World Scientific, Singapore (1992), 153 - 173. 6. T. Lusala: Tchebychev hypersurfaces of S n + 1 ( l ) , Geometry and Topology of Submanifolds, X, W. Chen et al. (eds.), World Scientific, Singapore (2000), 154 - 170. 7. T. Lusala: Cubic form geometry for surfaces in S 3 (l), Preprint Reihe Mathematik No. 676 (2000), TU Berlin, Fachbereich Mathematik, in Beitr. Alg. Geom. (2002) to appear. 2000 Mathematics Subject Classification: 53 B 25. Keywords: Tchebychev surfaces, surfaces of revolution of S 3 (l).

ON HOLOMORPHICALLY PROJECTIVE MAPPINGS R I E M A N N I A N ALMOST-PRODUCT SPACES

J. MIKES Department of Algebra and Geometry, Palacky University Tomkova 40, Olomouc, Czech Republic E-mail: [email protected]

ONTO

Olomouc,

O. POKORNA Department of Mathematics, Czech University of Agriculture, Kamyckd 129, Praha 6, Czech Republic E-mail: PokornaQtf.czu.cz We state the theorem which specifies fundamental equations of holomorphically projective mappings of a space with an affine connection and with an almostproduct structure onto a Riemannian almost-product spaces.

1

Introduction

Several results concerning holomorphically projective mappings (HPM) and its generalizations were described in 1 . 4 . 7 40,i2,i3,i4,i5 Q n e 0 f ^ n e topics studied here is H P M of special Riemannian spaces with almost complex and almost product structures, especially of Kahlerian spaces. Some properties of Riemannian almost-product spaces were described for example in 2 , n . In the present work we study some properties of holomorphically projective mappings from spaces with affine connection and with almost-product s t r u c t u r e onto Riemannian almost-product spaces. First we give t h e definitions of a Riemannian almost-product structure. D e f i n i t i o n 1. An almost-product structure on a differentiable manifold M is an affinor F ( ^ i d ) which satisfies F2 = id. D e f i n i t i o n 2. A Riemannian almost-product structure an almost-product structure F on M such t h a t g(FX,FY)=g(X,Y),

on a manifold M is

(1)

where g is a metric tensor field on M and X, Y are any tangent vector fields of TM. A manifold M with a Riemannian almost-product structure F is called a Riemannian almost-product space and denoted by Vn(g,F). T h e classification of Riemannian almost-product spaces is described by

211

212

A.M. Naveira in n . This is the analogue of the classification of almost Hermitian spaces by A. Gray and L. Hervella in 3 . Definition 3. Let An be a manifold with an affine connection V and an almost-product structure F . A diffeomorphism / from An onto a Riemannian almost-product space Vn(g,F) is called a holomorphically projective mapping if there exist a linear operator ip such that the following conditions hold V(X, Y) = V(X, Y) + X^(Y)

+ Yil>(X) + FX^(FY)

+ FY^{FX),

F(X) = F(X),

(2) (3)

where V and V are affine connections of An and Vn, X, Y are any tangent vector fields from TAn and their images in TVn. The condition (3) means that the holomorphically projective mapping preserves an almost-product structure. Hence, in the following we suppose F = F. It is known that holomorphically projective mappings preserve F-planar curves. Here a F-planar curve is a curve 7(t) such that any tangent vector 77? (£i), when subjected to a parallel transport r t l ) t 2 , remains in the tangent plane spanned by the vectors -^(£2) and F{-^{t2))Hence, this type of holomorphically projective mappings could be denned by a more natural way, as mappings which preserve F-planar curves and satisfie the condition VXF{Y) = X7XF(Y), for all X,Y. It follows from the equations (2) that holomorphically projective mappings are analogous to F\- and F2-mappings studied in 7 ' 5 . 2

Study of fundamental equations of holomorphically projective mappings of An(\7,F) onto Riemannian almost-product spaces

Let An be a manifold with an affine connection V and almost-product structure F . We shall study the problem of finding all Riemannian almost-product spaces Vn(g,F) such that there exists a holomorphically projective mapping of An onto Vn. Originally, similar problems were solved for geodesic mappings of Riemannian spaces, holomorphically projective mappings of Kahlerian spaces, hyperbolic and parabolic Kahlerian spaces 6>7>12>13 and F-planar mappings onto Riemannian spaces 5 ' 8 . In the following we use the local tensor notation which is traditionaly used in this field.

213

By a direct calculation we check that the equation (2) is equivalent to 9ij,k = 2ipk9ij + ip{i9j)k + 2i>k9ij + 4>(i9j)k >

(4)

where g^, F^, ipi are local components of the metric tensor g, the almostproduct structure F and the operator ip. We shall use the notation "," for the covariant derivative on An, and (ij) denotes a symmetrization of indices. We introduce the following notation: T,,,-i... = T . . . a . . . F f ,

=T-a--Fi.

T-*-

Let A„(V, F) be a manifold with an affine connection V and an almostproduct structure F and let g be a bilinear form with the components g^. Then the solution g^ of the equations (4) in the space An(V, F) which fulfills the conditions (a)

gjj = g~ij ,

(b)

\gij\ ^ 0 ,

(5)

determines the metric of a Riemannian almost-product space Vn(g, F). We remark that An(S7,F) admits a holomorphically mapping onto Vn(g,F). The equations (4) are the equations with unknown functions g^ and ipi. Because holomorphically projective mappings are a special case of F-planar mappings, the equations (4) can be reduced to a Cauchy system of differential equations 8 . In the following main theorem we show that there is a further simplification for some type of almost-product structures. Theorem. Let An be a manifold with an affine connection V and an almostproduct structure F satisfying F)M ± ±F«kFia ,

(6)

and Fj k cannot be expressed in the form

Fj,k = a%k ,

(7)

for any tensors a1 and bjk • Then An admits a holomorphically projective mapping onto a Riemannian almost-product space Vn if and only if the system of equations (4) of Cauchy type is solvable with respect to the unknown functions g^, where & = 9a0T20g7i

.

(8)

Here T^0 (see (21)) are the components of a tensor T which can be expressed in terms of the affine connection V and the almost-product structure F, \\gt:i\\ =

WSaW-1-

214

Proof. We apply covariant differentiation to the formula giaga^ = 5?: aj 9ia,k9 + 9ia9aj,k = 0. We get g" ,fc = -gap,kgaig0:>'• Applying (4) we check that

t\k = -2Vfe5ij - V ( i # " 2 ^ - i>H], l

(9)

ai

where ip = tpag . The conditions (5a) are equivalent to giaFl = g>aFi .

(10)

By covariant differentiation of (10) in An and applying (9) we have 9iaFJa,k=9JaFU-

(11)

By further differentiation and simplification we obtain

vn,i+stt>a*ij - vn,i - sim.i - ^Fl - *ii>aFii+ rKi+6&aFa,i=9iaFU-9ian,ik

(12)

We used an evident formula Khfc — -F^k which is true for almost-product structures. Contracting equations (12) for the indices i and k we obtain

nrn^ -mrfi,i=sa0Fi,i0 - 9i0Flh.

(1 3 )

where m = Fg. For an almost-product structure, the integer m belongs to the interval [—n + 2, n — 2]. Contracting (13) with Fj 8 and changing the index we get

a0F rFi m rn.1=9 Lw - ^ n ^ • n tl

w

Because m ^ n, (13) and (14) yield

rFjaJL = rp^v

( 15 )

wh e

- Kn - ^ ? v u - #*n*)+y^^Flw - sin*,) • By the aid of (15) we get from (12)

vH,i - vH,i - *lH,i + ^Fh = sa0 Tim, where

Hm - -** Ki+5i Kn+^ %* - si K9i+^u

(i6)

- m,ik •

Under the condition (6) we check easily, that there exist vectors e* and rf such that the vector a1 = Fl lskr^ is not collinear with the vector a1. Hence,

215

there exists £j such that ^a1 = 1, ^al = 0. Contracting (16) with r]kjf we obtain V>V - V j a i - V'a' + ^a1 Contracting (17) with ^j

= ga0 T ^ e V

•

(17)

we check that

and contracting (17) with £j we find

^^ea' where

3

(18)

+ g^T','a/3

T ^ = ( T ^ - £a* T ^ W e V f c -

*=

^ a

Applying (18) to (16) we have 4,

eia'Ftj

- aPF^ - JF^

+ a>X«) = 5 a / S I ^ u '

(19)

T where Tafjkl = X ' a m ~ TamH,& LfiklF^l + Ta0klFk,l~ Kcphl^lThe bracket on the left-hand side of (19) must be nonvanishing, otherwise there would be F*k = cfbjk, which is in contradiction with (7). So there exists a tensor field Qkl, satisfying

oFMKi - aJFh - "Hi + aiFh) = i4• •

Hence from (19) it follows that e = ga0 1*^klQfj

and further

^=ga0Tla0,

(20)

u^a^aiT%km-

(2i)

where

T

Because rp1 = giaipa, w e have ^ formula (8). This finishes the proof.

= i/>7<77i = g^T^g^

which is the

Acknowledgments Supported by the grant No. 201/02/0616 of The Grant Agency of Czech Republic.

216

References 1. D.V. Beklemishev, in Geometria. Itogi Nauki i Tekhn., Ail-Union Inst, for Sci. and Techn. Information (VINITI), Akad. Nauk SSSR, Moscow, 165-212, (1965). 2. O. Gil-Medrano and A.M. Naveira, in Rev. Roum. Math. Pures Appl. 30, 647-658 (1985). 3. A. Gray and L.M. Hervella, in Ann. Mat. Pura Appl., IV. 123, 35-58 (1980). 4. I.N. Kurbatova, in Ukr. Geom. 56., Kharkov 27, 75-82 (1984). 5. J. Mikes, in Vestn. Most Univ. 3, 18-24 (1994). 6. J. Mikes, in J. Math. Sci., New York, 18 3, 311-333 (1996). 7. J. Mikes, in J. Math. Sci, New York, 89 3, 1334-1353 (1998). 8. J. Mikes, V. Malickova and O. Pokorna, in Steps in Differential Geometry, Proceedings of the Colloquium of Differential Geometry. 25-30 July, 2000, Debrecen, Hungary, 209-217. 9. J. Mikes and N.S. Sinyukov, in Izv. Vyssh. Uchebn. Zaved., Mat. 1(248), 55-61 (1983). 10. J. Mikes and G.A. Starko, in Proc. of Winter School Geometry and Physics, Srni, January 1996, Rend, del Circ. Mat. di Palermo, Ser. II, 46, 123-127 (1997). 11. A.M. Naveira, in Rend. Math. Appl. VII. Ser. 3, 577-592 (1983). 12. N.S. Sinyukov, Geodesic mappings of Riemannian spaces. Moscow: Nauka, 1979, 256p. 13. N.S. Sinyukov, in J. Sov. Math. 25, 1235-1249 (1984). 14. N.S. Sinyukov, I.N. Kurbatova, J. Mikes, Holomorphically projective mappings of Kdhlerian spaces. Odessa State University, 1975, 69p. 15. K. Yano, Differential geometry on complex and almost complex spaces. Oxford-London-New York-Paris-Frankfurt: Pergamon Press. XII, 1965, 323p.

A C H A R A C T E R I S T I C P R O P E R T Y OF T H E C A T E N O I D

PABLO MIRA Departamento

de Matemdtica Aplicada y Estadistica, Universidad Cartagena, E-30203 Cartagena, Murcia, Spain E-mail: [email protected]

Politecnica

de

This paper studies rotation surfaces of R n , that is, surfaces which are invariant under the 1-parameter group of rigid motions of R n that fix a certain affine (n — 2)plane. It is shown that if the first normal space of a rotation surface that does not meet its base space has (not necessarily constant) dimension < 1 at every point, then the surface lies in an affine 3-dimensional space of R n . Along with this result, we classify the minimal rotation surfaces of R™ and study the case of rotation surfaces with parallel mean curvature vector.

1

Presentation

One of the most famous results in differential geometry characterizes the catenoid as the only non-planar minimal rotation surface in R 3 . This result was proved by Meusnier in the 19th century, and has been generalized in many ways. Besides, the interest of the geometry of minimal surfaces in R n is well known (see Hoffman and Osserman 3 and references therein). With this, the question of classifying the minimal rotation surfaces in R™ arises naturally. A surface \ '• M2 —• R™ is said to be a rotation surface if E = x(M) is invariant under the group of rigid motions of K™ leaving pointwise fixed a certain affine (n — 2)-plane P. In this case P will be called the base space of E. Note that there is a degenerate type of rotation surfaces, namely those which are contained in the base space. We will usually exclude this situation. The classification of the minimal rotation surfaces in R n tells what follows. Proposition 1.1 Any minimal rotation surface inW1 which is not contained in its base space must be an open piece of a plane or a catenoid. Besides, recall that if / : Mm —> Nn is an isometric immersion with second fundamental form a, the first normal space of / at p € M m is defined as Ni(p) = span{(Tp(X, Y) : X, Y G TpM}. Proposition 1.1 will be proved in Section 3 by means of the following theorem, that presents a codimension reduction result for rotation surfaces in R™. Theorem 1.2 Let E c R™ be a rotation surface that does not meet its base space. If at every point of E the first normal space has dimension < 1, then E lies in an affine 2,-space ofW1.

217

218

The proof of the Theorem, which will be presented in Section 2, is constructed in two steps. First, we find a specific isothermal parametrization for any rotation surface that does not meet its base space, and afterwards we show how the condition on the dimension of the first normal space turns into a restriction for the profile curve of the rotation surface. Let us remark that we are not assuming the first normal space to have constant dimension on E. This Theorem should be compared with the results on reduction of codimension appearing in Dajczer 1 and references therein. See also Dajczer and Tojeiro. 2 2

Proof of the Theorem

Let x • M2 —> E C R™ be a rotation surface in R" with base space P, and suppose that E n P is empty. Next we find a parametrization for E. Choose a hyperplane II containing P, with unit normal a. Then E n II is non-empty, what means that if we consider the smooth function / e C°°{M2) given by f(q) = (x{q),a), then 7 = / _ 1 ( c ) C M 2 is non-empty for a certain c € R. Moreover, we have the standard formula V/(g) = aT, where aT is the tangent part to M2 at q of the vector a. Along with this, it is easy to check that the surface meets II transversally at every point of T = xil) = E n II. This shows that 7 is a regular curve on M2. Hence T is a regular curve in II that does not meet P. Let us identify, composing with a rigid motion of R™ if necessary, the base plane P with the Xi,...,x n _2 coordinate plane, and the hyperplane II with the coordinate xn — 0 hyperplane. Then T is parametrized by T(s) = (ai(s), ...,a n _ 2 (s),fr(s),0), with b(s) / 0 for all s. We may also suppose that the condition | r ' ( S ) | 2 = a[(s)2 + ... + a'n_2(s)2 + b'(s)2 = b(s)2 holds for all s, simply by considering if necessary the alternative parameter u = u{s)=

I

\T'{r)/b(r)\dr.

Besides, let Gp be the set of all rigid motions of W1 leaving P pointwise fixed. It is easily shown that the matrix expression in canonical coordinates of any $ G GP is of the form $ = / „ _ 2 x At, being At the rotation in R 2 of angle t. Thus Gp = { $ t : t € K} is a 1-parameter group. Furthermore, we have S = GP(T) = {* t (p) : p G r and * t e GP},

219

what shows that E may be parametrized as x(s,t) = (ai(s),...,an-2(s),b(s)cost,b(s)smt).

(1)

The first fundamental form of this immersion is written as

i = | r ' ( s ) | W + b(s)2dt2 = b(s)2(ds2 + dt2), and hence s,t are isothermal parameters for E. Once here, a standard use of the Gauss formula V°XY = VxY (see Dajczer 1 for instance) gives
-b'2/b)cost,(b"

+ a(X, Y)

-b'2/b)smt),

<x(dtA) = {•.•,b,a'j/b,...,(b'2/b-b)cost,(b'2/b-b)smt),

(2)

a(ds,dt)=0. Since dimiVi < 1, there exist functions \(s,t),[i(s,t) such that \
without common zeroes

= 0-

(3)

If A = 0 at some point, it must hold a(dt, dt) = 0. That is, it must hold b = ±b' at this point. In particular, b' ^ 0 whenever A = 0. Besides, equations (2) and (3) yield Xa'j + {b'/b)(n - X)a'j =0 = K(s,t)a'!(s)

+ L(s,t)aj( s )

for all j 6 {1, ...,n — 2}. From the above comment we get that K(s,t), L(s,t) cannot vanish at the same time. Hence, if a is the projection of T on P, the vectors a"(s) and a'(s) are collinear for all s. Thus a is contained in a straight line, and S lies in an affine 3-space of Rn. This completes the proof. 3

Remarks

To begin with, let us prove Proposition 1.1. Consider a minimal rotation surface S in R" such that it is not contained in its base space P. Then U = E \ P is non-empty, and it is a minimal rotation surface that does not meet P. If we parametrize U as in (1), the minimality condition shows that a{d\, 9j) and cr(d2,d2) are collinear. From there, using that (7(81,82) = 0, we obtain that the first normal space of U has dimension < 1 at every point. Applying now Theorem 1.2 we get that U lies in an affine 3-space A3. Furthermore, A3 intersects P along a straight line £, and U is invariant under all rigid motions of W1 that fix P. Therefore U is invariant under all rigid motions of A3 that fix £. In this way, U must be an open piece of a plane or a catenoid. Since E

220

is a minimal surface, it is real analytic, and finally we get that E itself is an open piece of a plane or a catenoid. This finishes the proof of Proposition 1.1. It comes clear that Proposition 1.1 is essentially a result about reduction of codimension. Nevertheless, it is easy to convince oneself glancing at (1) that there are rotation surfaces lying fully in R n , i.e. that are not contained in any proper affine subspace of K n . The last part of this short note is devoted to clarify the role of minimality in this result. For this, we will study the case of rotation surfaces with parallel mean curvature vector. First of all, we note that if S C K™ is a rotation surface which does not meet its base space, and we parametrize it as in (1), the previous computations in equations (2) tell that the mean curvature vector is given by H

= ^ 2 K > - . a n - 2 . ( & " - b)cost,(b" - b)smt).

(4)

We say that H is parallel provided V ^ H = 0, where here V"1" is the normal connection of the immersion. Equivalently, H is parallel if V^-H e 3£(M) for all X € X(M). In this manner, H has constant length on S. Proposition 3.1 Any rotation surface in R™ with parallel mean curvature vector and which does not meet its base space is contained in an affine 4-space

ofW1. Proof. /.From (4) it is easily seen that Vg t H = Hi is collinear with xt, and that Vg s H = H 5 is orthogonal to x t . Hence H is parallel if and only if there exists a function X(s, t) such that H s = Xxs. Taking the first n —2 coordinates of this equality we get a';1 = 2{b'/b)a] + 2Xb2a'j for all j e {1,..., n - 2}. Thus, if a is the projection of T on P, we find that a'"{s) is a linear combination of a'(s) and a"(s) for all s. Therefore a is a plane curve and the rotation surface is contained in an affine 4-space of R™.

•

1

We finally note as a complement to this last result that the flat tori § (r) x S 1 (v / 1 — r2) C S 3 C R 4 are all rotation surfaces that lie fully in R 4 and have parallel mean curvature vector. Acknowledgements The author wishes to express his gratitude to Luis J. Alias for his guidance during the preparation of this work. He also wants to thank Ruy Tojeiro for valuable comments regarding some of the topics treated here.

221

This work was partially supported by DGICYT, MECD, Spain (Grant N° BFM2001-2871-C04-02) and by Fundacion Seneca, CARM, Spain (Grant N° PI-3/00854/FS/01). References 1. M. Dajczer, Submanifolds and isometric immersions. Based on the notes prepared by M. Antonucci, G. Oliveira, P. Lima-Filho and R. Tojeiro. Mathematics Lecture Series, 13. Publish or Perish, Inc., Houston, 1990. 2. M. Dajczer, R. Tojeiro, Submanifolds with nonparallel first normal bundle, Canadian Math. Bull. 37 (1994), 330-337. 3. D. Hoffman, R. Osserman, The geometry of the generalized Gauss map. Mem. Amer. Math. Soc. 28 (1980).

C O N V E X I T Y A N D SEMIUMBILICITY FOR SURFACES I N B,5 S. M. MORAES Departamento de Matemdtica, Universidade Federal de Viosa, 36571-000 Viosa - MG, Brasil e-mail: smoraes2000@hotmail. com M. C. ROMERO-FUSTER Departament de Geometria i Topologia, Universitat de Valencia, 4.6100 Burjassot (Valencia) Espanya e-mail: [email protected]

1

Introduction.

A surface in Mn is said to be locally convex provided it admits some locally support hyperplane at each point. Clearly, surfaces contained in a hypersphere are locally convex. Local convexity at every point of a surface M in M4 is equivalent to the global existence of two special tangent direction fields, as seen in [4]. These directions are called asymptotic and can be determined through the study of the contacts of the surface with hyperplanes. The curvature ellipse at a point p of a surface M C Mn, n > 3 is defined as the locus of all the end points of the curvature vectors of the normal sections along all the tangent directions to M at p. This ellipse lies in the normal subspace of M at p and is completely determined by the second fundamental form of the surface at it. Points at which this ellipse degenerates to a segment or a point are called semiumbilic or umbilic respectively. A surface for which the curvature ellipse is degenerated at every point is said to be totally semiumbilical. Totally semiumbilical surfaces substantially immersed in JR4 have two globally defined asymptotic fields and thus are locally convex. On the other hand, it has been shown in [14] that for a locally convex surface immersed in 4-space, the mutual orthogonality of asymptotic fields is equivalent to total semiumbilicity. Furthermore, it was also proven in [14] that total semiumbilicity is equivalent to umbilicity of the second fundamental form with respect to some globally defined normal field v on the surface. Now, results of B.Y. Chen Q1]) tell us that if such a field v is parallel, M must lie in a 3-sphere. On the other hand, surfaces contained in 3-spheres are seen to be totally semiumbilical ([14])Our aim here is to extend this study to the case of surfaces immersed in JR5. A special property of these is that all of them are (at least generically)

222

223

locally convex. This suggests that the requirement of local convexity, useful in the former case, is too mild for our purposes. To overcome this, we introduce a new concept: essential convexity. We see that this is equivalent to the existence of special asymptotic direction fields that we call essential. Essential asymptotic directions at non semiumbilic points lie on the plane determined by the curvature ellipse at this point. We characterize the total semiumbilicity in terms of ortho-gonality of the essential asymptotic directions. It is worth to observe that the concept of essential normal field was introduced in [10]. It was there shown that these fields give rise to all the non trivial principal configurations on any surface immersed in JR", n > 5. The integral curves of the essential asymptotic direction fields are the principal configurations associated the special essential normal called here binomials. We finally analyze the relation between hypersphericity and total semiumbilicity for surfaces in 5-space and see that, in contrast to the case of 4-space, hyperspherical surfaces do not need to be totally semiumbilical, unless they sa-tisfy some extra condition. We observe that the concepts and results introduced in this paper can be naturally extended to surfaces immersed in 2R™, n > 6. 2

Curvature ellipses and semiumbilic points.

Let M be a surface immersed in Mn, n = 4,5, and let V denote the Riemannian connection of Mn. Given vector fields, X and Y, locally defined along M, we can choose local extensions X and Y over Mn, and define the Riemannian connection on M as V ^ V = (V%Y) , that is, the tangent component of V ^ on M. If we denote by X(M) and Af(M) respectively the spaces of tangent and normal fields on M, the second fundamental form on M is defined as follows: a : X(M) x X(M) —> 7V(M)

(X,Y)

^Vx?-vxy,

This is a well defined bilinear symmetric map and induces, for each p G M and v G NPM, v ^ 0, a bilinear form on the tangent space TPM given by Hv(v,w) — (a(v,w),v). The corresponding quadratic form IIv{v) = Hu(v,v) = (a{v,v),v) is known as the second fundamental form in the direction v. Given p G M, consider the unit circle in TpM parametrized by the angle 9 G [0, 2IT\. Denote by 79 the curve obtained by intersecting M with the hyperplane at p composed by the direct sum of the normal subspace NPM

224

and the straight line in the tangent direction represented by 6. The curvature vector T]{0) of 70 in p lies in NPM. Varying 0 from 0 to 2n, this vector describes an ellipse in NPM, called the curvature ellipse of M at p (see [3] and [8]). We can take M locally as the image of an embedding / : fft2 —• M5. Let {x, y} be isothermal coordinates and {ei,e2, -.-,65} an orthonormal frame in a neighbourhood of a point p = /(0,0) € M, in such a way that {ei, e 2 } is the tangent frame determined by these coordinates and {63,64,65} is a normal frame. Then the second fundamental form of M at p is given by "«1

af{p) =

61

Cl"

a2 62 C2 _a3 63 C 3 .

where aj = af(e1,e1)-ei+2 = ^^{p)-ei+2,bi = af(e1,e2)-ei+2 = E^EG-F* (E£ik(P} ~ F 0 O > ) ) ' e*+2 = i 0 ( P ) • e*+2 a n d Ci = af(e2,e2) • ei+2 = I P ^ ( £ 2 0 ( P ) - USFgfgip) + J " & ( p ) ) • ei+2 = | g ( p ) • ei+2 for 2 2 2 i = 1,2,3, and ATpM, given by rj(9) = H + ^u[ cos26 + u2sin20, where H = | ( ( a i + ci) • e 3 + (02 + C2) • e4 + (a 3 + c3) • e 5 ), til = (ai - Ci) • e 3 + (a 2 - c2) • e 4 + (a 3 - c3) • e 5 and ii2 = h • e 3 + 62 • e4 + 63 • e5 (see [10]). A point p G M at which the curvature ellipse degenerates to a segment is said to be a semiumbilic. An inflection point is a semiumbilic point for which the curvature ellipse is a radial segment. Points at which the curvature ellipse reduces to a point are called umbilic. The semiumbilic points of generic surfaces in M4 form closed curves at which the inflection points are isolated (see [4] and [7]). On the other hand, generic surfaces in M5 only have isolated semiumbilic ([ 10 ]). Moreover, umbilic points may be avoided over generic surfaces in both M4 and 2R5. The affine span of the curvature ellipse at a point p is an affine subspace, Hp c NPM. We denote Ep the vector subspace of NPM parallel to Hp. For surfaces in 4-space, the subspace Hp — Ep coincides with the whole normal plane at a non semiumbilic point. If p is a semiumbilic point, Hp is an affine line in NPM. Lemma 1 The curvature ellipse r)(6) is non degenerate if and only ifu[ and u~2 are linearly independent vectors. Moreover, Ep is generated by u[ and u2. Proof: If u[ and u2 are linearly dependent, then there is A € Ft such that ai — Ci = Xbi for i = 1,2,3. But this implies that r](8) = | Y^=i ((ai +

225

Ci) + bip(8)) • ei+2, so n{8) is a segment. The converse follows easily from the expression of the curvature ellipse n{8) = H + | u } cos 28 + u~2 sin 28. Clearly if rj(9) is a segment then v\ and U2 must be linearly dependent vectors. • R e m a r k 1 p is an inflection point or an umbilic if and only if rank a / (p) < 1. 3

Binormal and Asymptotic directions.

The family of height functions on M associated to an immersion / : M2 —> Mn of a surface M in n-space, is defined as A(/) : M x Sn~x —y JR (p,v)

i—> (f(p),v)

=

fv(p).

The height function /„ has a singularity at p e M if and only if v is normal to M at p. This singularity is of Morse type (nondegenerate) for most normal directions. Nevertheless, it is impossible to avoid, even generically, all the normal directions for which the corresponding singularity is degenerate. A basic result is the following. L e m m a 2 Given a point p in a surface M immersed in lRn,n > 4, and a nontrivial vector v € NPM, the quadratic forms IIv(p) and Hess(fv(p)) are equivalent (in the sense that we may find coordinate systems in which they coincide). Proof: Given p e M, suppose that the local embedding / : 2R2 —» M5 is given in the Monge form at p and an orthonormal frame {ei, e-i, e3, e^, es} in a neighbourhood of p such that {ei, e^\ is a tangent frame and {e^, e^, e 5 } is a normal frame in this neighbourhood. Then, for any normal vector v e NPM, we can write v = vie^ + V2&A + ^365 and the height function in the direction v is given by fv{x,y) = Vifi{x,y) + v2f2{x,y) + v3f3(x,y). We then have that -driP) = E i = i aivi> 5i&(P) = E i = i biVi, and -g^(p) = £ \ Therefore Hess(fv(p))

=

E 1 = i Wi E j = i bM .Ei=iM>» E i

On the other hand the coefficients of the second fundamental form with respect to the normal direction v at p are given by ev(p) = -^tip) • v = E i = l aivi> fv(P) = -£di(P)-V And thus 1

UP)

= Ei=l Mi

E n—2 E n—2l Mi i i=

and

9v(p) = 0 ( p ) '« = E i = l °ivi-

\~\n—2 7 \r-*n—2

}_i=l

Hess(/ w (p)).D Wi.

226

In case that the Hessian determinant vanishes we have that Hess(fv(j>)) is a degenerate quadratic form and the normal direction v is said to be a binormal direction at p. Surfaces in M4 admit at most two of these directions at any point p ( 4 ). Yet at most points of a surface immersed in JR5, we may have an infinite number of such directions. In fact, we can consider at each point p 6 M, a linear map Ap : NpM —» Q(2), where Q(2) represents the three-dimensional vector space of quadratic forms in two variables, given by Ap(v) = IIv(p). If we denote by C the cone of degenerate forms in Q(2), we have that A~ 1(C) determines the binormal directions at p. A preliminary study from the viewpoint of contacts of the surface with tangent hyperplanes has been made in 6 , where we must warn that these directions were called degenerate, and the name of binormal was reserved for special ones at which the contact was of higher order. A point of M C M5 is said to be of type M» provided rank(Ap) = i, for i = 3,2,1,0. Semiumbilic points of ordinary type represent a particular case of type Mi. Inflection and umbilic points correspond to the type M\, whereas at the points of type Mo the curvature ellipse reduces to the origin ( 9 ). It was shown in 6 that a generic surface M C M5 satisfies that M = M3 U M2, where M3 is open and dense and M-i is an embedded closed curve. We observe that if p £ M3, A~ 1(C) is a cone in NpM, whereas when p e M 2 , A~l{C) may be either two planes with a common line, one plane, or just a line (according to the image of Ap cuts the cone C in two lines, one line or just the origin). Given a binormal direction b € NPM we have that Ker(Hess(fb(p))) ^ {0}. The nontrivial directions lying in this kernel shall be called asymptotic directions associated to b. For a surface in 4-space, these directions coincide with the conjugate directions introduced by Little in 3 and there are exactly two, one or none according the point is hyperbolic, parabolic or elliptic (see 4 and 5 ) . Following the definition of essential normal fields given in 10 we say that a binormal direction b at a non semiumbilic point p of a surface in 5space is called essential provided it lies in the plane Ep. So, essential binormal directions at p are those lying in A~l{C) n Ep. It is not difficult to see that there are at most two essential binormals at each point p e M3. In fact, it can be shown that similarly to the case of surfaces in 4-space, there are exactly two, one or none according the point is hyperbolic, parabolic or elliptic. The corresponding asymptotic directions shall also be called essential. Moreover, for any p e M 3 the height function to has corank 1 for all v € NpM and thus given any binormal we have a unique asymptotic direction. On the other hand, given v G KerAp, the height function fv has corank 2 at p and hence all tangent directions are asymptotic at p. We say in this case that v is a degenerate binormal direction. We observe that if p is a semiumbilic or a

227

umbilic point of a surface M C M5, the subset A~X{C) C NPM may either consist of two planes with a common line (= KerAp), just a plane, a unique line, or the whole NpM. Proposition 1 If p e M is a semiumbilic or umbilic point then all the non degenerate binormals lying in the same linear subspace in A~ 1 (C) share the same asymptotic direction. Proof: We have that KerAp = A'1^) C A~l{C). If p is a semiumbilic or a umbilic point then rankAp < 2 and hence KerAp is non trivial. Assume first that rankAp = 2. Then ImAp n C may be either the origin, a line or 2 lines. So A~ 1 (C) is either the line KerAp, a plane containing this line, or a couple of planes with this common line. In the first case, there is a unique binormal direction and it is clearly degenerate. Suppose therefore that Apx{C) is either a plane, or two planes with the commom line KerAp = (vo)- Let vi be a binormal direction orthogonal to VQ. Any non degenerate binormal v may be written as v = Xv\ + fivo, where X,fi e Ft and A ^ 0 . Therefore Hess(/„(p)) = A Hess(/ Ul (p)) + / z Hess(/„ 0 (p)) = A Uess(fVl(p)). And it follows that u G Ker(Hess(fv(p))) if and only if u £ Ker(E.ess(fVl(p))). If rankAp < 1, we have that A~ 1 (C) is either the plane KerAp or the whole NpM and the same argument applies. • It thus follows that at each semiumbilic point of ordinary type (i.e., lying in M 2 ) there are at most two asymptotic directions. These asymptotic directions shall be called essential in this case. At an inflection point p we always have a plane of binormal directions at p all whose corresponding height functions have corank 2 (determined by A~ 1 (0) ) and hence all tangent directions are asymptotic at such points. Proposition 2 Given a point p of type M3 in M C M5, let np : M5 —> TpM@Ep = M4 be the orthogonal projection. Thennp\M is a local embedding atp that takes the essential asymptotic directions ofM atp into the asymptotic directions o/7r p (M) atirp{p). Proof: That ITP\M is a local embedding at p follows from the fact that Ker(irp(p)) is transversal to TPM. Now, given an essential binormal b € Ep, we have that fb{x) = /b(7rp(a:)),Va; in a neighbourhood of p in M, for b is orthogonal to the kernel of the projection irp. Moreover, ITP\M is a local embedding at p and we have that Ker{Hess{fb{p))) = Ker(Hess(fb(iTp(p))).

•

228

4

Convexity and asymptotic lines.

Locally convex surfaces immersed in JR4 admit globally denned fields of asymptotic lines ( 4 ). In fact, under some genericity conditions over the kind of contacts that the surface has with hyperplanes (which can be translated into some mild conditions on the coefficients of the second fundamental form), it was shown in 4 that the local convexity is equivalent to existence of two globally defined asymptotic direction fields whose critical points are the inflection points of the surface. If we increase the codimension by one we have: Proposition 3 Any surface M in 1R5 is locally convex at every point of type M3. Proof: To see this we prove that a sufficient condition for local convexity at p is that the image of Ap meets the inner part of the cone C (composed of elliptic quadratic forms), which is automatically true at the points of M3. In fact, suppose that v € NpM is such that Ap(v) is a nondegenrate elliptic form. In this case Hess(fv(p)) is a (positive or negative) definite nondegenerate quadratic form. But this implies that the hyperplane orthogonal to v passing through p is a local support hyperplane for M a t p . • The M3-points of a generic surface in 5-space fill an open and dense submanifold whose complement is a regular closed curve made of M2-points ( 6 ). Therefore, we can say that generic surfaces in M5 are locally convex nearly eve-rywhere, in opposition to surfaces in 4-space that may have open locally convex (hyperbolic) regions separated from non locally convex (elliptic) open regions by the parabolic curve. Moreover, in virtue of Thorn's Transversality Theorem ( 2 ), we have that the generic surfaces form a residual subset in the set of all the immersions of surfaces in M5 with the Whitney C°°-topology. Thus, the non generic surfaces can always be seen as "limits" of generic and hence locally convex surfaces. This tells us that, local convexity is a very mild requirement in this case. We propose here a stronger concept which, as we shall see below is related to the existence of essential asymptotic directions. We say that a surface M C M5 is essentially convex at a point p if there is some support hyperplane for M at p whose normal vector v belongs to the normal subspace Ep and such that fv defines a Morse (nondegenerate) function at p. Proposition 4 M C R5 is essentially convex at p if and only if TTP(M) is locally convex at p in the 4-space TPM © Ep. Proof: Notice that 7rp|M : M -> TpM @ Ep = M4 is a local embedding at p that takes Ep to Ep. The result follows easily. • Proposition 5 If M c M5 is an essentially convex surface then it admits two globally defined essential binomial directions at every point of M3 U M2 •

229

Proof: We observe that essential convexity at a non semiumbilic point p is equivalent to asking that the plane Ep meets the cone in two lines. If p is semiumbilic of type M 2 , this implies that the line AP(EP) lies in the inner part of the cone. Therefore, the plane ImAp must meet the cone in two lines, these give us the two binormal directions. • As we pointed out previously, under appropriate genericity conditions we have that M = M3 U M 2 , with isolated semiumbilics on M 2 . Therefore, generic essentially convex surfaces admit two globally defined essential binormal fields and consequently the two essential asymptotic fields associated to them. Moreover, the fact that 7TP|M takes essential asymptotic directions of M at p to conjugate directions of irp(M) in M4 = Ep © TpM, implies that these two fields cannot coincide. Their critical points can be seen to be semiumbilic points of type M 2 at which the binormal is degenerate (or also inflection or umbilic points in the non generic situations). These questions shall be studied with more detail in a forthcoming paper (9) for the general class of essential principal configurations on surfaces. Totally semiumbilical surfaces are non generic in the above sense. Nevertheless, it can be shown that there is an open and dense set of surfaces in the subspace of totally semiumbilical ones with the Whitney C°°-topology, for which all the points are ordinary semiumbilics (type M 2 ) except by isolated inflection and umbilic points. Then, under the assumption of essential convexity we also find two globally defined (essential) binormal fields with their corresponding asymptotic fields. We remark that it can be deduced from some reduction of codimension results in 13 that surfaces completely made of inflection points must lie in a 3-space. 5

Semiumbilicity and asymptotic lines.

Given a surface M C Mn,n > 4, and a normal field v on M, the shape operator associated to v is a linear map Sv : TPM —» TpM defined by SV(X) = — ( V ^ P ) T , where P is a local extension to JRn of the normal vector field v at p and T means the tangent component. This operator is bilinear, self-adjoint and for any v,w G TPM satisfies the following equation: (Sv(v),w) = Hv(v,w). So Sv is related to the second fundamental form as follows: II„(X) = (SU(X),X). Thus for each p e M, there exists an orthonormal basis of eigenvectors of Sv G TPM. The corresponding eigenvalues k\ and fc2 are the maximal and minimal v-principal curvatures, respectively. The point p is a v-umbilic if the ^-principal curvatures coincide. Let Uv be the set of z/-umbilics in M. For any p 6 M\UV there are two ^-principal directions defined by the eigenvectors of Su, these fields of directions are smooth and

230

integrable and its integral lines define two families of orthogonal curves which are called the v-principal lines of curvature. The two orthogonal foliations with the z/-umbilics as its singularities form the v-principal configuration of M. We say that the surface M is v-umbilical if each point of M is ^-umbilic. The differential equation of //-lines of curvature is S„(X(p)) = \(p)X(p)

(1)

Suppose that <j>, U C M is an open neigborhood with local coordinates (u,v). Let E, F, G be the coefficients of the first fundamental form in this coordinate chart. The coefficients of the second fundamental form are e„ = H„{du) = fu = {a{du,dv),v)

(a(du,du),iy), = {a(dv,du),i/),

gv = IIv{dv)

(a(dv,dv),v),

=

where du = -J^ and dv = ^ . Equation (1) has the following expression in this coordinate chart, (see and 1 2 ).

10

( / „ £ - evF)du2 + (guE - evG)dudv + {gvF - f„G)dv2 = 0. Assume that this coordinate chart is isothermal: E = G > 0, F = 0. Then this equation has the form fvdu2

+ (gu - ev)dudv - fudv2

= 0,

(2)

It was proven in 10 that if v is a normal field on M which is orthogonal to the subspace Ep at every point then M is ^-umbilical. Remark 2 If p € M is an umbilic point, we have that Ep reduces to a point and thus u[ = u~2 = 0. But this implies that aj = Q and 6j = 0 for i — 1,2,3. Therefore p is v-umbilic for any normal field u on M. We denote by U{M) the set of umbilic points of M. Proposition 6 Given M C M5, we have that M is totally semiumbilical if and only if there exist linearly independent normal fields vx and v2 locally defined for each p G M\ U(M) such that M is vi -umbilical for j = 1,2. Proof: First observe that if p is semiumbilic then there exist a neighbourhood, Up, of p such that dimEq > dim£^p for all q € Up. That is, dim£^9 = 1 for all q eUp. We consider the line bundle £ determined by Ep in Up. Clearly we can find two linearly independent normal fields vx and v2 locally defined at p and orthogonal to £. But then ev, — gv, and fvj = 0 for j = 1,2. So M is z/-3-umbilical for j = 1,2.

231

Now suppose that vl and v2 are linearly independent normal fields locally defined at some p e M \U{M) such that M is ^-umbilical, j = 1,2. In isothermal coordinates we have evj (p) = g„i(p) and fui(p) = 0 for j = 1,2, for all p e M. Taking a normal frame {e3,e$,e^}, we have i/J = J2i=i l'iei+2 and and thus '£^=iai'4 = YA=\CM S L i ^ i = °> f o r J - !> 2 - T n a t i s .

Ei=i (ai - ci)vl = °

and

Ei=i ^

= °> f o r J = i . 2 '

where a

i = i0(p)

•

et+2, h = -E~g^{p) • ei+2 and a = ^-gfi(p) • ei+2 for i = 1,2,3, and E is the coefficient non trivial of the first fundamental form in isothermal coordinates. Hence both, v1 and v2 are perpendicular to the generators u\ = Y^=i(ai ~ Ci) • ei+2 and u2 = 5Zi=i h • ei+2 of the subspace Ep. But this implies that dimE p < 1 and thus p is semiumbilic. Since this holds for any p e M\U(M) we can conclude that M is totally semiumbilical. D Proposition 7 Given M C 2R5, M admits two linearly independent normal fields i/1 and i/2 locally defined for each p e M such that M is i/i -umbilical, j — 1,2, if and only if M admits a unique (non trivial) principal configuration. Proof: Given p € M consider isothermal coordinates in a neighbourhood U of p and suppose that v1 and v2 are such that M is ^-umbilical. Consider a normal field £ such that {£{p), v1(p), v2{p)} defines an orthonormal frame on NU. Then given any normal field r\, we can write n = h£ + fav1 + k2v2, for appropriate smooth functions h, fa, k2 • U —* M. The coefficients of the second fundamental form in the direction of r\ are given by ev = (^A,h£ + favl + k2v2) = he^ + faevi + k2eu2, U = {-MiM + hv1 + fav2) = hfr + kiUi + k2fu2, 9ri = ( 0 > 9 ^ + ^ i i / 1 +h2v2) = hgi + kigui +k2g„2. And the equation of the curvature lines in this coordinates becomes (see 10 ) h(f(dx2 + (g£ - e{)dxdy - fedy2) + fa (fui dx2 + {gvi-e„i )dxdy - fui dy2) + k2(fi/2dx2 + (gu2 — ev2)dxdy — f^dy2} = 0. Since M is ^-umbilical, for j = 1,2, we have that evj(p) = guj{p) and fvi{p) = 0 for all p e M and thus fujdx2 + {gvi — evj)dxdy - fvidy2 = 0, j = 1,2. Therefore the principal configuration associated to n is given by h ( fedx2 + (g$ — e^jdxdy — fedy2 J = 0. So both fields n and £ have the same principal configurations. Conversely, given M C Ft5, we know that it is ^-umbilical, where v is a locally defined normal field orthogonal to Ep at every point (see 1 0 ). Take p € M and a neighbourhood Up on which v is defined. Let r\\ and r\2 be normal fields, orthogonal to v that are linearly independent on Up. Their respective principal configurations are given by the equations fVidx2 + (gVi — eVi)dxdy — fmdy2 = 0, for i — 1,2. Since M admits a unique principal

232

configuration, we must have that fm = XfV2 and gm — em = X(gV2 — e^ 2 ), for some function A on Up. Taking v = rji — Xr}2 we have that fD = fVl — XfV2 = 0 and gp—e0 = gm — em — X [gV2 —em) = 0. Therefore M is v- umbilical. Since v and v are linearly independent on Up the proof is done. • Theorem 1 Suppose that M is an essentially convex surface immersed in 5space with isolated umbilics. Then M is totally semiumbilical if and only if the two families of essential asymptotic lines are mutually orthogonal everywhere except at set of critical points (which contains the set U{M)). Proof: That M admits two globally defined essential binormal fields bi,i = 1,2 follows from the Proposition 5 above. Consider the corresponding asymptotic fields 9i,i = 1,2. We have that #j G ivTer(Hess(/(,i(p))) \ {0}, i = 1,2 and since this hessian coincides with shape operator S^ip), i = 1,2, it follows that &i is an eingenvector of S^ corresponding to a null eigenvalue. Hence the corresponding asymptotic lines are lines of curvature. On the other hand, if M is totally semiumbilical it follows from the Propositions 6 and 7 that the principal configurations associated to b\ and 62 coincide. Since the asymptotic directions 0\ and 62 cannot coincide we have that the asymptotic lines associated to the binormal 6j must be mutually orthogonal. Conversely, if M admits two families of orthogonal essential asymptotic fields, we have that these must be the lines of curvature with respect to some essential binormals 61 and 62. Therefore there is a real valued function k defined on M for which the following equalities hold: fbt = kfb2 and g^ —e^ — k(gb2-eb2)- SoO = fbl-kfb2 = \{bi-kb2)\fu andO = gb1-kgb2-(ebl-keb2) = \bi — kb2\{gL/ — e„), which implies that M is ^-umbilical. Since 61 and 62 are binormal we can assume that ef,, = gb = 0 and thus k = — ^-. So M is vCb2

umbilical for the field v = ^-61 + ^-62- Since 61 and 62 are essential binormals it follows that v(p) e Ep, for all p e M. On the other hand, we know that M is 7?-umbilical where 77 is a (locally defined) vector field orthogonal to Ep at every point ( 10 ). Therefore M is umbilical with respect the two linearly independent fields r] and v and the Proposition 6 tells us that M is totally semiumbilical. • 6

Hypersphericity and semiumbilicity.

B.Y. Chen proved in x that a submanifold M of Mn, n > 3 is hyperspherical if and only if there is some parallel normal field v on M such that M is vumbilical. (In fact, this field is the restriction to M of the radial field of the hypersphere). It follows from this together with the fact, proven in 14 , that surfaces in 4-space are totally semiumbilical if and only if they are v umbilical

233

for some non trivial normal field v, that Corollary 1 A surface M contained in S3 is totally serniumbilical. This result does not hold in the case of surfaces contained in the 4-sphere: The Veronese surface represents an embedding of the real projective plane in S 4 , V : P1{2) —> - 5 4 ( ^ ) C M5 (see 3 for a precise definition), satisfying that the curvature ellipse is a circle at each point of V. All the points are parabolic, and the radial field is everywhere perpendicular to the plane defined by the curvature ellipse. Therefore V does not contain semiumbilic points. Nevertheless, we have the following Theorem 2 A surface M in S4 is totally serniumbilical if and only if it is v-umbilical for some normal field v on M which is tangent to S4 at every point. Proof: I f M c S 4 and let rj be the radial field of S4 restricted to M. Then M is ^-umbilical. Since M is totally serniumbilical it follows from Theorem 1 that there must be some other normal field v, linearly independent of n, such that M is P-umbilical. The fields rj and v define a rank 2 subbundle of NM such that M is umbilical with respect to any of its sections. We can take v tangent to S4 and lying in this plane. Conversely, we know that M is 77-umbilical, 77 being the radial field to S 4 . On the other hand M is v-umbilical for some normal field v tangent a S4. Since 77 and v are linearly independent it follows from Theorem 1 that M is totally serniumbilical. D In the case of the Veronese surface we have that v coincides with the radial field and Theorem 2 does not apply. Remark 3 Given two curves 7$ : S 1 —> 1RN, i = 1,2, the translation surface associated to 71 and 72 is defined as the image of the map 'Yi 'yi

'

^

^

"^^^

(s,t) •— I(7l(s)+72(t)) It can be seen that for most pairs (71,72,) the corresponding surface is a torus immersed with isolated doble points if N = 4,5 or embedded if N > 6. In the particular case that the curves 71 and 72 are contained in orthogonal subspaces of Mn it can be shown that T 7l)72 is a totally serniumbilical torus. Moreover, 71 and 72 can be chosen in such a way that T 7l )T2 does not lie in a hypersphere of Mn (yx). Acknowledgements We would like to thank A. Montesinos Amilibia for helpful comments and useful programs to analize the geometry of surfaces in M4 and M5.

The first author has been partially supported by CAPES grant no. BEX 1191/99-3. The second author has been partially supported by DGCYT grant no. BFM2000-1110. References 1. B. Y. Chen and K. Yano, Integral formulas for submanifolds and their applications. J. Differential Geometry 5 (1971), 467-477. 2. M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. GTM 14, Springer-Verlag, New York (1973). 3. J. Little, On singularities of submanifolds of higher dimensional Euclidean space. Annali Mat. Pura et Appi, (ser. 4A) 83 (1969), 261-336. 4. D. K. H. Mochida, M. C. Romero-Fuster and M. A. S. Ruas, The geometry of surfaces in 4-space from a contact viewpoint. Geometriae Dedicata 54 (1995), 323-332. 5. D. K. H. Mochida, M. C. Romero-Fuster and M. A. S. Ruas, Osculating hyperplanes and asymptotic directions of codimension 2 submanifolds of Euclidean spaces. Geometriae Dedicata (1999). 6. D. K. H. Mochida, M. C. Romero-Fuster and M. A. S. Ruas, Inflection points and nonsingular embeddings of surfaces in M5. Rocky Mountain Journal of Maths, (to appear). 7. J. A. Montaldi, On contact between submanifolds. Michigan Math. J. 33 (1986), 195-199. 8. C. L. E. Moore and E. B. Wilson, Differential geometry of twodimensional surfaces in hyperspaces. Proc. of Amer. Acad, of Arts and Sciences 52 (1916), 267-368. 9. S. M. Moraes and M. C. Romero-Fuster, Semiumbilics and normal fields on surfaces immersed in Mn, n > 3. Preprint (2001). 10. S. M. Moraes, M. C. Romero-Fuster and F. Sanchez-Bringas, Principal configurations and umbilicity of submanifolds in MN. Preprint (2001). 11. S. M. Moraes, M. C. Romero-Fuster and F. Sanchez-Bringas, Translation Surfaces. Preprint (2001). 12. A. Ramfrez-Galarza and F. Sanchez-Bringas, Lines of Curvature near Umbilical Points on Surfaces Immersed in M4. Annals of Global Analysis and Geometry, 13 (1995), 129-140. 13. L. Rodriguez and R. Tribuzy, Reduction of codimension of regular immersions. Math. Z. 185 (1984), no.3, 321-331. 14. M. C. Romero-Fuster and F. Sanchez-Bringas, Umbilicity of surfaces with orthogonal asymptotic lines in M4. Differential Geometry and Applications (to appear).

T H E GAUSS M A P OF MINIMAL SURFACES ANTONIO ROS Departamento de Geometria y Topologia, Universidad de Granada, 18071 Granada, Spain E-mail: [email protected] To Antonio M. Naveira, on his 60*'1 birthday We give a new approach to the study of relations between the Gauss map and compactness properties for families of minimal surfaces in the Euclidean three space. In particular, we give a simple and unified proof of the curvature estimates for stable minimal surfaces and for minimal surfaces whose Gauss map image omits five points.

The Gauss map of a minimal surface in the Euclidean space R3 is a conformal map. This fact has deep consequences in the behavior of these surfaces and has allowed a massive presence of complex variable techniques in the classical theory of minimal surfaces. In 1959 Osserman 19 started a systematic study of this map, showing that the Gauss map of a non-flat complete minimal surface must be dense in the unit sphere S 2 . He also proved 20 the following curvature estimate, which clearly implies the result above: There exists a positive constant C > 0 such that, for all minimal surfaces ip : E —• R 3 whose Gauss map omits a fixed geodesic disc in S 2 and for all point p G E, we have \K(p)\d(pf
(1)

where K is the Gauss curvature of ip and d(p) is the geodesic distance from p to the boundary of E, that is, d(p) is the infimum of the lengths of the divergent curves in E leaving from p. Improving some intermediate results 21,16 and specially the paper by Xavier 32 , Fujimoto 6 proved that given five distinct points of the sphere, there is a positive constant C such that any minimal surface whose Gauss map omits these points satisfies the estimate (1). In particular, the only complete minimal surface satisfying this property is the plane. Since examples are known 22 of complete minimal surfaces whose Gauss map omits precisely any arbitrarily prescribed set of S 2 with less than five points, the result of Fujimoto is sharp. Another interesting property of minimal surfaces which has been studied in detail is stability. Minimal surfaces are characterized as critical points of the area functional. A minimal surface is said to be stable if its second

235

236

variation formula for the area is non negative, for any compactly supported infinitesimal deformation, see [4] for more information. It is a remarkable fact that stability depends only of the Gauss map of the surface. Some results, involving stability, which look like those obtained by Osserman and Fujimoto have been showed. Do Carmo and Peng 1 , Fisher-Colbrie and Schoen4 and Pogorelov27 have proved independently that the only complete stable minimal surface in R 3 is the plane. Later Schoen30 proved, the stronger result, that the stability assumption implies the estimate (1). In this paper we develop an unified approach to the study of these two problems: the Gauss map image and the stability of minimal surfaces in R 3 . We prove that, modulo general arguments, the curvature estimates for minimal surfaces which are either stable or whose Gauss map omits five points, follows from the uniqueness of the plane among complete surfaces in the above families. In particular it we give a new proof of the Schoen's curvature estimate for stable minimal surfaces in R3, see [30]. An unexpected consequence of our work is the fact that in order to prove the above uniqueness of the plane in the complete case, it is enough to show it for surfaces with bounded Gauss curvature. Using this property we simplify the proof of Fujimoto results in

n

Our approach is inspirated in a paper by Zalcman 34 . He takes a similar point of view for the study of properties of meromorphic maps. In §1 we give Zalcman's result, adapted to our setting, and using it we obtain a new proof of Picard theorems. Another important remark is that all the results stated in §1 are true for harmonic maps from Riemann surfaces to compact n-dimensional Riemannian manifolds. In this paper we will only work with orientable surfaces. 1

Compact properties for meromorphic maps

Let £ be a Riemann surface , C the finite plane, S 2 the unit sphere in R 3 and D(r) = {z e C : \z\ < r} the Euclidean disc of radius r > 0. The unit disc will be noted simply by D. Let M(E) be the set of meromorphic maps defined over S, that is the set of holomorphic maps from E to S 2 C R 3 . Projecting stereographically from the north pole of the sphere we obtain an identification of S 2 with C. Unless otherwise is stated, we will consider meromorphic maps as S 2 -valued. If F S M(E) and / is its associated C-valued map, then we have the relation Of

F

|f|2 _ 1

= ^jf^vWT-1^s2cCxR

= m3-

(2)

237

We consider on .M(£) the topology of the uniform convergence on compact subsets, which is equivalent to the C^-convergence on compact subsets for any natural number k. If V is an arbitrary property for meromorphic maps, we put P ( £ ) = {F £ M ( £ ) : F satisfies the property V}. Given a property V, we consider the following assertions: I) For any two Riemann surfaces E and Si and for any holomorphic map without branch points : £ —* S j , if F e 7>(£i), then F o <j> e V{T). II) Let £ be any Riemann surface and F 6 .M(E). If for any relatively compact domain O of £ one has F|n € V(il),

then F e P ( £ ) .

III) For any Riemann surface, 'P(E) is a closed subset of .M(E). IV) "P(£) is compact for any Riemann surface. If V satisfies the axioms I), II) and III) we will say that V is a closed property , and if it satisfies I), II) and IV) we will say that it is a compact property. The following lemmae are known or reformulations of known results, see for example 13>14. We include proofs for sake of completeness. If F e M(£l), where Q is a domain of C, we denote the length of the Euclidean gradient of F by | V F | e . If / is the C-valued map associated to F, using (2) we have the relation IVFI 2 = 1

le

8

^'l

(l + l/l 2 ) 2 '

Lemma 1 Let T C Ai{D) be a family of meromorphic maps defined on the unit disc. Then T is relatively compact if and only if the family {|VF| e : F € T~\ is uniformly bounded on compact subsets of D. Proof. As our topology is equivalent to the C1-convergence on compact subsets, if T is relatively compact, then the gradients of the maps of T are uniformly bounded on compact subsets of D. Conversely, if {|VF| e : F G J-} is bounded uniformly on compacts subsets, then it follows from Ascoli-Arzela theorem that T is relatively compact. •

238

If E is a Riemann surface, then, by the uniformization theorem, there is on E a complete metric compatible with its Riemann structure and constant Gauss curvature equal to 1, 0 or —1. We will call this metric the canonical metric of E, and we will denote it by ds2c. We write A c and dAc to denote the Laplacian and the canonical measure of this metric. If F £ M(T,), we denote by |VF| C the length of the gradient of F with respect to ds2. In particular for the complex plane ds2c = \dz\2, and for the unit disc we have

ds2c = ( j ^ f

\dz\2

|VF| C = ^ H V F | „ ,

(3)

for any F G M(D). Lemma 2 Let P be a closed property. Then the following assertions are equivalents: i) V is a compact property. ii) P(D) is compact. iii) The family {|VF| C : F € P ( E ) } is uniformly bounded by a constant which is independent of the Riemann surface E. Proof. The equivalence between i) and ii) follows from axiom I) and the fact that a family of meromorphic maps over E is relatively compact if and only if its restriction to a neighborhood of each point of E is relatively compact. If V satisfies iii), then we have from (3) that the family

{\VF\e:FeV(D)} is uniformly bounded on compact subsets of D, and using Lemma 1 we conclude that V(D) is compact. Suppose now that V is compact. Passing to the universal covering and using I), it is enough to prove iii) for the canonical metrics of E = D, C or S 2 . As the group of direct isometries of these metrics acts transitively on E in the three cases, the uniform bound follows from the punctual bound and the axiom I) applied to the above transformations group. • Lemma 3 Let V be a compact property. Then i) V(C) contains only constant maps. ii) If F € V(D — {0}), then F has a meromorphic extension to the disc D. Proof, i) Let F e V{C) and z e C. Define the sequence Fn : C -> S 2 , n = 1,2,... by Fn(w) = F(z + nw), for any w 6 C. From I) we see that Fn lies in P(C). So, |VF„| e (0) = n|VF| e (z) is a bounded sequence, that is |VF| e (z) = 0 for any z. Then F is constant.

239

ii) The canonical metric on D — {0} is given by ds2c = (. , /

\\z\log\z\J

)

\dz\2.

Hence D(l/2) - {0} has finite area with respect to this metric. Let F G V(D - {0}). From Lemma 2 there is a positive constant C, such that |VF| C < C in D - {0}. So / Jn(±)-{0}

\VF\2edxdy = [

C2dAc < oo,

|VF|*di4 c < /

JD(±)-{0}

JD(i)-{0}

where the first equality follows from the conformal invariance of the Dirichlet integral. As F has finite energy in a neighborhood of the origin, then it has a meromorphic extension to the disc (reason for instance as follows: As the energy of F is also the area of its spherical image counted with multiplicity, for small r > 0, F(D(r) — {0}) cannot be all the sphere. So the associated C-valued map, / : D(r) — {0} —> C, can be assumed holomorphic, and we conclude using the normality of the sequence fn{z) — f(z/2n), n G N, as in 14 p. 258). • Zalcman 34 theorem states that a closed property which satisfies V(C) C {constant maps} must be compact. We will need this result in the following, more precise form: T h e o r e m 1 3 4 . Let V be a closed property. Then we have the following alternative: i) V is a compact property or ii) there exists F G V(C) with |VF| e (0) = 1 and \VF\e(z) < 1 for any zeC. Proof. Suppose that V is not compact. Let ds\ be the canonical metric on D. Using Lemma 2 we find sequences {Gn} C V{D) and {zn} C D such that \VGn\c(zn) —> oo. By the homogeneity of the metric ds2, we can suppose that zn = 0 for any n. From I) we have that the maps Hn : D —> S 2 , defined by Hn(z) = Gn(z/2) for any z in D, are in V(D). Moreover |Vi?„| c (0) -> oo and, using (3), \VHn\c = 0 on dD for each n G N. So, composing if necessary with conformal transformations of D, we can assume that max|Vi?„| c = |V#„| C (0), for any n. Put Rn = 2|Vi/„| c (0). In terms of the Euclidean metric \dz\2, the properties of the maps {Hn} can be written as |V#„|e(0)=-R„-^oo

and

\WHn\e(z) < r - ^ y ^ , 1

\Z\

240

for any z € D and n e N . We define a new sequence of meromorphic maps Fn : D(Rn) -> S 2 , by Fn(z) = Hn(z/Rn) for each z in D(Rn). Then Fn is in V(D(Rn)), |VF n | e (0) = 1

and

\WFn\e{z) < ( l - ^ - )

,

for any z in D(Rn) and n in N. So |VF„| e is uniformly bounded on compact subsets of C and then, taking a subsequence, we can assume that {Fn} converges to a certain map F € M(C). As V is closed, using III), we see that F | D ( r ) e V{D(r)) for each r > 0, and from II) we have that F € V(C). Finally, from the properties of {Fn} we conclude that |VF n | e (0) = 1

and

| V F n | e < 1 in C.

Then F satisfies the conditions of ii).

• Now we will study some concrete compact properties. We start with the most famous one. EXAMPLE 1. Let X be a subset of S 2 . We can consider the property Vx defined for any Riemann surface £ by p x ( S ) = {F £ M(T.) : F(E) n X = 0} U {constant maps}. Vx satisfies trivially the axioms I) and II). It also verifies III) thanks to the Hurwitz theorem. So Vx is closed. If X has three or more points, then Vx is a compact property. This facts contains the classical Montel-Caratheodory theorem, the little Picard theorem and the great Picard theorem. Using Theorem 1, we will give a new proof of these results. Theorem 2 Let X = {a,b,c} C S 2 be a subset with three distinct points. Then Vx is a compact property. Proof. Suppose that Vx is not compact. Then, from Theorem 1, there exists a non constant map F e Vx(C). Clearly, we can assume the normalization X = {(0,0,-1), (0,0,1), (1,0,0)}. If / is the C-valued map associated to F, then / is an holomorphic map / : C —> C with 0 , 1 ^ / ( C ) . Hence / has a 2"-th holomorphic root, / „ , for any natural number n. Moreover U € VXn(C), where Xn = {9 e C : 0 2 " = 1} C C = S 2 . Then we see that Vxn is not compact, and Theorem 1 says us that, for any n, there is F„ e VXn(C) such that |VF„| e (0) = 1 and | V F n | e < 1 in C. Hence, there is a subsequence of {Fn} converging in our topology to some meromorphic map G € .M(C), with |VG| e (0) = 1. So G is non constant. On the other hand, as

241

Xn C Xn+i for each n, we conclude using Hurwitz theorem that G e "Py(C), where Y = UnXn. But G is and open map and Y is dense in the horizontal equator of S 2 . Then G(C) is disjoint with this equator and so it is contained in a open half sphere, which contradicts Liouville theorem. • EXAMPLE 2. Let a be a positive number, E a Riemann surface and ds2 a metric on E compatible with its conformal structure. Let dA and A be the canonical measure and the Laplacian of ds2, and \Vv\ the length of the gradient of v with respect to this metric, where v is any smooth function. We define the property Va by 7>„(E) = {F € M(£)

: / (|Vv| 2 - a|VF|V) 0, for all v e C0°°(E)},

where C Q ° ( E ) denotes the space of compactly supported smooth functions on E. Note that VaiX) is independent of the metric ds2. We have trivially that Va satisfies the hypotheses II) and III). One can see for instance in Fisher-Colbrie and Schoen4 that F € 'Pa(E) if and only if there exists a smooth positive function u on E such that Au + a|VF| 2 u = 0 (although completeness of the metric is assumed in 4 , this hypothesis is unnecessary in this concrete result). In particular u is a positive superharmonic function, and from Liouville theorem we have that Va(C) = {constant map}. Moreover, if S i is another Riemann surface and 4> : Ei —> E is an holomorphic map without branch points, taking on Ei the metric ds2 = *{ds2), we have Ai(u o 0 ) + o|V(F o <j))\2(u o 4>) = 0, and so F o e Va(T,i), Ai and |Vv|i being the Laplacian and the length of the gradient of the function v with respect to ds2. Then Va verifies I) and we have proved that Va is a compact property. Remark. Consider the space of harmonic maps, W(E,M"), from the Riemann surface E into a fixed compact Riemannian manifold Mn (which can be viewed as an embedded submanifold of the Euclidean space). Of course .M(E) c W(E,S 2 ). It follows from standard elliptic theory, use for instance Theorems 11.4 and 6.2 of 10 , that in this space of harmonic maps the C 1 - convergence on compact subsets is equivalents to the Cfe-convergence on compact subsets of E, for any k > 1. Lemma 1 remain true if we take H(D,M) instead of M(D). Lemma 3 is also true for closed properties of harmonic maps (in this case one must invoke Sacks and Uhlenbeck28 in order to conclude that an harmonic map with finite energy on a punctured disc extends, in an harmonic way, through the puncture). Then all the results in this section (except those in Example 1 which depend of the openness, and in

242

particular of Hurwitz theorem, of meromorphic maps) extend, with the same proof, to harmonic maps from Riemann surfaces into Mn. Example 2 give us compact properties for harmonic maps. Also, Example 1 give us closed properties if we take X a open subset of Mn (some of those may be compact properties). From Theorem 1, in order to prove the compacity of a certain closed property for harmonic maps V, it is enough to prove a Liouville type Theorem for maps in V(C) C H(C, M) with bounded gradient. Note that, in particular, the holomorphic map associated to such a F,

will be constant. A test example to decide the utility of this general point of view in the study of properties of harmonic maps, could be the its application to the solution of the following problem (related to a conjecture of Do Carmo about the Gauss map of constant mean curvature surfaces in R 3 ): Are there non-constant harmonic maps F : C —> S 2 whose image does not contain any equator of the sphere? 2

Conformal metrics on the disc

Let ds2 = e2u^ \dz\2 be a Riemannian metric on the disc and denote by K and A its Gauss curvature and its Laplacian. Let r = \z\ and p be the Euclidean distance and the ds 2 -distance from the points of D to the origin respectively. In this section we will also assume that (D, ds2) is a ds 2 -disc with center at the origin of D and radius R > 0 which is diffeomorphic via the (is2-exponential map based at the origin with the corresponding Euclidean disc in the tangent space at 0 € D. In particular p < R and p = R on 3D. Now we study the influence of bounds of the Gauss curvature over the distance function p. Lemma 4 i) If K < 0, then p < Rr in D. ii) / / —1 < K, then CR{T) < p in D, where CR : [0,1] —> R is given by . . , eR + 1 + (eR - l)r cR(r) = log „ -J-B rr-eK + 1 - (e K - l)r Proof, i) By hypothesis p is smooth in D — {0}. As K < 0, then a standard comparation theorem gives us Ap > 1/p, and so, Alogp > 0 in D — {0}. On the other hand, A l o g r = 0 in D - {0}. Hence \og{p/r) is a subharmonic function on D with \og{p/r) = log R on dD. So the lemma follows from the maximum principle.

243

ii) As K > — 1, comparing ds2 with the hyperbolic metric we have Ap > (coshp)/(sinhp) and by direct computation we obtain ep - 1 A log6 — < 0 in D - {0}. l J eP + 1 _ So log ff^xj is superharmonic in D. Moreover it takes on the boundary of the disc the value log JJT^J • From the maximum principle we have 1 ep - 1 eR - 1 reP + 1 ~ eR + 1' or, equivalently, Cfj(r) < p in D.

•

Note that CR is an increasing function with CR(0) = 0 and CR(1) = R. We consider now a sequence of metrics ds^ = e2u"\dz\2 on £>, each of which satisfies the hypothesis of the beginning of this section with R = Rn, where {Rn} is an increasing sequence diverging to infinity. Let Kn be the Gauss curvature of ds^. Lemma 5 If {Kn} is uniformly bounded and the sequence {ds2,} converges, uniformly on compact subsets of D, to a smooth metric ds2 = e2u\dz\2, then ds2 is complete. Proof. We can suppose that —1 < Kn in D for each n. Let 7(i), 0 < t < 1, be a divergent curve in D with 7(0) = 0. Prom Lemma 4 we have

CRn(Mt)\) < Pn(l(t)) < / V ° V ( « , JO

where pn denotes the ds^-distance to the origin. Taking limits when n goes to infinity, as 7([0,£]) is compact, we obtain

iog

s

™ r^

iyKi

*

As 7 is a divergent curve we have that \^{t)\ goes to 1 when t —» 1. So 7 has infinite ds 2 -length and therefore ds2 is complete.

• 3

Minimal surfaces

Let ip '• E —> R 3 be a conformal minimal immersion of a Riemann surface E in R 3 . If z is a local complex parameter of E, defined on a simply connected relatively compact domain Q C E, then, from the Weierstrass representation 22 ,

244

we know that ip is determinate by a pair of C-valued meromorphic maps f,g G M.{Q). Moreover / has no poles and, in order to avoid branch points of ip, its zeroes occur only at the poles of g, and the order of the zero is twice that of the pole. If N : £ —•> S 2 is the Gauss map of ip, then N is meromorphic and its corresponding C-valued map is precisely g. If we denote by ds2 and K the metric induced by ip and its Gauss curvature, then we have

<>/ = Ztj (±(l-g*)f,l-(l+g*)f,gf\

ds2

dZ,

,i/i(i+\9\ 2 )Y l d Z f = m ] d z f a n d K=-(

4|ff/|

V = -M!i Vl/I(i + M ) / ivvi 2 2 2

(4)

(5)

(6) ()

L e m m a 6 Let ipn : £ —> R 3 be a sequence of conformal minimal immersions, {Nn} C .M(£) the sequence of their Gauss maps and Kn the Gauss curvature of ipn. Suppose that {Nn} converges to a meromorphic map N G M(T,), that the sequence {Kn} is uniformly bounded and that {4>n(Po)} converges for some point po G S. Then we have the following possibilities: i) JV is a constant map, or ii) a subsequence {Kni} of {Kn} converges to zero, or hi) a subsequence {ip'n} of {ipn} converges to a conformal minimal immersion tp : £ —» M3 whose Gauss map is N. R e m a r k . Of course, the above cases are not incompatible. The topology in i) and ii) is the uniform convergence on compact subsets. Proof. Suppose that N is non constant and that — 1 < Kn in £ for each n G N. Let p G £ be a point and (Qp, z) a complex local coordinate centered at p. Let gn and / „ be the pair of maps given by the Weierstrass representation of ipn and g the C-valued map corresponding to iV. Take first a point where N is unbranched and with N(p) ^ (0,0,1). So g(0) G C and g'(0) ^ 0. By choosing £lp and e > 0 sufficiently small, we have that g is holomorphic and without branch points on Qp, and that 'ff ' ' > 2s2, in ft„. (i + l g, for n large enough, we have

^

2 2

(1 + M )

* >e2, in aP>

245

and so

Finally we obtain e < | / „ | in Q,p, for large n, and then {/„} is relatively compact in M(QP). Therefore the sequence of globally defined holomorphic 1-forms {fndz} is relatively compact on Ei = {p € S : iV(p) 7^ (0,0, l ) a n d p i s notabranch point of N}, because it is relatively compact in a neighborhood of each of their points. Observe that E — Ei is a discrete set. Taking a subsequence we can assume that {fndz} converges on E i either to an nonzero holomophic 1-form fdz or to infinity. We discuss each case separately. a) {fndz} converges to infinity on Ei. Let p be a branch point of N with N(p) T£ (0,0,1). Hence in a small disc D(2e) of Q p , g is holomorphic and so the same holds, from Hurwitz theorem, for gn with n large. Hence fn has not zeroes on D(2e) and converges to infinity on dD(s). From the maximum modulus principle we conclude that {/«.} converges to infinity on D(e). Suppose now that N(p) = (0,0,1), that is g(0) — oo. On a small disc D(2e) C tip the map g has neither zeroes nor poles other than the origin. So gnfn is an holomorphic map without zeroes, for n large. As g\fn converges uniformly to infinity on dD(s), the maximum principle implies that gnfn converges to infinity on D(e). Therefore it follows from (5) that |Vt/>„|e converges to infinity on Qp for each p e E . As |VJVn|e converges to |VAT|e on fip, we conclude finally, using (6), that {Kn} converges to the zero function on E. b){fndz} converges to an holomorphic 1-form fdz on Ei. Let p e E — Ei. If D(e) is a small disc contained in Clp, as / „ —> / on dD(e), we see that {fn} is uniformly bounded on dD(e), and then, using the maximum modulus principle, it is also bounded on D(e). So {/„} is relatively compact on D(e). We conclude easily that fdz extends in a holomorphic way to E and that the global 1-forms fndz converge to fdz on E. Moreover, from the Hurwitz theorem, we have that the zeroes of fdz occur precisely at the poles of g, and that the order of the zero is twice the order of the pole. Then gfdz and g2fdz are holomorphic in E, gnfndz —> gfdz and gnfndz —> g2fdz. As {ipn{Po)} converges for some point po of E, we conclude finally, from (4), that the sequence of harmonic maps {ipn} converges uniformly on compact subsets of E to an unbranched conformal minimal immersion ip : E —> R 3 , whose Weierstrass representation is given by the pair (g, fdz). In particular N is the Gauss map of ip. •

246

Let V be a compact property. We say that V satisfies a curvature estimate if there exists a positive constant C = C(V) such that any conformal minimal immersion ip : E —> R 3 whose Gauss map lies V{Ti) verifies the curvature bound \K\d2 < C

(7)

in E where K is the Gauss curvature of ip and d is its geodesic distance to the boundary of E. Note that we are assuming that C does not depend on E. Theorem 3 Let V be a compact property for meromorphic maps. Then the following alternative holds: i) V satisfies a curvature estimate, or ii) there is a conformal complete minimal immersion ip : D —> R 3 , whose Gauss map lies in V{D) and whose Gauss curvature K satisfies \K(0)\ = 1 and \K\ < 1 in D. Proof. Suppose the assertion in i) does not hold. We will construct a minimal surface satisfying ii). There exists a sequence of (non complete) minimal surfaces tpn : E„ —> R 3 and points pn e E„ such that Kn{Pn) = —z,

foreach

n

and dn(pn) -> oo,

(8)

where Kn is the Gauss curvature of ipn and dn its geodesic distance to the boundary. Now we will made several normalizations. In each step we will translate to the new surfaces and new points the notation ipni E„, pn, Kn and dn. a) We change the surface E„ by its geodesic disc of center at pn and radius dn(Pn) — 1- So the new surfaces satisfy (8) and, for each n, the new function |iir n |d^ is bounded and vanishes at the boundary. b) We change pn by the point of E„ where the function l-ft'nl^ attains its maximum and we change E„ by the geodesic disc of center at the new pn and radius dn(pn). Note that the new sequence \Kn(pn)\dn(pn)2 diverges to infinity. c) Using homotheties of R 3 we expand conveniently our surfaces in order to obtain tpniPn) = 0 and Kn{pn) — - 1 / 2 , for each n. So the surfaces satisfy again (8). Moreover, using the maximizing property of pn, we obtain

for each p e E „ and for each n e N.

247

d) If we take p € S n such that its geodesic distance to the point pn is smaller than Rn = ^dn(pn) then dn(p) > .R2 - i? n and, using (9), we have \Kn(P)\

<\ ( ^ " ^ J

' for each n.

(10)

So, as Rn —> oo, if we change £ „ by its geodesic disc centered at pn and radius Rn, we have that \Kn\ < 1 in £ n for each n large enough, dn(pn) = Rn and our surfaces satisfy (8). e) The exponential map of the surfaces S n with base point pn is defined on a Euclidean disc of radius Rn in the tangent space at pn. Moreover, as the Gauss curvature of a minimal surface is non positive, this map is a local diffeomorphism. Hence, taking pull-back with these exponential maps, we obtain a new sequence of minimal surfaces tpn : £ „ —> R 3 , where the surfaces E„ are topological discs. From the uniformization theorem S n (viewed as a Riemann surface) is the disc D (the case E n = C is impossible because the surface is non flat and V is compact): Moreover we can suppose that pn = 0 for each n. By construction, the surfaces ipn : D —> K 3 are geodesies discs with center at the origin of D and radius Rn and their exponential maps are diffeomorphisms from the corresponding Euclidean discs in the tangent space at 0 onto the disc D. f) As V is a compact property we can assume, passing to a subsequence if necessary, that the Gauss maps Nn £ V(D) converge to a map N € M(D). As resume we have constructed a sequence of conformal minimal immersions ipn : D —> R 3 satisfying the hypothesis of Lemma 6. If the option iii) of this lemma holds, then Lemma 5 says that the metric induced by the limit conformal minimal immersion if) : D —> R 3 is complete. Moreover we obtain from (10) that the Gauss curvature of tp satisfies K(0) = - 1 / 2 and K > - 1 / 2 in D and we conclude easily by scaling. So, in order to finish the proof of the theorem, it is enough to show that options i) and ii) of Lemma 6 cannot hold. The case ii) is impossible because Kn(0) = —1/2 for each n. Suppose, reasoning by contradiction, that the Gauss map N is constant. Fix r, 0 < r < 1, and put dn{r) = ^ n -geodesic distance from 0 to dD(r). Then there is a natural number n(r) such that for n > n(r) we have that Nn(D(r)) is contained in the half sphere centered at N. So by the result of Osserman 20 , there is a positive constant C such that, for n > n(r), \Kn(0)\dn(r)2

< C.

(11)

As |fCn(0)| = 1/2 for each n and, from Lemma 4, c/jn (r) < dn(r), taking r close to 1 and n large enough, we can made CRn(r) as large as you want. This

248

fact contradicts (11) and we conclude that the option i) of Lemma 6 cannot hold. • Corollary 1 Let V be a compact property. Then the following assertions are equivalents: i) V satisfies a curvature estimate. ii) The plane is the only complete minimal surface in M3 whose Gauss map lies in V. iii) There are not complete minimal surfaces conformally equivalent to the disc with bounded Gauss curvature and whose Gauss map lies in V(D). Proof. It is clear that i) =4> ii) =Mii), and from Theorem 3 we conclude that iii) => i). • Now we apply the above results to the concrete examples we are interested to: minimal surfaces whose Gauss map omits some points of the sphere and stable minimal surfaces. EXAMPLE 3. We follow the notations of Example 1 above. Fujimoto6 proved that if X c S 2 has more than four points then Vx satisfies a curvature estimate. This result is the best possible, see 22 p. 72. Using our results above we can give a simplification of the arguments of 6 . Theorem 4 (fj). If X has more than four point, then Vx satisfies a curvature estimate. Proof. We can assume that the points a i , . . . , 04 S C and 00 lie in X C C. Suppose, reasoning by contradiction, that there exists a complete minimal immersion ip : D —> R 3 whose Gauss curvature satisfies \K\ < 1 and such that its Gauss map lies in Vx(D)- Let g and / be the pair of holomorphic maps determinate by the Weierstrass representation of ip. From the boundedness of the curvature we conclude, using Yau 33 version of the Schwarz lemma and (5), that ^ q ^ r ^ l

+ bl 2 )

in D.

(12)

On the other hand, from the Fujimoto6 main lemma in , see also Mo and Osserman 17 , given e and e' satisfying 0 < 4e' < e < 1, there is a positive constant C depending only of dj, j = 1 , . . . , 4, such that (i + | g | » ) ( 3 - ) / y |

c

n L b - ^ i - -i-N2'

(13)

249

Solving for 1 + \g\2 in (12) and substituting its value in (13), we obtain B

i - N 2 < A,

(14)

where B is a positive constant and A is given by

'i/r-^nj-^-a,! 1 -'

n—Pi

2/(l-e)

)

•

(I5)

If we consider the metric ds^ = X2\dz\2 on D' = {z 6 D : g'(z) ^ 0}, we have easily that ds2) is flat and complete at the punctures. Moreover from (14) we conclude that ds% is complete at the boundary of the disc. So the universal cover of D' must be C and this contradiction proves the theorem.

• EXAMPLE 4. With the notation of Example 2 above, a minimal surface is stable if and only if its Gauss map lies in V\. After the results mentioned in the introduction about stability of minimal surfaces, Kawai 15 proved that if a > 1/8, then the only complete minimal surface whore Gauss map lies in Va is the plane and so, from Theorem 3, Va satisfies the curvature estimate (7). We do not know if this result is the best possible. In particular we can reprove in a simple way the curvature estimate for stable minimal surfaces obtained by Schoen30 in the case of minimal surfaces in R3, compare also with Nitsche 18 p. 450. Theorem 5 (I30])- There exists a positive constant C such that for any stable minimal surface ip : S —> K3 we have \K{p)\d{j>? < C,

for all p e S ,

where K is the Gauss curvature of the surface and d is the geodesic distance to the boundary. Proof. Do Carmo and Peng 1 , Fisher-Colbrie and Schoen4 and Pogorelov27 have proved independently that the plane is the only complete orientable stable minimal surface. Then the theorem follows directly from Corollary 1.

• Remark. There is another relevant similitude between stable minimal surfaces and minimal surfaces whose Gauss map omits some points of the sphere which involves complete minimal surfaces with finite total curvature. These surfaces represent, in the context of relations between complex variable and minimal surfaces, the part which correspond to the compact Riemann

250

surface theory, see . It is proved by Mo and Osserman 17 that a complete minimal surface which takes five prescribed values of S 2 only a finite number of times, has finite total curvature. On the other hand, it was proved by Fisher-Colbrie 3 and Gulliver and Lawson 11,12 that, if the index of a complete minimal surface is finite (this means that the index of the second variation formula on compact subdomains of the surface is bounded uniformly) then the surface has finite total curvature. In both cases the finiteness assumption can be rewritten as outside of a compact subset, the Gauss map of the surface omits five points of the sphere (resp. is stable). It seems natural to hope that if V satisfies a curvature estimate , then we can also conclude that a complete minimal surface whose Gauss map verifies the property outside a compact subset must have finite total curvature. However we have been unable to prove it. Addendum This paper was written some years ago and has circulated in preprint form. It has been cited by Fujimoto 9 , Osserman 23 ' 24 ' 25 , Osserman & Ru 26 , Zalcman 35 and in the books by Fujimoto8 and SchifT29. Acknowledgments This research was partially supported by DGICYT grant no BFM2001-3318. References 1. M. do Carmo & C. K. Peng, Stable minimal surfaces in R 3 are planes, Bull. Amer. Math. Soc, 1 (1997) 903-906. 2. R. S. Earp & H. Rosenberg, On the values of the Gauss map of complete minimal surfaces in R 3 , Comment. Math. Helv. 63 (1988) 579-586. 3. D. Fisher-Colbrie, On complete minimal surfaces with finite Morse index in three manifolds, Invent. Math. 84 (1985) 121-132. 4. D. Fisher-Colbrie & R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm. Pure Appl. Math., 33 (1980) 199-211. 5. H. Fujimoto, Value distribution of the Gauss map of complete minimal surfaces in R m , J. Math. Soc. Japan, 35 (1983) 663-681. 6. —, On the number of exceptional values of the Gauss map of minimal surfaces, J. Math. Soc. Japan, 40 (1988) 237-249.

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7. —, Modified defect relations for the Gauss map of minimal surfaces, J. Diff. Geom. 29 (1989) 245-262. 8. —, Value distribution theory of the Gauss map of minimal surfaces inR™, Aspects of Mathematics, E21. Friedr. Vieweg & Sohn, Braunschweig, 1993. 9. —, Nevanlinna theory and minimal surfaces Geometry, V, 95-151, 267272, Encyclopaedia Math. Sci., 90, Springer, Berlin, 1997. 10. D. Gilbarg & N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, New York, 1977. 11. R. Gulliver, Index and total curvature of complete minimal surfaces, Proc. Symp. Pure Math. 44 (1986 ) 207-211. 12. R. Gulliver & H. B. Lawson, The structure of a stable minimal hypersurface near a singularity, Proc. Symp. Pure Math. 44 (1986 ) 213-237. 13. W. K. Hayman, Meromorphic functions, Oxford University Press, London, 1964. 14. E. Hille, Analytic function theory, II, Chelsea Publishing Company, New York, 1987. 15. S. Kawai, Operator A — aK on surfaces, Hokkaido Math. J. 17 (1988) 147-150. 16. F. J: Lopez & A. Ros, On the Gauss map of complete minimal surfaces, (1988) unpublished. 17. X. Mo & R. Osserman, On the Gauss map and total curvature of complete minimal surfaces and an extension of Fujimoto's theorem, J. Diff. Geom., 31 (1990) 343-355. 18. J. C. C. Nitsche, Lectures on minimal surfaces, vol. 1, Cambridge Univ. Press, New York, 1986. 19. R. Osserman, Proof of a conjecture of Nirenberg, Comm. Pure Appl. Math., 12 (1959) 229-232. 20. —, On the Gauss curvature of minimal surfaces, Trans. Amer. Math. Soc. 96 (1960) 115-128. 21. —, Global properties of minimal surfaces in J53 and Em, Ann. of Math. 80 (1964) 340-404. 22. —, A survey of minimal surfaces, Dover, New York, 1986. 23. —,A new variant of the Schwarz-Pick-Ahlfors /emma,Manuscripta Math., 100 (1999) 123-129. 24. , —,From Schwarz to Pick to Ahlfors and beyond, Notices Amer. Math. Soc. 46 (1999) 868-873. 25. , —,A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc, 128 (2000) 3513-3517. 26. R.Osserman & M.Ru, An estimate for the Gauss curvature of minimal

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27. 28. 29. 30.

31.

32.

33. 34. 35.

surfaces in Rm whose Gauss map omits a set of hyperplanes J. Differential Geom. 46 (1997) 578-593. A. V. Pogorelov, On the stability of minimal surfaces, Soviet Math. Dokl., 24 (1981) 274-276. J. Sacks & K. Uhlenbeck, The existence of minimal immersions of 2spheres, Ann. of Math. 113 (1981) 1-24. J. L. Schiff, Normal families, Universitext, Springer-Verlag, New-York, 1993. R. Schoen, Estimates for stable minimal surfaces in three dimensional manifolds, Ann. of Math. Studies, 103 Princeton University Press, Princeton NJ, (1983) 111-125. A. Weitsman & F. Xavier, Some function theoretic properties of the Gauss map for hyperbolic complete minimal surfaces, Michigan Math. J. 34 (1987) 275-283. F. Xavier, The Gauss map of a complete non-flat minimal surface cannot omit 7points of the sphere, Ann. of Math. , 113 (1981) 211-214, Erratum, Ann. of Math. 115 (1982) 667. S. T. Yau, A general Schwarz lemma for Kaehler manifolds, Amer. J. Math. 100 (1978) 197-203. L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Montly 82 (1975) 813-817. —, Normal families: new perspectives, Bull. Amer. Math. Soc. (N.S.) 35 (1998) 215-230.

THE FERMI-WALKER CONNECTION ON A RIEMANNIAN CONFORMAL MANIFOLD B. SALVADOR ALLUE Departamento de Geomatria y Topologia, Facultad de Ciencias Universidad Complutense de Madrid 28040 Madrid, SPAIN E-mail: beatrizsalvadorQmat. ucm. es

Matemdticas,

In this paper a new notion of connection will be presented as the naturally associated to any conformal Riemannian structure C over the differentiable manifold M. Since this connection is defined along parametrized curves in the manifold, it is of a special nature. However, it proportionates an adequate tool for better understanding of the structure. Classical invariants associated to conformal structures turn out to be naturally defined by means of this new point of view.

1

Introduction

Let C be a Riemannian conformal structure over the m-dimensional differentiable manifold M , with m > 2. T h e structure C is usually regarded as t h e associated family of conformally equivalent Riemannian metrics over M, where two metrics differ by a factor t h a t is a differentiable positive function on M. We will also consider in the conformal structure all t h e local metrics defined on open sets U of M, compatible with t h e restricted conformal struct u r e on U. Hence, if g is a global metric on M preserving t h e initial conformal structure C, t h e following identification will be assumed C = {e2f

g| M : / € C°°(W), U open set of M}

(1)

Each metric in t h e initial conformal structure gives rise to a Riemannian manifold t h a t can be completely studied by means of its associated Levi-Civita connection, t h e key concept in Riemannian geometry. It is known t h a t for any pair of metrics conformally equivalent, g and g = e2-^g, t h e corresponding Levi-Civita connections V and V , are strongly

related by the equation given below, dependant on the 1-form df, VxY-VxY

= X(f)Y + Y(f)X-g(X,Y)F

(2)

being F = gradsf e X{M), so that g{F,X) = df(X), VX e 3E(M). Indeed, the conformal structure C defined on the manifold M determines a distinguished family of symmetric linear connections on M, rigidly controlled by 1-forms. If an extra condition of coherence with a given parametrized

253

254

curve is required, the conformal structure turns out to be rigid enough to distinguish a unique connection along the initial curve. In fact, at any point, the dependence on the parametrized curve can be reduced to a dependence on its 2-jet (see Lafuente-Salvador1). According to this result, this paper is concerned to the development of the theory associated to such canonical conformal connection in (M,C), and offers an alternative presentation of conformal invariants along curves. 2

The Fermi-Walker connection

Let 7 : / 3 t i—• j(t) € M be a regular parametrized curve in the conformal manifold (M,C). As stated above, the conformal structure defined on the ambient space M, will provide a characteristic linear connection canonically defined along the curve, that, in essence, will be determined by the property of preserving the conformal structure of the ambient space and the velocity field of the given parametrization. Note that the requirement of 7(f) being a curve with velocity field not null assures the existence, for any point 7(^0) in the curve, of several compatible metrics g € C such that g(7'(£)>7'(*)) = 1, in a small open interval Jto centered in to- Then, from the relation given in (2), a conformal metric g = e2-^g € C having 7 ^ (t) as a geodesic curve can be easily obtained from the initial metric g by means of a differentiable function / satisfying the following condition along the curve: (gradgfiMt))

= -jtl'(t),

Vi € Jt0

(3)

Henceforth, it is always possible to obtain conformal metrics in C having the restriction of the curve 7(f) to a sufficiently small open interval centered in to, as a geodesic curve. Moreover, each pair of conformal metrics sharing a given parametrized curve as a common geodesic, will be related by a differentiable function / which exterior differentiation df is identically null at the points of the coincident geodesic curve. According to the formula (2) above, this last property implies that their respective Levi-Civita connections, V and V, will be coincident at the points of the curve; in particular, the induced connections along the parametrized curve, V/dt and V/dt, will define the same operator. These last results allow us to present a new notion of connection associated to the conformal structure C, defined along any (regular) parametrized curve 7 : / 3 t 1—> 7(i) G M, that can be characterized by the property of being coincident with the Levi-Civita connection induced along the curve for any

255

conformal metric in C having (a restriction of) the initial parametrized curve 7(t) as a geodesic. It will be called the Fermi-Walker connection along the parametrized curve j(t) in the conformal manifold (M, C), and will be denoted by Dn/dt. The classical notion of Levi-Civita connection associated to the Riemannian (M, g), being g e C a conformal metric, and the Fermi-Walker connection on the conformal manifold (M, C), are related by means of the following result. Proposition 1. Given a conformal metric g € C and a parametrized curve 7(t) whose velocity field is g-unitary g{j'(t), 7'(t)) = 1 Vi. The FermiWalker connection D"1 jdt of the conformal manifold (M,C), can be obtained from the Levi-Civita connection W/dt along the curve, given by the Riemannian manifold (M, g), by means of the following rules:

(i)

i£7'(i)=0,Vt6J

(u)%V{t)=pr{%V(t)) W(t) G £ ( 7 ) x orthogonal to i{t), and being pr : T^WM -> 7 / ( i ) ± the orthogonal projection over 7'(t)~L- Or equivalently, D"1 jdt is defined by the formula: ^V(t)

= j

t

{g(V(t),7'(*))}i(t) + pr(jt(pr

o V)(t))

for any held V(t) e 3£(7). In Relativity theory, the formula (2) above is given to define the classical Fermi-Walker connection along any observer curve, i.e. along any time like unitary curve in the Lorentzian model (this can be found, for instance in the work of Sachs-Wu 2 ). This connection has a remarkable importance since it is used to define a standard of non-rotation along curves which can be accelerated. The results obtained above, that can be easily reformulated in the semi-Riemannian case for any parametrized curve whose velocity field is nowhere in a light direction, demonstrate that the notion of this Fermi-Walker transport, however being classically defined by means of the auxiliary presence of metric tools, is indeed a notion associated to the underlying conformal structure of the metric space where Relativity is modeled. 3

The Schouten tensor

Since the conformal structure C given in the ambient space M, is manifested by means of the natural notion of the Fermi-Walker connection, it seems

256

natural to develop the theory of tensors associated to connections, in order to obtain useful tools for the understanding of the initial conformal structure. In that sense, it will be seen that it is possible to associate to the Fermi-Walker connection an induced tensor, that turns up to be a fundamental reference for conformal invariants along curves. The construction of this special tensor is based on the already known notion of the Schouten tensor associated to any Riemannian structure (M, g), given by the formula LHX,Y)

= ^

(Ric?{X,Y)

-

^g(X,Y))

where Rics is the Ricci tensor and Sc^ the scalar curvature of the metric g. Remember that this tensor provides a significant tool for conformal theory. It is introduced to define the Weyl curvature tensor and check local flatness of the underlying conformal structure. Since the development in this section will be analogous to the previous one, observe that for a pair of conformally equivalent metrics, g and g = e 2 ^g, the computation of the difference between their respective Schouten tensors, Ls and L s , concludes with the following equation (Ls - L*) (X, Y)=g

(VjtF, Y) - df (X) df (Y) + ±g (F, F) g (X, Y)

(4)

with the notation F = gradsf G X(M). In case the pair of conformal metrics have in common (a restriction of) the curve 7(f) as a geodesic curve, it is easy to see that the property obtained above (^/7(4) = 0, equivalent to F 7 ( ( ) = 0), implies the identity LZ(Y(t),V(t))

=

L*(1>(t),V(t))

for any field V(t) defined along (a restriction of) the curve 7(t). Therefore, it is possible to associate to the Fermi-Walker connection D1 /dt along 7 : 7" 3 t i-> 7(4) G (M,C), a unique well defined differentiable 1-form L 7 , that can be defined by means of the following characterization: Vt e / , V& G Tl{t)M ^(6):=Lg(7'(*).&)eR

(5)

being g a conformal metric inducing the Fermi-Walker connection D1/dt in an open neighborhood of t. It will be called the Schouten tensor LP1 associated to the Fermi-Walker connection D'1 /dt in the conformal manifold (M,C), and it provides an useful tool to characterize already known conformal invariants for curves in (M,C).

257

4

Conformal invariants along curves

As promised in previous sections, the theory already developed in this paper offers an alternative characterization of classical invariants along curves, induced from the conformal structure of the ambient space. 4-1

Conformal geodesies

A parametrized curve j(t) on the conformal manifold (M,C) is said to be a conformal geodesic of the structure if the following conformally invariant third order equation is satisfied dtydt1)

g(7',7')

dt7

2

g(7',7')

7

+ g(7',7')L(7,,-)Tg-£(7',7')7'

(6) 3

for some/any conformal metric g e C (see Bailey-Eastwood ). In particular, if the conformal metric is locally chosen to define the Fermi-Walker connection and the Schouten tensor along the parametrized curve, the following result is obtained. Proposition 2. Let j(t) a parametrized curve in the conformal manifold (M,C), then, the following conditions are equivalent: (i) 7(i) is a conformal geodesic curve, (ii) t i e Schouten tensor along the parametrized curve f(t) is identically zero: L7=0. 4-2

Conformal parameter

The parametrization of a curve *y(s) on the conformal manifold (M, C) is said to give its conformal parameter if the following conformally invariant third order equation is satisfied

for some/any conformal metric g G C(see Bailey-Eastwood 3 ). In particular, since the Fermi-Walker connection and its Schouten tensor along the curve can be attained by the chosen conformal metric, the following characterization is obtained.

258

Proposition 3. Let j(t) a parametrized curve in the conformal manifold (M,C), then, the following conditions are equivalent: (i) t = s is the conformal parameter of the curve

j(t),

(ii) the Schouten tensor Ly is zero over the tangent direction of the curve: IP{j)

= 0.

Moreover, an easy calculation allows us to compare the associated Schouten tensor of two different parametrizations 7(t) and 7(s) = 7(t(s)) of a given curve in M. Two conformal metrics g and g = e2-^g, having, respectively, the parametrized curves -y(t) and 7(s) as unitary geodesies curves, are related through a function / satisfying

^-) = (-^)v(t(-)) along the curve. So, computing the formula (4) above, we conclude that the relation between the corresponding Schouten tensor is given by ^(7'(5))=t'(S)2^(7'(t(S)))-2(

1 t"'(s)

3 t"(sf

2 t'(s)

4 t'(s) 2

= t'(S)2^(7'(t(S)))-2{t}s

(7)

where the term m i i s

=

^ 1 _ 3 («M.\2 V(s) 2\f(s)J

=

ft"{8)\ \t'(s)J

_ l

(t"(s)\2 2\t'(s)J

denotes the Schwarzian derivative of the function t(s) respect to s (see Schwarz 5 ). This derivative is associated to the projective structure of S 1 = K 1 U {oo} = P 1 , and it determines the function t, except composition with transformations '

at + b

ct + d'

when

\

c d

.

(ab,)eSL(2,R)

We are now in a position to regain the conformal parameter of a given curve from a general parametrization j(t), just by means of an easy integration of the action of the Schouten tensor over the velocity field. Proposition 4. Let j(t) be a parametrized curve in the conformal manifold (M,C), then, the conformal parameter s associated to the curve will be

259

reached by means of any reparametrization s(t) satisfying the equation,

{s}t = - W W ) . The work of Bailey-Eastwood3 is given here as a reference for the initial definitions of the conformal invariants appearing above, since their presentation gives a direct justification of our results. However, these notions are originally presented by Elie Cartan 4 in his own model of tangent spheres for conformal geometry. To any conformal metric, the associated Levi-Civita connection of the linear model is extended to an associated normal Cartan connection, in Cartan's model. The conformal invariants above are then naturally defined by means of the geometry of this last connection. The Fermi-Walker connection presented in this paper is also susceptible of being extended to a normal Cartan connection, providing a canonical way of transporting tangent spheres along parametrized curves. In this context, the results stated in the last Propositions can be as well obtained from the original notions of conformal invariants given by Cartan. 5

Resemblance between conformal and metric structures

For any conformal manifold, the naturally defined Schouten tensor L1 provides a criterion of curvature along the associated curve 7(4), measuring the failure of being a geodesic of the conformal structure. It defines the conformal analogue to the notion of acceleration associated to parametrized curves in Riemannian manifolds. In fact, given a manifold endowed with a Riemannian metric g, and using the notation of1 — g ( 7 ' , •) for the acceleration form of a parametrized curve 7(i), the analogues of Propositions 2 and 3 for metric invariants along curves can be expressed as follows: (i) geodesic curves ^(t) of g are defined by the condition:

(ii) parametrizations of a given curve with g-constant velocity are characterized by the equation: c^ (
260

In these conditions, a computation with formula (4) above gives rise to the following identity:

i

-M-i«(|y.^v)-i*'

being k(t) the first curvature of ^(t) in (M, g). The conformal structure associated to the metric g defines a conformal parameter s along the curve. By Proposition 4, the relation between this latter and the metric parameter t defining the arc length, is given by the formula {s}t=Ifc2(t).

(8)

Through the same procedure, it can be also demonstrated that geodesic curves of the conformal structure are characterized by the condition of being curves with metric curvature k(t) = K constant, parametrized by its conformal parameter. It is easy to see that in that case, equation (8) above admits the following solution: s(i) = tanATi/2, for K^O, &{t) = t, for K = 0. So, for the conformal structure defined by g, we can conclude that the family of geodesies of the structure has been extended to those parametrized curves 7(s) of constant metric curvature K, with a parameter s satisfying: For K^O, s(t) =

a sin Kt/2 + b cos Kt/2 csin Kt/2+ dcos Kt/2

For K = 0,

s(t) =

at + b ct + d

being t the metric parameter given by the arc length, and (

. 1 € SL(2, K).

Observe that for K = 0 and s(t) = at + b, the family of metric geodesies is also included. References 1. J. Lafuente, B. Salvador, "From the Fermi-Walker to the Carton connection", in "Proc. of the 19th Winter School on Geometry and Physics, Srni 1999", Rendiconti Circ. Mat. Palermo, Supp 63, 149-156 (2000).

261

2. R.K. Sachs, H. Wu, General Relativity for Mathematicans, ed International Press (1994). 3. T.N. Bailey, M.G. Eastwood, "Conformal circles and parametrizations of curves in conformal manifolds", in Proc. of the Am. Math. Soc, 108, 215-221 (1990). 4. E. Cartan, "Les spaces a connexion conforme", Ann. Soc. Pol. Math., 2, 171-221 (1923). 5. H.A. Schwarz, "Uber einige Abbildungsaufgaben", in J. fur riene und angewandte Math., 70, 105-120 (1869) or "Gesammelte Mathematische Abhandlungen", Vol II, Julius Springer, 65-83 (1890).

O N T H E V O L U M E A N D E N E R G Y OF SECTIONS OF A CIRCLE B U N D L E OVER A COMPACT LIE G R O U P MARCOS SALVAI FOMAF-CIEM, Ciudad Universitaria, 5000 Cordoba, Argentina E-mail: [email protected] Let G be a compact simply connected semisimple Lie group endowed with a bi-invariant Riemannian metric and let E —• G be a vector bundle with twodimensional fibers and a G-invariant metric connection (generically, it has no parallel unit sections). We prove that if E carries the Sasaki metric, then the constant unit sections are exactly those of minimum volume and minimum energy among all smooth sections of the associated circle bundle.

Gluck and Ziller proved in the much cited paper 5 that Hopf vector fields on 5 3 are exactly those having minimum volume among all unit vector fields. This article motivated the study of the volume, and later of the energy n ' 1 2 of unit tangent fields on various Riemannian manifolds, mainly the critical points of the functionals (see the abundant bibliography on the subject for instance in the references to 4 ) . Recently, Brito * proved the analogue of the result by Gluck and Ziller for the energy instead of the volume. We are interested in a natural generalization, namely, volume and energy of sections of sphere bundles. An important source of examples is the following: Let G be a Lie group with Lie algebra g. Let V be a finite dimensional vector space with an inner product and o (V) the set of all skew-symmetric endomorphisms of V. Let £ = G x V - * G b e the trivial vector bundle. For v e V, let Lv : G —> E be the "constant" section Lv (g) = (g,v). Given a linear map 0 : 0 —> o (V), there exists a unique connection V on E —• G such that (V Z L„) {g) = Le{Z)v (g) for all g £ G and all left invariant vector fields Z on G. Moreover, the connection is metric. If G has a left invariant Riemannian metric, one can define on E the canonical Sasaki metric induced by V, in such a way that the map (dTr, £ ) € : T(.E -» TgG x Eg is a linear isometry for each £ € E (here g = -K (£) and /C is the connection operator associated with V). The vector bundle E —» G with the connection above and the Sasaki metric is called the Riemannian vector bundle over G induced by 6.

262

263

For example, the situation in the paper of Gluck and Ziller is a particular case of the preceding: Think of S3 as the Lie group of unit quaternions with the canonical bi-invariant (round) metric. If E is the Riemannian vector bundle over 5 3 induced by e = I ad = ±dAd

(1)

and tg denotes left invariant multiplication by g, then

F:E^TG,

F(g,v)=deg{v)

is an affine vector bundle isomorphism, and moreover an isometry if E and TG carry the corresponding Sasaki metrics. Via this isomorphism, Hopf vector fields are congruent to constant unit sections. Notice that the involved vector space is three-dimensional, so the result of Gluck and Ziller is not a consequence of the result in this paper, which deals with circle bundles. As far as we know, the following results are the unique ones concerning minima of the volume and energy of sections of sphere bundles, apart from the trivial case where parallel unit sections exist. a) A detailed study on the minimum of the volume and energy of unit vector fields on tori can be found in 8 and u , respectively. b) Hopf vector fields on Sz are exactly those unit vector fields on S3 with minimum volume 5 and minimum energy 1. c) In 9 we prove an analogue of the main result of 5 , also in the setting above, for the Riemannian vector bundle over S 3 induced by 6 = \ dp (cf (1)), where V is the algebra of quaternions and p is the representation of Sd on V given by p(g)h = hg^1 (quaternionic multiplication). We also use calibrations. d) For n > 1, no unit tangent vector field on S2n+1 has minimum energy 2 ' 3 . e) The two distinguished left invariant unit vector fields on a Berger threesphere are exactly those of minimum energy 6 . In this note we obtain one more result in this direction: T h e o r e m . Let G be a compact simply connected semisimple Lie group endowed with a bi-invariant Riemannian metric and let E —> G be the Riemannian vector bundle over G induced by a linear map 6 : Q —> o (V), where V is a two-dimensional vector space with an inner product. Then the constant unit sections are exactly those of minimum volume and minimum energy among all smooth unit sections.

264

Corollary. Under the hypotheses of the theorem, if 6 ^ 0 there are no parallel unit sections. Remark. In fact, the proof shows that a stronger statement is true: The result is still valid if the metric is only left invariant and some (or any) generator Z of (Ker 6) satisfies that exp (tZ) acts on G on the right by isometries (or equivalently, Z is a Killing vector field of G). Critical points of the energy of unit sections of Riemannian vector bundles induced by a map 0 as above (and more general) have been studied in 10 . Given a smooth unit section V : G —> E, the volume of V is defined to be the volume of the submanifold V (G) of E, with the induced metric. A section is in particular a map from a Riemannian manifold to another, hence one has the notion of the energy £ (V) of the section V. As in the case of vector fields, there exist constants c\ and c 2 , depending only on the dimension and the volume of G, such that £(V)=Cl+c2B(V),

(2)

where B{V) = JG | | W | | 2 w is the total bending of V on G (here ( W ) f f : TgM -> Eg, || W | | 2 = tr ( W ) * (VV), the star denotes transpose and integration is taken with respect to the volume form u> associated to the Riemannian metric of G). Lemma. Let G be a compact Riemannian manifold and f : G —> R a smooth function satisfying JGfiv = Q. Then I y/l + (f + cfu and

> vol (G) Vl + c2

[ (f + c)2 u>> c2 vol (G)

(3) (4)

JG

for any constant c € R. Moreover, equality holds if and only if f vanishes identically. Proof. We may suppose without loss of generality that G has unit volume. We verify (3): The function
>4>(J(f

+ c)w\

= Vl + c*.

265

The other inequality follows in a similar manner, using instead of the convex function tp (x) = x2. In both cases, since and tp are strictly convex, equality holds if and only if / + c (or equivalently / ) is constant, and this clearly happens if and only if / = 0. • We prove the Theorem by carefully computing and taking appropriate lower bounds, as in 1. Proof of the Theorem. We may suppose that 9^0, otherwise for a unit section it is equivalent being parallel, constant and realizing the minimum of both functionals. Let {u,v} be an orthonormal basis of V and A e g* satisfying 9 (Z) u = A (Z) v and 9 (Z) v = -A (Z) u for all Z € g. Let V be a smooth unit section of E —* G. Since G is simply connected, there exists a smooth function a : G —> R such that V (g) = {g, cos (a (g)) u + sin (a (g)) v)

(5)

= cos a (g) Lu (g) + sin a (g) Lv (g). By the definition of V, we have for any left invariant vector field Z on G that V z L u = L6(z)u — ^ iz) Lv

and

SI ZLV = LB(Z)V =-\(Z)LU.

(6)

Prom (5) and (6) we have VZV

= {Z (a) + A (Z)) ( - (sina) Lu + (cos a) Lv).

On the other hand, let Z\ be a unit generator of (Ker#) compute Z\ (a) w = da(Zi)

u = da A iz^

= d{aiz1u))

(7)

= (KerA) . We

— a.d

(iz^)

(i denotes interior multiplication). By Stokes' Theorem, / Zx (a) w = - / a.d (iZl to) = - / a. (divZi) w = 0, JG

JG

(8)

JG

since divZi = 0 {Z\ is a Killing vector field on G because exp(tZi) acts on G on the right by isometries). At each j e G w e have the operator ( W ) g : TgG -> Eg = {g} x V £* V. In the following we do not write explicitly the basepoint g. Let {Z\,..., Zn} be an orthonormal left invariant parallelization, where Zx is as above, and let A be the matrix of W with respect to the bases {Z\,... ,Zn} and {u,v} . By (7) and the choice of Z\, we have

A-

(~(sma)X\ \ (cosa)X J '

266

with X = (Zi (a) + A {Zx), Z 2 ( a ) , . . . , Z„ (a)). Hence A*A = X*X and so, if Xi is the i-th component of X, we have (A*A)iJ=xixj.

(9)

E n e r g y : By (2), instead of the energy of V, we may consider its total bending, which is given by B{V) = [ H W f u ; = / tr {A* A) w = f JG

] T :xju>. JGi=l

JG

Therefore, by (8) and (4) with f = Z\ (a) and c = A {Z\), we have B(V)>

[ x\ u = I (Zi (a) + A (Zi)) 2 w > vol (G) A ( Z ^ 2 , JG

JG

which equals the total bending of any constant unit section. Volume: As for tangent bundles, we have vol (V) = J

v/det(ld

+ (VV)* (VV)) w,

where Id is the identity on TG. By (9), if e$ is the i-th element of the canonical basis of R n , we have det (I + A* A) = det (ei + xxX, ...,en

+ xnX)

n

= det ( e i , . . . , e n ) + ^ d e t ( d , . . . ,ei-i,xtX,

e»+i,... ,e n )

t=i

= 1 + arf + • • • + x2n, since det is n-linear and vanishes whenever two entries are proportional. Therefore, by (8) and (3), with / = Z\ (a) and c = A (Zi), we have vol (V) = J

y,l+ii+-' + ^

> IG

> vol (G) y / l + A(Z 1 ) 2 which equals the volume of any constant unit section. By the Lemma, both for the total bending and the volume, equality holds if and only if Zj (a) = 0 for all i = 1 , . . . ,n, that is, if and only if a (or equivalently the unit section V) is constant. •

267

Proof of the Corollary. Looking at the expressions for the volume and the energy of unit sections, it is clear that these functionals attain the minimum at parallel unit sections, provided they exist. By the Theorem they can exist only if 9 — 0, since otherwise the constant unit sections are not parallel. • Acknowledgments This work has been partially supported by FONCyT, CIEM (CONICET) and SECyT (UNC). I would also like to thank Tomas Godoy for his help. References 1. F. Brito, Total bending of Hows with mean curvature correction, Diff. Geom. Appl. 12 (2000) 157-163. 2. V. Borrelli, F. Brito and O. Gil-Medrano, An energy minimizing family of unit vector fields on odd-dimensional spheres, in Global Differential Geometry: The Mathematical Legacy of Alfred Gray. M. Fernandez and J. Wolf (ed), Amer. Math. Soc. Contemp. Math. 288 (2001) 273-276. 3. —, The infimum of the energy of unit vector fields on odd-dimensional spheres, preprint. 4. O. Gil-Medrano, Relationship between volume and energy of vector fields, Diff. Geom. Appl. 15 (2001) 137-152. 5. H. Gluck and W. Ziller, On the volume of a vector field on the threesphere, Comm. Math. Helv. 61 (1986) 177-192. 6. J. C. Gonzalez-Davila and L. Vanhecke, Energy and volume of unit vector fields on three-dimensional Riemannian manifolds, to appear in Diff. Geom. Appl. 7. E. Hewitt and K. Stromberg, Real and abstract analysis, Graduate Texts in Mathematics 25, Springer Verlag, New York, Heidelberg, Berlin, 1975. 8. D. L. Johnson Volume of Bows, Proc. Amer. Math. Soc. 104 (1988) 923-932. 9. M. Salvai, An analogue of calibrating unit vector fields on the threesphere, preprint. 10. —, On the energy of sections of sphere bundles, preprint. 11. G. Wiegmink, Tota] bending of vector fields on Riemannian manifolds, Math. Ann. 303 (1995) 325-344. 12. C. M. Wood, On the energy of a unit vector Geld, Geom. Ded. 64 (1997) 319-330.

O N M I N I M A L G R O W T H IN G R O U P THEORY A N D R I E M A N N I A N GEOMETRY ANDREA SAMBUSETTI Dipartimento

di Matematica "G. Castelnuovo", Universita di Roma P.le A. Mow 5 - 00185 Roma, Italy E-mail: [email protected]

"La

Sapienza"

After surveying some classical problems in growth theory, we present an algebraic result about minimal growth, explaining the geometry behind it. We have decided to give a parallel presentation of concepts in Riemannian geometry and geometric group theory, in order to point out similarities and relations between them. Complete proofs of all the facts reported below can be found in the references.

1

Growth of groups and manifolds

Let G be a discrete group endowed with a finite generating set S. These data give rise to a left-invariant distance dg on the group which is known as the word metric of G: the distance ds{gi,g2) is simply the length of the shortest word on S U S~l representing gf x<72- This makes of (G,S) a discrete metric space; therefore, one can consider balls in (G,S), and the function /3(G,S)(R) given by the cardinality of balls of radius R is called the growth function of (G,S). A first, rough distinction between groups may be then introduced by examining the growth type of G: this can be finite, polynomial, exponential etc., according to the growth type" of the function P(G,S) ; the group is said to be of intermediate growth if Rn •< /3(G,S)(-R) f° r ah n > 0, but (3(ats){R) ^ ciR for all e > 0. Notice that the growth type of a group G does not depend on the choice of the generating set S, since the metric spaces (G, S) and (G, S') are quasi-isometric b for any two finite generating sets S,S', and the growth type is clearly invariant by quasi-isometries. "Formally, one can introduce an equivalence relation on the space of non-negative functions on R + : one says that / •< g when there exist A, B > 0 such that f(R) < Ag{BR) for R > 0 , and that f — g when / ;< g •< f. Then, one says that / has polynomial growth if / ^ Rn for some n > 0, and that / has exponential growth if / ~ eR. 6 Two metric spaces (X, dx) and (Y, dy) are said to be quasi-isometric if there exist constants A, B > 0 and (not necessarily continuous) maps / : X —» V and g : Y —• X such that, for all i , i j € X and y,yi € Y, one has: (i) dY(f(xi)J(x2)) < Adx{xi,x2) + B and dx(g{yi),g(y2)) < AdY(y1,y2) + B (ii) dx{g{f{x)),x) < B and dY(f(g(y)),y) < B.

268

269

There exists a corresponding notion of growth for a closed (i.e. compact without boundary) differentiable manifold X endowed with a Riemannian metric ds2. One looks at the Riemannian universal covering X of X and then defines the growth function (!)(x,ds2)(R) as the function given by the volume of balls of radius R in X (the choice of the center being not influent). Then, according to the growth type of the function /3(x,ds2)i the manifold X is said to be of polynomial, exponential or intermediate growth. Again, the growth type of X is a characteristic independent of the choice of the Riemannnian metric ds2: it is, in fact, a topological property of the underlying differentiable manifold. Actually, the analogy between groups and manifolds is not purely formal: A.S. Shvarts 22 in the 50's, and later J.W. Milnor 15 , independently, showed that the growth type of a closed differentiable manifold X is the same as the growth type of its fundamental group G = ^ ( X ) . In modern terminology, this can be explained by saying that the Riemannian universal covering X of X and 7Ti(X) are quasi-isometric, for any choice of a Riemannian metric on X and of a word metric on ni(X). Moreover, it is classical that every finitely presented group can be realized as the fundamental group of a closed manifold of dimension at least 4, therefore questions about growth types of finitely presented groups and closed manifolds are equivalent. On the other hand, geometric methods in group theory have spread so widely, mostly after M. Gromov's work, that tools derived from geometry (geodesies, curvature, boundary at infinity, Gromov-Hausdorff distance etc.) have become fundamental for the investigation of discrete groups (the reader is recommended to look at the huge papers of Gromov 9 and 10 , or to the book of E. Ghys and P. de la Harpe 4 for a panorama). Before skipping to entropy, we would just mention here a beautiful, classical problem (originally asked by J. Milnor for finitely generated groups) which has been solved under special geometric assumptions, but that still holds unsettled in general: 1.1 P R O B L E M . DO there exist finitely presented groups (or closed manifolds) of intermediate growth type? In 1972 J. Tits 23 showed that any linear group (more precisely: any finitely generated subgroup of a connected Lie group) either is solvable up to finite index, or it contains a free nonabelian subgroup and then it has exponential growth (see also the Appendix 8 ) . On the other hand, J. Milnor 16 and J.A. Wolf 2 4 proved that any solvable group either contains a nilpotent subgroup of finite index, and then it is of polynomial growth, or it does not and it

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is of exponential growth. Thus, at least for linear groups, an alternative holds: polynomial or exponential growth type. Surprisingly enough, in 1984 R.I. Grigorchuk 5 constructed a finitely generated group G of intermediate growth; Grigorchuk's example, however, does not admit a finite presentation, and this is the reason why Milnor's problem above has been reformulated specifically for finitely presented groups. 2

Entropy

Entropy is a finer invariant than simple growth type, defined for finitely generated groups (f.g. groups, for short) and Riemannian manifolds of exponential growth. The entropy of a f.g. group G, endowed with a finite generating set S, is just its exponential growth rate: Ent(G, S) =

lim R'1 • log P(G,s)(R) R—>+oo

In other words, when (3(c,S)(R) ~ e>lH, then the entropy is given by the coefficient h. Similar notions can be defined 1 for groups of polynomial growth, taking the polynomial degree of the growth function (3(G,S){R)> but henceforth we shall restrict our attention to groups and manifolds of exponential growth. Analogously, for a closed Riemannian manifold (X, ds2), the (volume) entropy oiX Is Ent(X, ds2) =

lim R'1 • log

0{X,dsMR)

R—>+oo

It is classical that this is the same as the exponential growth rate of the function /3(iri(X),«fa')(fl) = card{ 7 € m(X) \ i( 7 ) < R} given by the number of geodesic loops of X, based at some point xo, of Riemannian length smaller than R, modulo homotopy rel. XQ (clearly, these limits do not depend on the choice of the base points x$ £ X, io € X). The reason of the name "entropy" given to these numbers is that, for a Riemannian manifold (X, ds2), the above limit gives a measure of the complexity of the geodesic flow in the unitary tangent bundle of X; moreover, when the Riemannian metric is nonpositively curved, then Ent(X, ds2) precisely coincides with the topological entropy of the geodesic flow, as defined in theory of dynamical systems (see Manning 1 4 ). It can be easily checked that entropy, differently from simple growth type, does depend on the particular generating set S or on the Riemannian metric ds2 chosen for G and X, since it is not invariant by quasi-isometries. Actually, even if entropy is simply an asymptotic invariant, it strongly reflects

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the geometric structure of the group and of the manifold. The following are excellent examples explaining this philosophy: 2.1 EXAMPLE: ENTROPY OF FREE GROUPS.

Let F(Sn) be the free group on a generating set Sn of n elements. It is readily computed that the number of distinct words on Sn U S " 1 of length smaller or equal than N (plus the identity e) is given by N

Pmsn),s„)(N)

= 1 + ] T 2n(2n - l ) * " 1 fc=i

hence, E n t ( F ( 5 n ) , 5 „ ) = log(2n - 1). On the other hand, if G is any group generated by a set Sn of n elements, one clearly has Ent(G, Sn) < log(2n — 1), since balls of radius N in (G, Sn) contain fewer elements than the corresponding balls in ( F ( 5 n ) , 5 n ) (as there exist relations). The remarkable fact is that free groups are precisely characterized by the condition of maximizing entropy: 2.2 ASYMPTOTIC CHARACTERIZATION OF FREE (NONABELIAN) GROUPS.

Let G be any group generated by a set Sn of n elements. log(2n — 1), then G is free on Sn.

If Ent(G, Sn) =

The above theorem is folklore (cp. Grigorchuk and de la Harpe 6 , or Koubi 1 2 ): if G = F(Sn)/H for some nontrivial H, one can show that Ent(G, Sn) is strictly smaller than log(2n — 1) by choosing a geodesic word h on Sn U S~x representing a nontrivial element of H, and then computing the number (3h{R) of distinct words not containing h (as clearly /?/j(i?) > /3(ois„)(i?))This simple remark is a particular case of a more general property of free products which we shall see in §4. 2.3 EXAMPLE: ENTROPY OF NEGATIVELY CURVED MANIFOLDS.

Let (X, hyp) be a closed, n-dimensional hyperbolic manifold: this means that X has constant sectional curvature k(hyp) = —1 and that the Riemannian universal covering of X is the hyperbolic space X = H " . One has that Ent(X, hyp) = n — 1, since the hyperbolic metric of H " can be written, in polar coordinates, as (sinhr) 2 can,sn-i @dr2 (where cans«-i is the canonical metric of the sphere), and the volume of balls in X is given by Vol(B(x0,R)=

[ f Jsi-1 Jo

{smhR)n-1drdaSn-i^yol(Sn-1)e{n-1)R

On the other hand, by Rauch's comparison theorem it follows that if (X, ds2) is a closed, n-dimensional Riemannian manifold of negative curvature, with sectional curvature normalized so that k(ds2) < — 1, then the density of the

272

Riemannian measure of X in polar coordinates is greater than (sinh R)n 1, and therefore Ent(X, ds2) >n—l. Again, the amazing fact is the following 2.4 ASYMPTOTIC CHARACTERIZATION OF HYPERBOLIC MANIFOLDS 2

2

.

2

Let (X,ds ) be a negatively curved manifold, with curvature k(ds ) < — 1. If Ent(X,ds2) = n — 1, then (X,ds2) is hyperbolic. 3

Minimal growth

We have seen that entropy strongly depends on the word (resp. Riemannian) metric chosen on the group (resp. manifold). In order to obtain an intrinsic invariant of groups, M. Gromov 7 denned the algebraic entropy of G as AlgEnt(G) = infEnt(G,5) where the infimum is taken over all finite generating sets S of G. Analogously, to define a topological invariant 0 of closed manifolds we may introduce the geometric entropy of X, defined by GeomEnt(X) = inf Ent(X, ds2) ds2

where ds2 varies among all metrics on X with diameter diam(X, ds2) = 1. This normalization condition is clearly necessary in order that the infimum is not trivially zero, as entropy is not a scale invariant (it is straightforward to verify that Ent(X, Xg) — A _ 1 / 2 Ent(X,g) for all A > 0). However, other normalization conditions are interesting as well, e.g. a condition on the sectional curvature k(ds2) < — 1 (for manifolds admitting negatively curved metrics), or the condition Vo\(X, ds2) = 1 on the volume, each different normalization giving a different invariant and a different variational problem, about the existence and characterization of metrics realizing the minimal value. For instance, the invariant defined by the infimum of the Ent(X, ds2)'s, when the volume Vo\(X, ds2) is fixed, is known as the minimal entropy d of X, and it has been recently widely investigated because of important relations with locally symmetric metrics (cp. Besson-Courtois-Gallot 2 and our paper 19 for a panorama of results about minimal entropy and applications). c

Even if the Riemannnian metrics involved in the definition of GeomEnt(X) are supposed to be compatible with a differentiable structure of X, it is possible to show that the geometric entropy actually is a topological invariant, as it may be equivalently denned by making use only of P.L. metrics on X (cp. proof of Lemma 2.3 in Babenko 1 ). d also called full absolute asymptotic volume by I.K. Babenko 1.

273

Again, the analogy between algebraic and geometric entropy is not exclusively formal: actually, M. Gromov 7 shows that for any closed manifold X one has the following

3.1

GEOMETRIC-ALGEBRAIC INEQUALITY:

GeomEnt(X) > \AlgEnt(iri(X))

The most disappointing fact about these invariants is that they are hardly computable. We do not know the exact value of GeomEnt(X) in any case where it is nontrivial; on the other hand, for f.g. groups, computations are known just in very few cases, such as free groups (the algebraic entropy of a free group is realized by any free generating set, thus AlgEnt(F(S' n )) = log(2n — 1)), or free products of some cyclic groups (for instance, AlgEnt(Z 2 * Z 3 ) = Ent(Z 2 * Z 3 , 5 ) = ^ , where S = {1 Z 2 ,1 Z 3 } is the canonical generating set), cp. de la Harpe n . Even worst, one is still not able to give an answer to the following question, which is one of the main open problems in growth theory and was originally asked by M. Gromov 7 : 3.2 PROBLEM. DO there exist f.g. groups of exponential growth with AlgEnt{G) = 0? The geometric analogue of this problem is: 3.3 PROBLEM. DO there exist closed manifolds of exponential growth with GeomEnt(X)=0? Notice that this might perfectly happen since, even for groups and manifolds of exponential growth, the algebraic and the geometric entropy are just infima of positive numbers. A group G (resp. a manifold X) such that AlgEnt(G) > 0 (respectively, such that GeomEnt(X) > 0) is called of uniformly exponential growth. Most groups and manifolds of exponential growth are known to be of uniform exponential growth, the widest classes being that of non-elementary Gromov hyperbolic groups 12 and that of manifolds admitting a negatively curved metric (by Inequality 3.1, since their fundamental group is Gromov hyperbolic). Another class of groups with uniform exponential growth is that of free products (different from the infinite dihedral group Z 2 * Z 2 , which is the only free product whose growth is not exponential); correspondingly, all reducible manifolds (i.e. connected sum of non-simply connected manifolds) have uniform exponential growth, again by Inequality 3.1. Remark that, by the geometric-algebraic inequality, a positive answer to Problem 3.3 would imply a positive answer to Problem 3.2; this means that,

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before trying to look for manifolds of non-uniform exponential growth, one should care first to answer to its algebraic version. An interesting question is whether Problem 3.3 is equivalent to the purely algebraic problem of finding finitely presented groups of non-uniform exponential growth. However, there is a preliminary question that one should investigate before Problem 3.2, as pointed out by R. Grigorchuk and P. de la Harpe 6 : 3.4 PROBLEM. DO there exist f.g. groups G such that Ent(G,S) > AlgEnt(G) for every finite generating set S ? When a group G does not admit a generating set S realizing its algebraic entropy, we shall say that the minimal growth of G is not achieved. A first reason of interest of Problem 3.4 is that a negative answer would rule out Gromov's problem. Moreover, as a matter of fact, in all cases where AlgEnt(G) is computable, this value is realized by a more or less "canonical" generating set. A further reason to conjecture that AlgEnt(G) is a true minimum is that, by enlarging the generating set S C S', the entropy tends to grow (as 5"-balls contain more elements than S-balls). In spite of that, in the next section we shall show that Problem 3.4 has a positive answer. Unfortunately, we are able to give examples which are finitely presented, but they all have uniform exponential growth, thus they do not answer Gromov's original question 3.2. There is a corresponding geometric problem about the realization of the minimal growth of a differentiable manifold: 3.5 P R O B L E M . Do there exist closed manifolds X such that Ent(X,ds2) > GeomEnt(X) for every Riemannian metric ds2 of unit diameter on X? The answer is probably positive (one guesses that the infimum, when the manifold is not too symmetric, is realized by singular Riemannian metrics) but we do not know an explicit counterexample. Finally, questions which are entirely analogous to Problems 3.3 and 3.5 may be asked for the minimal entropy of manifolds (as defined before, by normalizing the volume). In that case, it is straightforward to see that the analogue of question 3.3 has a positive answer: for instance, any manifold X which is the product of a hyperbolic manifold with a simply connected manifold S satisfies Ent(X,ds2) > 0 for all ds2, but MinEnt(X) = 0 (in fact, one can arbitrarily dilate the hyperbolic factor and shrink S obtaining metrics ds2. on X of constant volume and such that Ent(X,ds2,) \ 0)). It is less easy to exhibit manifolds with nonzero minimal entropy and such that Ent(X,ds2) > MinEnt(X) for every metric ds2: in 19 we produced such examples.

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4

Growth tightness

4.1 T H E O R E M 20 . Let G = G\*G2 be a nontrivial free product off.g. groups, G •£ Z2 * Z2. Then, G is growth tight with respect to any word metric, which means: given any finite generating set S of G, one has Ent(G, S) > Ent{G/N, S/N) for every nontrivial normal subgroup N < G. A sketch of proof of this theorem will be given in the next section. An analogue of Theorem 4.1 holds for groups of exponential growth which are amalgamated products G\ *F G2 of residually finite groups Gi over a finite subgroup F, but we shall not enter in the discussion of this more general result, since it is more technical and unnecessary for our purposes. The meaning of growth tightness of a f.g. group (G, S) is that an asymptotic invariant is sufficient to characterize the group among all of its quotients. This generalizes the asymptotic characterization 2.2 of free groups with respect to free bases. Notice that growth tightness is a metric property, and it generally depends on the generating set 5 chosen for G. For instance, let G be a simple group of exponential growth, and let S any generating set for G. Then G' = G x G is growth tight with respect to the metric relative to the generating set 5" = (S U 1) x (5 U 1), as the only non trivial quotients of G' are its factors, and it is easy to show that Ent(G', S") = 2Ent(G, S). On the other hand, G' is not growth tight with respect to the generating set S" — (S x 1) U (1 x S), which defines the product metric ds x ds on G'; in fact, in this case one has that Ent(G', S") = Ent(G, S), since the entropy of a product, with respect to the product metric, is always the maximum of the entropies of factors (we thank L. Bartholdi for this example). The point of Theorem 4.1 is that it holds for every word metric: this is a peculiar property of free products. As a direct application of this theorem, we can produce a large class of groups of exponential growth whose minimal growth is not achieved, thus answering positively to 3.4: 4.2 COROLLARY. Let G be the free product of a non-Hopfian group with any nontrivial group. Then, the minimal growth of G is not achieved. Recall that a group G is called Hopfian if it is not isomorphic to a proper quotient of itself; equivalently, if every surjective endomorphism G —» G is injective. Thus, for instance, by the structure theorem of abelian groups, all f.g. abelian groups are Hopfian; free groups also are Hopfian (this is essentially equivalent to the Subgroup Theorem), and Z. Sela 21 recently proved that all torsionless Gromov hyperbolic groups are Hopfian. Thus, a non-Hopfian

276

group is, in some sense 6 , an uncommon object; however, such groups exist, the simplest example being Baumslag-Solitar group, which is defined as BS= and which admits a surjective, but non-injective, homomorphism : BS —> BS, given by 4>{a) = a,{b) = b~1a~1ba (cp. 1 3 ). Admitting Theorem 4.1, the proof of Corollary 4.2 is very simple: Proof of Corollary 1^.2. Let G = G\ * Gi with G\ non-Hopfian (in particular, different from Z 2 ) and G2 ^ (1). Then, G is growth tight with respect to any generating set by 4.1. Moreover, G is not Hopnan, since if i : Gi —> G\ is a surjective homomorphism with nontrivial kernel, then <j>\ can be obviously extended to a surjective homomorphism : G —> G with nontrivial kernel H. But then, for any generating set S of G, we have that (5) is a new generating set which yields a smaller entropy, because Ent(G.S) > Ent(G/H,S/H) = Ent(G, 4>(S)) by growth tightness of (G, S) and since induces an isomorphism G/H = G. This clearly shows that the minimal growth of G is not realized by any generating set S. 4k 4.3 EXAMPLE Let G = BS*Z2 =< a,b,c | a~lb2a = b3,c2 = 1 >. Then, the minimal growth of G is not achieved. More precisely, let cf> : G —> G be defined by (b) = b~1a~1ba, (c) = c; then, for every generating set S, one has that the entropies Ent(G, Sn) = n(5')) form a strictly decreasing sequence. We do not know whether this sequence really converges to AlgEnt(G). 5

Sketch of proof of Theorem 4.1

Recall the method used to prove growth tightness of free groups with respect to a free generating set, explained in 2.2, which is based on counting words on S U S - 1 not containing a geodesic representative of h G H. Clearly, this method cannot be applied successfully to a general group, since in general distinct words do not yield necessarily distinct elements, and therefore estimating Ph may not be sufficient. To follow this computational approach one should restrict to strongly Markov groups (that is, group G for which there exist a unique geodesic representation for every element, and an algorithm, e

I t is known that Gromov hyperbolic groups are, in a statistical sense, dense in the class of all finitely presented groups. This was conjectured by M. Gromov 1 0 and proved, independently, by A. Y. Olshanski 1 7 and C. Champetier 3 .

277

described by a finite automaton, producing this language), but finding good automata with respect to S may be very difficult, for S generic. As we deal with free products of any groups (which do not need to be Markov) we shall use a more flexible method. It consists of two main steps and runs as follows (see 20 for a detailed proof). Step 1. Let (G, S) be a group of exponential growth endowed with a finite generating set. Let (Z 2 , de) be the group of order two endowed with the metric defined by assigning the real length I > 0 to its generator 1. Now, consider the free product G * Z2 endowed with the "product norm", which is defined, for g i l • • • lg„+i E G*Z2 (with possibly g\ = e or gn+\ = e) as: ||5ilff2l-l5n+i||s** = Y^\\9i\\s + ne. i

This norm defines a left-invariant distance on G*Z2 by setting 9* & G * Z2. Notice that this metric is not a word metric; however, the entropy of the discrete metric space (G * Z2, ds * de) is welldefined, since balls of finite radius are finite sets, and it can be estimated as follows: logfl + e-Ent(G,S)e\ Ent{G*Z2,ds*de) > E n t ( G , S ) + 8V '-. Step 2. Now let G = Gi * G2 be a nontrivial free product of exponential growth and let S be any finite generating set. For any nontrivial normal subgroup H of G, set G = G/H and S = S/H. The second step consists in finding a contracting immersion j : (G*Z2,d-g*de) --> (G,ds) for £ » 0 i.e. an injective, Lipschitz map of Lipschitz constant 1 (not necessarily a homomorphism). Then, i?-balls of (G, S) will contain more elements than corresponding balls in (G * Z2,d-g* de), therefore it will result, by step 1: Ent(G, S) > Ent(G * Z 2 , % * dt) > Ent(G, S) 4 Now, where does geometry come in? First, there is a general fact about word metrics which plays a fundamental role: 5.1 P R O P E R T Y O F WORD METRICS ON GROUPS: a group (G,S) endowed with a finite generating set and with its word metric is almost a length space f. This means that it stays at bounded Hausdorff-Gromov distance from a true •f Recall that a metric space (X, d) is called a length space if the distance between any two points xi,a;2 is equal to the infimum of lengths of (Lipschitz) paths joining x\ t o X2-

278

length space, and precisely one has du ((G, ds),C(G, S)) < 1/2, where C(G, S) is the Cayley graph of (G, S), which is a complete length space. This property insures a certain regularity of balls and annuli of (G, S): namely, one can show that, for the annulum A(GfS)(R) = B(GiS-)(R + l)\B(GS-)(R-l) (centered at the identity) one has a(R) = c&rdA{GtS){R) > eBnt(-G's)R for all R > 0 (1) This permits to compute the entropy of the space (Gr*Z2, ds*di) by dissecting every ball B{R) of G * Z2 in products of annuli of G:

B((£+2)N) D | J Bn((£+2)N) D (J

\J

*?=1

lA(GtS)((£+2)ki-£)

where Bn(R) are the elements of B(R) which can be written as l^il • • • \gn, with gi ^ e. Since this is a decomposition in a disjoint union, using (1) a simple computation gives the estimate announced in step 1. Secondly, there is a fact specific of free products which makes things work. Recall that a geodesic segment a of (G, S) is a minimal path, in the Cayley graph C(G,S), between points of G. This is the same as the data of the initial point p and the endpoint q of a, and of a reduced word on SU S~x of minimal length representing p~xq (one also says that a, represents p~xq); we shall write II 01 \\s= ds(p, q) for the length of a. Now, given two geodesic segments a and /?, one can consider the composition of a with (3 just by translating (3 by the appropriate element of G, in order that its initial point coincides with the endpoint of a; clearly, the resulting path a/3 need not to be geodesic. However, for a free product G = G\ * G2 (G ^ Z 2 * Z2) the following holds: 5.2 P R O P E R T Y OF GEODESICS OF FREE PRODUCTS: there exists a finite set E = {ei, 62, £3, £4} of geodesic segments, which we call a complete set of joints, with max, ||ej \\s= $> such that: (i) if a,/3 are geodesic segments, then the curve ati(3 is a S-quasigeodesic, for some £j e E: that is, if p and q are the initial and the endpoint of aerf, then ds(p,q) >\\ a ||s + || j3 ||s -5; (ii) every geodesic segment from p to q passes through the 5-neighborhood of (every point) of the subsegment ti. The first condition means that, in some sense, geodesics can be "enchained", up to possibly inserting a small piece which turns the second segment in a new direction: we shall denote (one of) the resulting 5-quasigeodesic(s) aeip by a * (3. Notice that this property is false, for instance, for abelian groups and flat metric spaces (try in the Euclidean plane!).

279

The second condition means that the joints ti of enchained geodesies are "corridors" of our group, as every geodesic with the same endpoints needs to pass through a relatively small place around £j. Moreover, joints are very common to construct: this is related to the normal form of elements of a free product, that is to the unique representation of an element g G G\ * G-x as a word #[1]#[2] • • • g[m] on the alphabet (Gi \ {e}) U (C?2 \ { e D- Namely, for every g G G there exist a complete set of joints {ti} whose normal forms contain j a s a subword. Now, the idea of the contracting map j : (G * Z2, d§ * de) •—• (G, ds) is quite natural. Let h be a nontrivial element of H, let Eh = {ti} a complete set of joints containing h, and set I = maxj || e* \\s- For every element
|| 7i lis +ni =\\ ff* ||s*«

On the other hand, the injectivity of j essentially stems from the fact that, by (ii), the endpoint of the quasigeodesic 7 keeps track of all the corridors which 7 passes through, hence of all the endpoints of the 7J'S (up to 5). The algorithm for reconstructing precisely each 7, (and, in turn, each ~g~i) is pretty technical (see 20 for details). Essentially, the joints e$ containing h can be opportunely chosen in order that their normal forms have no nontrivial overlapping with each other; then, if || h \\s^ 0, it is not difficult to show that when a, (3, a', (3' are geodesic segments of G modulo h (i.e. they also represent geodesic segments in G/), then a*(5 = a'*(5' implies a = a' and /? = /?' necessarily. This argument clearly implies the injectivity of j , by induction. References 1. I. K. Babenko, Asymptotic invariants of smooth manifolds, Russian Acad. Sci.hv.Math. 41 no.l, 1-38 (1993). 2. G. Besson, G. Courtois, S. Gallot, Lemme de Schwarz reel et applications geometriques, Acta Math. 183 no. 2, 145-169 (1999). 3. C. Champetier, Proprietes statistiques des groupes de presentation finie, Adv. in Math. 116 no. 2, 197-262 (1995). 4. E. Ghys, P. de la Harpe, Sur les groupes hyperboliques d'apres Gromov, Birkhauser (1990). 5. R. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 48 no. 5,

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6.

7. 8. 9.

10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

939-985 (1984). R. Grigorchuk, P. de la Harpe, On problems related to growth, entropy and spectrum in group theory, J.Dynam.Control Systems 3 no.l, 51-89 (1997). M. Gromov in Structures metriques pour les varietes riemanniennes, Cedic/Fernand Nathan, Paris (1981). M. Gromov, Groups of polynomial growth and expanding maps, Publ.Math. IHES 53, 53-73 (1981). M. Gromov, Asymptotic invariants of infinite groups, in Geometric Group Theory eds. G.Noble and M.Roller, London Math.Soc.Lecture Notes Series 182, Cambridge Univ. Press, 1-295 (1993). M. Gromov, Hyperbolic groups, in Essays in group theory, ed. S.Gersten, MSRI Publications 8, Springer, New York, 75-263 (1987). P. de la Harpe, Topics in geometric group theory, The University of Chicago Press (2000). M. Koubi, Croissance uniforme dans les groupes hyperboliques, Ann. Inst. Fourier 48 no. 5, 1441-1453 (1998). W. Magnus, D. Solitar, A. Karras in Combinatorial group theory: presentation of groups in terms of generators and relations, Pure and Applied Mathematics, Interscience Publishers, New York London (1966). A. Manning, Topological entropy for geodesic flows, Ann.Math. 110, 567-573 (1979). J. Milnor, Problem 5603, Amer.Math. Monthly 75, 685-686 (1968). J. Milnor, A note on curvature and fundamental group, J.Diff.Geom. 2, 1-7 (1968). A. Y. Olshanskii, Almost every group is hyperbolic, Internat. J. Algebra Comput. 2, 1-17 (1992). A. Sambusetti, Minimal entropy and simplicial volume, Manuscripta math. 99, 541-560 (1999). A. Sambusetti, On minimal entropy and stability, Geom. Dedicata 81, 261-279 (2000). A. Sambusetti, Growth tightness of free and amalgamated products, to appear in Ann.Sci. Ecole Norm. Sup. Z. Sela, Endomorphisms of hyperbolic groups: I. The Hopf property, Topology 38 no. 2, 301-321(1999). A.S. Shvarts, A volume invariant of coverings, Dokl. Akad. Nauk. SSSR 105, 32-34 (1955). J. Tits, Free subgroups in linear groups, J. Algebra 20, 250-270 (1972). J. A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J.Diff.Geometry 2, 421-446 (1968).

D E F O R M A T I O N OF LIPSCHITZ R I E M A N N I A N M E T R I C S IN T H E D I R E C T I O N OF THEIR RICCI CURVATURE.

Eckerstrasse E-mail:

M.SIMON 1, D-79104 Freiburg im Breisgau [email protected]

The purpose of this manuscript is to evolve non-smooth Riemannian metric tensors by the dual Ricci-Harmonic map flow. This flow is equivalent (up to a diffeomorphism) to the Ricci flow. We use this flow to evolve metrics which arise in the study of spaces whose curvature is bounded from above and below in the sense of Aleksandrov, and whose curvature operator (in dimension three Ricci curvature) is non-negative. We show that such metrics may always be deformed to a smooth metric having the same properties in a strong sense.

1

Introduction a n d s t a t e m e n t of results

Let (Mn,D) be an n-dimensional manifold with a smooth (C°°) structure D. We say that a tensor S on a smooth manifold (M, D) is Ck or S is in Ck((M, D)) if in local co-ordinates (which come from the structure), S = {Sij}, is Ck. To avoid any confusion we will fix the differentiable structure D of M and do not consider other structures (M, D). For this reason we will suppress the D in (M, D). When considering a Riemannian metric tensor g = {gtj} on a compact manifold M we often assume g is C2. This allows us to define the Riemannian curvature tensor which is then continuous. Given a C°° Riemannian metric o0 on a manifold M, we can always find a T > 0 and a 1-parameter family of C°° Riemannian metrics {g(t)}te[o,T] on M, denoted (M,g(t)), such that —g(t) = -2Ricci(5(t)), for all t e [0,7] 9(0) =

90,

(1)

where g is C°°(M x [0, T]) (C°° on the manifold (M x [0, T]) with the induced structure), and Ricci(g(i)) is the Ricci curvature of the Riemannian manifold (M, g(t)). Notice that 1 makes no sense if g is not twice differentiable in space for all t € [0,T]. The family (M, Sl(t))t€[o,T] is called a solution to the Ricci flow with initial value g 0 . Ricci flow was invented, and used by Hamilton to prove that every compact three manifold which admits a C°° Riemannian metric g0 with Ricci (g0) > 0 also admits a metric g^ of constant positive sectional curvature 15 . The flow was constructed in such a way that various

281

282

geometrical conditions are preserved by the flow, and so that it is "nearly" a gradient flow for the Yamabe quotient

UMV°19)

»

where R(#) is the scalar curvature of (M, g) and volg is the volume form with respect to g on M. The gradient flow of E(g) would be jtg{t)

= -Ricci(5(t)) + (\R(9(t))

-

^p-r(t))g(t),

where r(t) = vJ(M(t)\(M)- Unfortunately this flow does not always have a solution (even for a short time). For this reason Hamilton studied a modified version of this flow ^g(t)

= -Ricci( S (t)) +

\r(t))g{t),

the so called normalised Ricci flow ( 1 5 ). Many metric tensors on manifolds arise from Riemannian metric tensors which are not smooth. For example the geometric object obtained by cupping a two dimensional cylinder off with two hemispheres ( 24 example 1.8) is a nice geometrical object sitting in R . As a manifold it is simply topologically S2, and we give this S2 the standard differentiable structure. It inherits a natural Riemannian metric g from the ambient space R 3 (along the joins we define g by continuity). This metric g on S2 is C 1 , 1 with respect to the standard differentiable structure of S2, but not C2. The curvature is defined away from the join and is bounded from above and below, but has a discontinuity at the join. This manifold with metric tensor is a well known example of a 'metric space with curvature bounded from above and below' studied initially by Aleksandrov 1 in connection with his investigation of the intrinsic geometry of convex surfaces, and later for it's own sake by Aleksandrov and his followers (see 6 for an overview of the theory and a good bibliography). Here the curvature bound from below is zero. If we take two copies of a two dimensional truncated cone imbedded in R 3 and join them at their boundary we obtain a nice geometrical object (as a manifold it is topologically equivalent to the infinite cylinder R x 5 1 ). The metric g inherited from the ambient space R 3 may be defined on the join by continuity and is then C 0 ' 1 (Lipschitz continuous), but not C1. Note that if we approximate this metric g by a family of metrics {Q g as a —> oo in the C° norm, then s u p x 6 M |Riem( Q #)| —* oo as a —• oo.

283

In this sense g has infinite curvature at the join, and (M, g) is not a manifold with curvature bounded from above and below. The third example is the cone. Let us consider the two dimensional cone sitting in R 3 as a graph over R 2 . This cone then inherits a metric g from the ambient space R 3 . Clearly g is C°° with respect to the standard coordinates in R 2 away from the point corresponding to the tip of the cone (for simplicity let this point be 0 = (0,0)), but g cannot be continuously extended to this point. We see this as follows. The cone C is a graph over R 2 , C = {(x,a\x\),x € R 2 } , where a > 0 is some fixed constant, and hence using the formula for the metric of a graph, we obtain 9ij

= Sij + a2 —t \x\ — 1 £ | ~

=SiJ+a2^: 1

Clearly xe = (e, e) —» 0 as e —> 0 and lim e _o 9\2{xt) = \ - Also ye = (e, —e) —» ~

2

0 as e —> 0, but km^o guiVe) = ~\- Hence there is no way to continuously extend g to the point 0. Note however that Si, < gn < (1 + a2)5ih for all x £ M - (0), in the sense of tensors. Later we shall see that metrics which fulfill such estimates, with 0 < a2 < e(n) small, can nevertheless be flown. We would like to have a way of evolving C° metrics go by something like Ricci flow, so that for all times t bigger than zero, the solution g(t) is smooth, and as time approaches zero from above, the metric g(t) approaches go uniformly on all compact subsets of M. The flow should also preserve various curvature conditions. Non-regular Example Let M = S1 x N, where N is a compact manifold which admits a positive Einstein metric 7, go be the warped product metric on M given by go(x,q) = ho(x) ®"f(q), where ho is a Riemannian metric on S1. Then the Ricci flow has the solution g(x,q,t) = ho(x) © (1 — 2ki)^{q), which has for all times t > 0 the same regularity as the regularity of ho • This means clearly that we cannot hope that the Ricci flow will 'smooth metrics out' on M with respect to the fixed differentiable structure. It is well known that if a metric is a C2 Einstein metric then one may introduce Harmonic co-ordinates for which the metric is C°° ( n ) . Such coordinates are only C2'a compatible with our fixed structure D on M, and so not admissible as smooth (C°°) co-ordinates for (M,D). We note that in example one, if we introduce Harmonic co-ordinates (change the structure) the metric will never be C2 (otherwise we could apply the result of n mentioned above, and introduce Harmonic co-ordinates which make the metric C°° which contradicts the fact that the scalar curvature has a discontinuity at the join).

284

In this paper we shall consider the dual Ricci-Harmonic Map flow (see section 6. 1 7 ). This leads to a more general version of the Ricci DeTurck flow, considered initially by DeTurck in 9 . We give here a short introduction to the these two flows. In the paper 3 the authors use Ricci flow to smooth out C 2 metrics by introducing harmonic co-ordinates at appropriately chosen times. Let g(t), t e [0,T] be an arbitrary one parameter family of smooth metrics, and (fit '• M —> M an arbitrary one parameter family of smooth diffeomorphisms. Then the metric g(t) defined by g(t) =
(^MP))0

where (V)t = ( # V ) 4 , Va(p) =

gpa, and V is the co-variant derivative with respect to the metric g (see 27 , proposition 1.4). In particular if g(t), t e [0, T] is a solution to Ricci flow, then 2Ricci($(t)) + ViVj +'VJ-Vj, (2) where

(V)j(p,t)

^^(p,i)^((^(p),t)^,

?rA4>t{p))

= V(4H(p),t).

(3)

We have now the freedom to choose the time dependent diffeomorphism (M x [0,T]), where / is the solution to the Harmonic map heat flow equation:

sLf(p,t) = f(p,0)

(^Af)(f(p,t)), =Id(p),

where h is some fixed smooth background metric. For an arbitrary function / : (M, g) —• (N, h) between two Riemannian manifolds, the Laplacian of / is then a vector field in TN defined in co-ordinate form by (^Af)(y)

= g^{x){^^p{x)

-

ra0(x)£,P(x))

+Yjk(y)g^(x)^p(x)^f^x), where f(x) = y. Since V(y,t) = —§if{x,t)

— -Af(y,t),

(4) we obtain

where g = f*g, in view of the way ChristofFel symbols and tensors change under a co-ordinate transformation. So we see that the system 3 may be

285

written •QI^A*)

V(x,t)i

= -2Ricci(ff(t)) +*V4V} +'V,-Vi, where =gij(x,t)gkl(x,t)(^TJkl-)?il)(x,t).

(5)

17

The reader is referred to section 6 for further discussion of the system 5 which is called the dual Ricci harmonic map heat flow, or 13 , 2 9 for further information about harmonic map heat flow. It is shown in 17 section 6 , that the evolution equation 5 for g(t) is a strictly parabolic system of equations. In particular if we choose h = go, then the evolution equation for g(t) in 5 is the Ricci-DeTurck flow, which was first introduced in 9 to prove the short time existence for Ricci flow on a compact manifold using standard parabolic techniques (short time existence for Ricci flow on a compact manifold was first proved by Hamilton 15 and relied upon the sophisticated machinery of the Nash-Moser Inverse function Theorem). The evolution equation for t in (1.2) may be written as a first order evolution equation in terms of g. That is, ^a(p,t)

= (&,VT(0(p,t)) = ^r(p,t)gjk(m^jk


- ^fe)(P,t),

=Id(p),

(6)

in view of the derivation of V given above. If we can solve the evolution equation for g(t),t £ [0,T] in 5, and the solution g(t) is sufficiently regular, then we may solve 6 and then define g(t) to be g(t) = ( i ^ 1 )*(
~

9^9bp9PqRacqd

• V b 3 q d + 2 V c 3 o p • Vqgbd

- 2 V c # a p • Vdgbq - 4:Vagpc • V d 3bq),

286

5(0) =

9o,

(7)

an

where Rabcd — Riem(/i)0(,C(j d V is the co-variant derivative with respect to h. Note that if h is not twice differentiable, then 7 makes no sense, since then Rabcd — Riem(/i)a&cd is not defined. If we choose h = go, that is we wish to examine the Ricci DeTurck flow, and go is not twice differentiable, then we cannot make sense of the above equation. For this reason we will always choose a smooth h not equal to go (but close to go m some to be specified C° sense) when examining 7. Restriction . We consider equation 7 only for fixed background metrics h which are C°° and have bounded curvature on all of M. The first part of the paper 28 is concerned with finding a sensible solution to the h flow for initial data go which is non-smooth. Theorem 1.1 (below) is the target theorem of this section. Definition 1.1 Let M be a complete manifold and g a C° metric, and 1 < 8 < oo a given constant. A metric h is said to be a 8 fair background metric for g, or 8 fair to g, if h is C°° and there exists a constant ko with sup |Riem(/i)(x)| = xeM -TMP)

ko < oo,,

< 9(p) < Sh(p) for all peM.

(8) (9)

Remark 1. By the result of Shi 26 , if g is a Riemannian metric and h a smooth Riemannian metric satisfying 8 and 9 then there exists a smooth metric h' which is 28 fair to g, and sup "|VRiem(/i)(a;)| = kj < oo, where V is the jth covariant derivative with respect to h (we simply flow the metric h by Ricci flow for a very short time: it then fulfills the required estimates). We will assume (without loss of generality) that our h always fulfills such estimates. Remark 2. Let M be a compact manifold, and g a C° metric on M for which (M, g) is complete. Then for every 0 < e < 1 there exists a metric h(e), for which h(e) is 1 + e fair to g. Proof (of Remark 2): We may use de Rham regularisation 10 , or a locally finite partition of unity and Sobolev averaging (6 ) to obtain a C°° metric h

287

which is C° as close as we like to g. A bound on the curvature follows from the compactness of M. • Theorem 1.1 Let go be a complete metric and h a complete metric on M which is 1 + e(n) fair to go, e(n) as in Lemma 2.4- There exists aT = T(n, ko) and a family of metrics g(t), t 6 (0, T] in C°°{M x (0, T]) which solves h flow for t e (0, T],h is (1 + 2e) fair to g(-,t) ,for t G (0,T] and limt_>0 sup x € Q , h\g(;t)-g0(-)\

= 0,

s u P : c e M l V g | 2 < C i ( " ' V - f c i ) , forallt£(0,T},ie{l,2,...},

(10) (11)

where Cl' is any open set satisfying Cl' CC Cl, where CI is any open set on which go is continuous (see 28 Theorem 5.2). Remark 3 . As a consequence of Theorem 1.1, we see that if the metric go in thm. 1.1 is continuous except for a set I C M of isolated points, then the distance function p{t) : M x M —> R, defined by p(t)(x,y) = dist(g(t))(x,y) is lipschitz, and smooth almost everywhere, for all t > 0, and satisfies limt_»o p{t){-, •) = p(0)(-, •) uniformly on any compact subset of M — I. Remark 4. / / M is not compact, g is C° on M, and g is a 'metric of curvature bounded from above an below' (see below) outside some compact set Cl, and satisfies the global bound sup

|Riem(#)(x)| 2 < ko,

x&M-Q

for some constant ko < oo, then for every 0 < e < 1 there exists an h(e) so that h is 1 + e fair to g. Proof (of Remark 4): We mollify g as in the proof of remark two to obtain a metric h which is C° as close as we like to g. One needs to check that sup M |Riem(/i)| < oo. On Cl this follows by compactness. Outside of Cl this is true because a metric with bounded curvature also has bounded curvature after it is mollified 6 . This paper is concerned with flowing metrics go of bounded curvature from above and below (initially studied by Aleksandrov 1, see 6 for a good overview), or locally Lipschitz metrics which satisfy (for example) Ricci(go) > 0, to obtain a smooth metric g which satisfies Ricci(g) > 0. The main theorem of this section is as follows. Let 1Z(g) be the curvature operator of g, and G(g) : A 2 (M) A 2 (M) —» R be the operator defined by G{g)(4>,Tp) = ii^kl9ik9ih

(1.8)

288

where A 2 (M) is the space of smooth two forms on M. 1(g) will refer to the Isotropic curvature in the case that Mn = M4 (see the proof of Theorem 6.7 28 for an overview of Isotropic curvature, and the discussion before Theorem 6.6 28 for an overview of the curvature operator). Theorem 1.2 Let A4(n,ko,d,v) be the set of (Mn,g) such that Mn is an n-dimensional compact manifold and g is a metric with curvature K(M, g) bounded from above and below which satisfies -k0 < K(M,g)

< k0,vol(M,g)

> v,diam(M,g)

< d.

There exists an ei(3,ko,d,v) > 0, £2(71, ko,d,v) > 0, and £3(4, ko,d,v) > 0 with the following properties. If (M3,g) is an element of M(3, ko,d, v) and satisfies Ricci(^) > — t\g, then there exists a smooth Riemannian metric g' on M 3 where (M3,g') has non-negative Ricci-curvature. If (M,g) is an element of M(n,ko,d,v) and satisfies 11(g) > —C2G(g), then there exists a smooth Riemannian metric g' on M where (M, g') has non-negative curvature operator. If (M,g) is an element of M(4,ko,d,v) and satisfies 1(g) > — e?, then there exists a smooth Riemannian metric g' on M where (M,g') has non-negative Isotropic curvature (see 28 Theorem 6.8 or Theorem 2.5 , section 2 of this paper for the Ricci curvature case in 3d). Remark 5 Theorem 1.2 may be used in conjunction with a standard scaling argument to show that there exists an 64 = £4(71) > 0, such that if sup M (|Riem((7)| + 4-)rf2 < £4, then g admits a smooth Riemannian metric with positive curvaV n

ture operator (scale so that d — I). The proof (see theorem 2.5 in this paper) may be trivially modified to show that «/sup M (|Riem(g)| + -^)d2 < 64, then g v readmits a metric with zero Riemannian curvature: that is (M, g) is a smooth flat Riemannian manifold (*). This is not in contardiction to the various examples of 'Gromov almost flat' manifolds which do not carry aflat metric (see section 1.4 of the book "Gromov's almost flat manifolds", by Peter Buser and Hermann Karcher, Asterique 81, Socie'te' mathe'matiqu de france, 1984 )• The reason is the term -\: Let M be a manifold which does not admit a flat Reimannian metric, but admits metrics gi, each of which has diam(M,gi) = 1 , and limj-.co |Riem(#j)| = 0. Then vol(M,gi) —> 0 (otherwise we would obtain a contradiction to the statement (*) ) . Remark 6 In dimension three, non-negative curvature operator is equivalent to nonnegative sectional curvature. In dimensions bigger than three, non-negative curvature operator implies non-negative sectional curvature.

289

Theorem 1.2 is proved by an application of Cheeger's finiteness Theorem and Gromov's compactness Theorem for metrics in M(n, k0,d, v) and a contradiction argument, and an application of the following theorem. Theorem 1.3 Let Mn be a manifold (compact or not compact) which admits a complete metric go of bounded curvature from above and below. If Tl(go) > 0 then Mn admits a smooth Riemannian metric g satisfying 11(g) > 0. If n = 3 and Ricci(go) > 0 then M 3 admits a smooth Riemannian metric g satisfying Ricci(g) > 0. If n = 4 and I(go) > 0, then M 4 admits a smooth Riemannian metric g satisfying 1(g) > 0 (see Theorems 6.2, 6.6 and 6.7 in 28 or Theorem 2.2 in section 2 for the Ricci curvature case in 3d). Theorem 1.3 is proved by flowing the metric go with the hflow from Theorem 1.1, and showing that the smooth solution g(t) also satisfies the curvature bounds from below. We may slightly weaken the hypotheses of the Corollary in the Ricci curvature case. We replace the bound on the curvature from above by a Lipschitz condition. Theorem 1.4 Let M 3 be a compact three manifold, and go be a locally Lipschitz metric on M which satisfies Ricci(go) •> 0, in the weak sense of definition 2.3. Then the solution g(x, t),t € (0, T] to h flow of go exists (for some smooth metric h) and satisfies Ricci(g(x,t)) > 0 for allt € (0,T] in the usual smooth Riemannian sense (see Theorem 2.4 in the next section or Theorem 6.5 in 2%). 2

Applications t o Lipschitz metrics

Assume that our initial metric go is a 'metric with bounded curvature' on a compact manifold M, in the sense of Aleksandrov ( 1 , see 6 for a good overview). Such metrics are locally C 1 , Q , and using a Theorem of Nikolaev, we may approximate go by a family of smooth Riemannian metrics whose sectional curvatures are bounded from above and below by constants which approximate the bounds for go- Furthermore the bounds from above and below for Ricci curvature and curvature operator of the approximating metrics are approximately the same as those for go- We state this more precisely. Lemma 2.1 Let g be a 'metric with bounded curvature' on a manifold M, with curvature K(g) C < K(g) < C in the sense of Aleksandrov. We may approximate g by smooth

Riemannian

290 metrics, {ag}aeN

so that C'--
a

and lim \ag-g\c^0(n)^>Q, v

a—>oo

'

l i m 'Tff - 9\c°(M) -> 0 a—>oo

/or open £1 C M whose closure is compact. Furthermore if the curvature satisfies B'g < Ricci( 5 ) < Bg, {B'G < K{g) < BG) then {B1 - ^)ag

< Ricci( a 5 ) <(B + ±)ag,

((B' _ I ) « G < K(ag) <(B + i)«G),

(12)

a Proof : The approximation is achieved by mollifying or regularising g . One may use Sobolev averaging and a partition of unity ( see 6 ) . • If the dimension of X is three, and (X, go) is a space with curvature bounded from above and below with Ricci(go) •> 0, then we may use the hflow to flow <7o and so obtain a family g(t), t 6 (0, T) of smooth metrics all of which satisfy Ricci(g(t)) > 0. Theorem 2.2 Let go be a complete metric with bounded curvature on a manifold M, —$< K < ko, such that Ricci(<7o) > 0 in the Aleksandrov sense. Then there exists a metric h which is 1 + e(n) fair to go (t(n) as in lemma 2.4 of 28 ) , a T(n, h, k0) > 0, and a family of smooth Riemannian metrics g(x,t),t G [0,T] such that g(x,t),t e [0,T] solves hflow, d(-it) —> go(-) uniformly on compact subsets of M as t \ 0, and 0 < Ricci(ff(x,<)) < c2(k0,n,S,h) a

for allte

(0,T\.

Proof :Let g0, be the approximating metrics for go (obtained from lemma 2, and let h be Ng0 a metric which is 1 + e(n) fair to all ag0 for a big enough.Also let ag(x,t),t G [0, T] denote the corresponding solutions to the hflow, and g(x,t),t G (0,T] the limit (as a —> oo) solution. Note that each a g0 satisfies sup M | V("So)! — ci> from 2, and hence we see, arguing as in the proof of lemma 4.1 (but without multiplying our test function by time t) that

291

SU

PMX[O,T) I ^( Q 0)l ^ c i- Calculating the evolution equation of the function

t(a+ '| v(ag)\2)

| V (ag)\2 as in the proof of lemma 4.2 - 3 , and arguing as in

the proof of lemma 4.2

28

, we get that sup M x [ 0 T i | V {ag)\ < •%, in view of

the fact that | V( a c/)| 2 is bounded. This then implies that the tensor V{ag) = agi:j{Yk3(ag) SU

PMX[O,T] H\V(g)\

~Tkj(h))

satisfies


h

supMx[o,T] "|Riem( 9 )| + |VV( f f )| < ft-

(13)

We wish to calculate the evolution of the curvature tensor of the metrics ag For a fixed point p, let (f> : Be(p) x [0,e] —> M be a time dependent local diffeomorphism satisfying the equation

§-/(p,t)

= (& *

V)(MPU)

V(p,t) (p,Q)

=

v(Mp),t),

={^Y)k-Yik){p,t), =Id(p),

(14)

and define g(t) = (fpt*9)(t)- As explained in the introduction, g(t) satisfies the Ricci flow equation 1. Also Ricciij(5(t))(p) = RiccUj((ptJ{t))(p)

= Ricci a/3 (g(t))(<Mp))^- a ^—

^

which gives us that £at Ricciy(0(t))(p) = ( |Ricci a / J (s(t))(0 t (p)) + ^ R i c c i a / 3 ( 3 ( t ) ) ( 0 t ( p ) ) | < A t ( p ) r ' ) ^ 7 0 Q ^ - ^ + Ricci^(5(t))(
+

m c z i a p m m t { p ) )

^

r

^ - ^

= ARicci(5)i;?- + 0(Ricci(5))ij + (VsRicci^V 8 + R i c c i j ^ V ' + RicciS7j V ,

(15)

where in order to obtain the last equality we have used the fact that g(t) satisfies the Ricci flow equation 1 (as explained in the introduction) and here 9 is a quadratic term coming from the curvature evolution equation (defined below). The operator 9 operates on Symmetric tensors on Riemmanian three manifolds as follows: Choose co-ordinates at a point so that the metric g is the identity matrix, and X is diagonal at this point. Then for i 6 {1,2,3},

292

9{X)u = (Xn^^i) - X(p(i),P{i))2 + Xu(Xn^in^ + -X"(p(i),p(i) - 2Xu), where n(i) p(i) are defined so that {i,n(i),p(i)} = 1,2,3. In dimension three, the evolution of the Ricci curvature for a family of metrics g evolving by Ricci flow is given by JjRicci(<7) = ^ARicci(g) -I- ^(Ricci(^)), where #(Ricci) is the quadratic defined above (See 1 5 ). More specifically, if we choose coordinates around Xo for given io so that Riccii;,(:ro, to) is diagonal, with values Riccin = A < Ricci22 = A* < Ricci33 = u, then 6>(Ricci)n = (n- vf + A(/z + v - 2A) > RRn

-

3gklRlkRu,

and similarly 8(Ricci)^ > RijR—3gktRikRjiClearly 6 satisfies the conditions of Theorem 7.3 28 (non-compact maximum principle) and so, in view of 13, 15,2 and the initial conditions, we may apply the corollary 7.4 28 (corollary to the non-compact max. principle) to the tensor N = Yl\cci{ag{t)) whose evolution equation is given by 15, to obtain Ricci{ag{x,t))

> - - , for all t € [0,T"], a

where T" = T"(n, ko). Similarly we may apply Theorem 7.3 28 to the function TV = |Riem( a 5)| 2 to obtain sup M x r 0 T //i \Riem(ag)\2 < c(n,k0,h). Letting a go to infinity gives us the result.D Hence, if M is compact, we may apply the result of Hamilton ( 16 ) to obtain M is diffeomorphic to a quotient of one of the spaces S3 or S2 x R 1 , or R 3 by a group of fixed point free isometries in the standard metric. When the dimension of M is two we obtain a similar result by examining scalar curvature and arguing as in the theorem above. Note that in dimension two, the scalar curvature evolves according to the equation -§^R = Ai? + R2 (see 28 for details). We can actually slightly weaken the hypothesis of 'curvature bounded from above' for Theorem 2.2 to a uniform Lipschitz condition on the initial sequence of metrics. Definition 2.3 Let M be a manifold, and g be a locally Lipschitz metric. We say that Ricci( 0, if there exists a family {ag}ae{i,2,...} of smooth metrics on M which satisfy Ricci(Q<7) > —-, and l i m a _ 0 0 s u p M a\ag — g\ = 0 , and | r ( a 5 ) — T(g)\ < k for all a G { 1 , 2 , . . . } , where k is some constant which does not depend on a. Theorem 2.4 Let M be a manifold and go be a complete locally Lipschitz metric on M which satisfies Ricci(<7o) > 0, in the sense of definition 2.3. Then there is a solution g(x,t),t G (0,T] to h flow of g0 for some smooth metric h and some

293

T = T(n, fco), and it satisfies Ricci(g(x,t)) > 0 for all t € (0,T] in the usual smooth sense. Proof : The proof is the same as for the case of bounded curvature from above. • We now show that the theorems of Hamilton for manifolds with nonnegative Ricci curvature in three dimensions, and non-negative curvature operator in n-dimensions can be epsilon improved. To do this we argue by contradiction and apply Cheeger's finiteness Theorem( 8 ), and Gromov's compactness Theorem (see 24 for an exposition), The argument here was inspired by the argument given in Berger 4 , where it is shown that there as a 8 < | such that any compact, even dimensional manifold which is 5 pinched is either homeomorphic to an n-dimensional sphere or is isometric to one of the symmetric spaces of rank one ( t h e complex projective space CPn, the quaternion projective space HP™, or the Cayley projective space CaP2). This is an epsilon improvement on the ^ pinched rigidity Theorem (which is the same result for \ pinched manifolds). See Berger 5 , Klingenberg 2 1 and Rauch 2 5 for the Sphere Theorem, rigidity Theorems and their generalisations. Theorem 2.5 Let M(n,ko,d,v) be the set of (Mn,g) such that Mn is an n-dimensional compact manifold and g is a metric with curvature K(M, g) bounded from above and below which satisfies -fc 0 < K(M,g)

< k0,vol(M,g)

> v,diam(M,g)

< d.

There exists an e\(3,ko,d,v) > 0, e2(n,ko,d,v) > 0, and £3(4, ko,d,v) > 0 with the following properties. If (M3,g) is an element of Ai(3,ko,d,v) and satisfies Ricci(g) > — t\g, then there exists a smooth Riemannian metric g' on M3 where (M3,g') has non-negative Ricci-curvature. If (M,g) is an element of M(n,ko,d,v) and satisfies TZ(g) > —£2G{g), then there exists a smooth Riemannian metric g' on M where (M,g') has non-negative curvature operator. If(M4,g) is an element of M(i,ko,d,v) and satisfies 1(g) > —62, then there exists a smooth Riemannian metric g' on M 4 where (M 4 ,g') has non-negative Isotropic curvature. Proof : All of these results are proved in the same way using Gromov's compactness result and Cheeger's finiteness Theorem for manifolds in M(n, k0,d,v). We prove the Ricci curvature result here. Fix k0,d and v. Assume, to the contrary that there is no such t\ > 0. Then we have for i € { 1 , 2 , . . . } , manifolds M, with metrics & such that (Mj, gi) e M(3, kQ, d, v), and Ricci(g'j) > — i , but there is *no* smooth g^ on Mi such that # / has non-negative Ricci curvature. By Cheeger's finiteness Theorem, after taking a sub-sequence if necessary, we may assume that Mj = M. By Gromov's com-

294 pactness Theorem, gi —> g in Cl'a for some g £ M(n,k0, d, v) which satisfies Ricci( 0 on M in the sense of Aleksandrov. We may flow this metric g using hflow (Theorem 2.2) to obtain a metric g' on M which is smooth and has non-negative Ricci curvature: a contradiction. • Acknowledgments The author would like to thank Gerhard Huisken, for his continuous interest in and support of this work, and Frank Duzaar and Ernst Kuwert for their useful comments and support. Thanks to Wilderich Tuschmann for introducing me to spaces of bounded curvature (in the sense of Aleksandrov). We would also like to thank the Humboldt Universitaet (DFG-Graduiertenkolleg " Geometrie und Nichtlineare Analysis") and the Albert-Ludwigs-Universitaet Freiburg, (DFG-Graduiertenkolleg "Nichtlineare Differentialgleichungen") for their hospitality and financial support. References 1. Aleksandrov, A.D., Intrinsic Geometry of convex surfaces. Godekhizdat, Moscow-Leningrad, (Russian), Zbl. 38352, German Trans.: AcademicVerlag. Berlin 1955, Zbl. 65,151. 2. Berestovkij, V.N.,Introduction of a Riemannian structure in certain metric spaces. Sib. Mat.Zh. 16,651-662 (1975) .Engl, transl.: Sib. Math. J. 16, 499-507 (1976) Zbl. 325-53059. 3. Bemelmans, Josef; Min-Oo; Ruh, Ernst A. Smoothing Riemannian metrics. Math. Z. 188 (1984), no. 1, 69-74. 4. Berger, M. Sur quelques Varietes Riemanniennes suffisamment pincees , Bull.Soc.Math. France, 88, 57-71 (1960). 5. Berger, M. Sur les Varietes Riemanniennes pincees juste du dessous de \ Ann.Inst. Fourier 33, 135-150, (1983) Zbl. 497.53044 (1983). 6. Berestovskij, V.N., Nikolaev , I.G. II. Multidimensional generalized Riemannian Spaces, Encyclopedia of Mathematical Sciences, Vol. 70, Geometry IV, (1991) Springer. 7. Calabi, E. ,An extension of E.Hopf's maximum principle with an application to Riem. Geometry, Duke Math J. 25, (1958), 45 - 56. 8. Cheeger, Jeff Finiteness theorems for Riemannian Manifolds , American J. Math. 92,61-74. (1970). 9. DeTurck, D.,Deforming metrics in the direction of their Ricci tensors, J.Differential Geometry 18 (1983) ,no. 1, 157 - 162.

295

10. de Rham, G.,Varietes differentiates: Formes courants, formes harmoniques. Herrmann, Paris, Zbl. 65,324. 11. DeTurck, D., Kazdan, J.L.Some regularity theorems in Riemannian geometry, Ann.Sci.Ecole Norm. Sup. (4) 14 (1981), 249 - 260. 12. Ecker, K. ; Huisken, G.,Interior estimates for hypersurfaces moving by mean curvature , Research report for the Centre for mathematical analysis, Australian National University, CMA-R15-90, 20-25, 1990. 13. Eells, J. ; Sampson, J.H. Harmonic mappings of Riemannian manifolds, American J. Math. 86 (1964) 109 - 160. 14. Gilbarg, D., Trudinger, N., Elliptic Partial Differential Equations of Second Order ,Springer, (1970). 15. Hamilton, R.S., Three manifolds with positive Ricci-curvature, J. Differential Geom. 17 (1982), no. 2, 255 - 307. 16. Hamilton, R.S., Four manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2 , 153 - 1710. 17. Hamilton, R.S., The formation of singularities in the Ricci flow, Collection: Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7-136. 18. Hamilton, R.S., Lecture Notes on Heat Equations in Geometry, Honolulu, Hawaii,(1989). 19. Hamilton, R.S., Four-manifolds with Positive Isotropic Curvature, Communications in Analysis and Geometry Vol. 5, Number 1, 1-92, (1997). 20. Huisken , G., Asymptotic behaviour for singularities of the mean curvature flow, J. Differential Geom. 31 (1990) , no. 1, 285 - 2910. 21. Klingenberg, Wilhelm., Ueber Riemannsche Mannigfaltigkeiten mit positiver Kruemmung , Ann. Mat. Pura Appl. (4) 60 1962: p 49-59. 22. O.A Ladyzenskaja, V.A. Solonnikov and N.N. Uralceva , Linear and quasilinear equations of parabolic type, Transl. Amer. Math.So c. 23 (1968). 23. Nikolaev, I.G. Smoothness of the metric of spaces with curvature that is bilaterally bounded in the sense of A.D. Aleksandrov. Sib. Math.J.24, 247-263,(1983) Zbl. 547-53011. 24. Peters, S., Convergence of Riemannian manifolds Compositio Mathematica62: 3-16 (1987). 25. Rauch, H.E.,yl contribution to Differential Geometry in the large , Ann.Math. 54.: 38-55, (1951). 26. Shi, Wan-Xiong.,Deforming the metric on complete Riemannian manifolds. J.Differential Geometry, 30 (1989), 223-301. 27. Simon, M.,PhD. Thesis : A class of Riemannian manifolds that pinch when evolved by Ricci flow, PhD. Thesis, University of Melbourne , 1998.

296

28. Simon, M.,Deformation of C° Riemannian metrics in the direction of their Ricci curvature to appear in Comm. Anal. Geom., 2001. 29. Struwe, M., Variational Methods Vol. 34, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer 2nd ed. (1991). 30. Schoen, R. and Yau, S.T. Lectures on Differential Geometry , International Press (1994), pp. 2 - 4.

S O M E CLASSIFICATION P R O B L E M S O N N A T U R A L B U N D L E S RELATED TO WEIL B U N D L E S JIRi T O M A S Department

of Mathematics, Faculty of Civil Engineering, Technical Brno, Zizkova 17, Brno, Czech republic E-mail: [email protected]

University

Let A be a Weil algebra of width k and M be a manifold, dim M = m > k +1. We determine all natural operators AM '• TM —» TT*TAM lifting vector fields on M onto T*TAM, t h e cotangent bundle to a Weil bundle. We also prove t h a t natural operators of this kind are obtained as a composition of the flow operator T*TA with a natural transformation TT*TA -> TT*TA over Jdj.,yA- For a monomial A, we find all natural T-functions / M : TAT* M —> R and determine all natural transformations aM • TAT*M -+ T*TAM.

1

Preliminaries

We present another contribution to the theory of Weil bundles. The starting point is a general result by Kolaf giving the full classification of natural operators T —» TTA lifting vector fields on manifolds to a Weil bundle TA (4 and 6 ) and the result classifying all natural T-functions defined on T*TA, the cotangent bundle to a Weil bundle TA. As for natural transformations TAT* —> T*TA, we follow some partial results by Doupovec, 3 . We use the basic terminology from 6 . The problem of the classification of all natural operators AM • TM -> TT*TAM is related to the already solved problem of the classification of all natural T-functions / M : T*TAM —* R as follows. Every natural operator AM '• TM —• TT*TAM is identified with a natural T-function gM : T*T*TAM -» R linear on fibers of T*{T*TAM) -» T*TAM. Applying the natural equivalences s : TT* —* T*T by Modugno-Stefani, 9 and t : TT* -> T*T* by Kolaf-Radziszewski, 8 , we obtain the identification of gM with a natural T-function / M : T*TTAM -> R given by / M = gM ° txA M ° ST\ M • Thus we investigate natural T-functions defined on A for D denoting T*TT>®AM t 0 determine all natural operators T -> TT*T the algebra of dual numbers. We remind the already mentioned couple of general results. The first one reads as follows All natural operators T —* TTA are of the form L(c)TA + XD for some c £ A and D € VerA, where TA denotes the flow operator associated to the natural bundle TA, L(c) denotes the natural affinor by Koszul on TTA determined

297

298

by c € A and XD is an absolute natural operator T —> TTA derivation D € VerA

associated to a

Let £ : M —> T ¥ be a vector field. Consider an operation ~ transforming £ into a function £ defined on T*M by £(w) = < £(q(cu)),u} > for the cotangent bundle projection q and w e T*M. Clearly for a natural bundle F and a natural operator AM '• J M —> TFM, the function AM • T*FM -> R defined by J W P O = AMX for any vector field X on M is a natural T-function. For some cases of A, e.g. D£, all natural T-functions T*TA —> R are of the form h(L(c)TA,\D)) for any smooth function h : R d i m A+dlm VerA —> R and c and D forming some bases of A and P e r A. We remind the construction of additional natural functions from nonnatural operators TM —> TTAM applying the operation ~ u . Operators and functions of this kind are constructed from natural operators T —> TT£ supposing A to be D £ / i \ Remind that Dj* = jQ(R f c ,R) is the Weil algebra of r-jets of smooth functions of k variables with the source at zero. Such a construction is given by the following assertion. Proposition 1 Let A = D£/7 be a Weil algebra, p : D£ —> A the projection homomorphism with its associated natural transformation p : T£ —> TA, f7) and L : A —> D£ a linear map satisfying pot = id A • For a manifold M and xo £ M let BXo be a minimal set of generators of the algebra J£ 0 (M, R)o = T£*M. Then there is an embedding Isx • TAM —> TJTM satisfying PM ° IBX = idTAM and

such that {lBX0{3Av))(irxJ)

= t{(jA(p){Jx0f))

f°r

an

V 3AV

e

T

xoM

JlJ: £ Bx0The embedding IQX from Proposition 1 enables to obtain operators A = : TM -> TTAM from natural operators A : T -> T£ as follows

AM ; B XO

AMJB^

=Tpo\o

lBxo

'(!)

It also enables to define coordinates on TAM and T*TAM by means of those on TjTM and T*T£M. Consider the polynomial form of elements from D£, namely -^xaTa for 0 < \a\ < r. Since Weil bundles preserve products, we have canonical coordinates x%a on T£R m = (D£) m for 1 < i < m and 0 < \a\ < r. Let S be a system formed by non-zero images p{ra) of all Ta G D£ forming its monomial linear basis. Take a maximal linearly independent subset SQ of S (a linear basis of A). Then any element d € <S - <S0 is uniquely expressed as d^a for a £ So- For any element b € S, select a monomial representative r' 3 having a minimal multiindex among all of them. Then there is such a basis

299

So C S for which any cf = csa satisfy \S\ > \a\ for minimal representatives ra of p~1{a) and TS of p - 1 ^ ) . Define the map t : A —» D£ by t('a) = r a for a minimal representative r a of a e So and t(d) = c* r a for other elements d £ S and their minimal representatives TS . Hence t is a linear map satisfying p o L = id^ from Proposition 1. It introduces the coordinates y'a on TAM by

f(p(^)) =i^r"

(2)

The formula yla — x%a + jfXgC^ gives the relation between coordinates yla of the element p{hx\r'y) and xla of the projected element of TkM. The transformation laws for the action of the jet group Grm on the standard fiber (T*TA)0B.m are of the form tfa = aj,... / s y^ .. .y1^ + lt.a?hl...htyhsl •••Vtslcafined by pfdyi, 1

^WT^h1...htys

For the

additional coordinates pf on T*TAM de-

they are of the form p] =

tc 1---y6 t %pf

(

-^~a)h...lsyX

• • .fcj>f

+

1

ifwe consider the action of G ^ . Using these

coordinates, we obtain the following expressions for AA/ ; B X A? = ( A j ) M ; B , 0 = ( ( ^ ^ M

+ (jqgjjWV^P?

(3)

Let i™ : Rfe —> R m defined by i™ = idRk x (0) m ~ fc be the canonical inclusion of Rfe into R m . Define i £ T 0 A R m by i = jAif. In coordinates, it satisfies yla = 0 whenever | a | > 2 and y] = 5j. Without loss of generality, we can suppose J to be normal, i.e. spanned by generators as follows. Let 7r£ : D£ —» D | be the canonical projection of Weil algebras. Then every generator of I maps under -K\ either onto a linear monomial or onto zero. Let us recall the identification of the jet group Grk = im;Jo(R fc ,R fe )o with the Lie group AutT>rk of algebra automorphisms of D£, 6 . It is given by Jo9(Jof) = 3a

1- Let St(i) C Grm be the stability subgroup of the immersion element i 6 T^H"1 under the canonical left action of Grm. In u we proved that GA = Stl(i) = kerp n GJ^, if we consider the restriction of PR.™ onto GJ„. Further, it holds G^/GA = (regT" 4 )oR m , x and there is a map Imm : T*(repT j 4 ) 0 R m -> {T*TA)0Km, n . In respect to this map all AM,B* are GJ^"1-invariant which means that any AM,BX is natural if and only if it is G^-invariant. Thus all natural T-functions in question are identified with G^-invariants denned on T*TAHm+1. Further, under the map Imm any Aj evaluates itself

300

over (pf) e T*TARm+1 as p?. Taking into account the fact that p°j can be annihilated by the action of G^1 leaving the values of i and other pf untouched, we obtain the bijection between A^ and canonical projections defined on T*TA~Rm+1. The coordinate expression for the action of GA on T*TAf\.m+1 is obtained as the restriction of the general transformation laws. It reads Pj A

~

a\0\

jaPl

S\/3\

m+1

jS

a0Pl

l

'

N

In fact, T*T H is identified with the space R endowed with this action and so we are searching for GA H G1m+1 -invariants on HN. Denote by B the base of all A^ and by B the base of the corresponding functions obtained from A? by their restriction to T*TARm+1. The following assertion describes an important property of (GA nker7r£)orbits used essentially for the construction of additional natural functions. Denote by Bs C B the set of all (GA H ker ^ - i n v a r i a n t s selected from B and denote by Ns the number of elements in Bs. Clearly, B\ C Bi C . . . C S r _i C Br. Further, denote B\ = Bs - Bt and Nts = Ns - Nt. Then we have P r o p o s i t i o n 2 Let w e RN and Orbs(w) be its (GA n ker TTT)-orbit. Then Bg+1(Orbs(w)) has the structure of an affine subspace in RN! , the modelling vector space of which is (JB^+i n GA)/H for a normal Lie subgroup H C Bm+i n GA • The canonical injection io of such a vector space into the vector space RN' and the sum of a point with a vector are given by io([3So+1
and

w + [j 0 s+ V]tf = £(js0+\,w)

(5)

for L?o+V]ff ^ (B^+i n GA)/H, any w of Bss+1(Orbs(w)) and the canonical left action £ of a jet group on the standard fiber. The second formula from 5, giving the definition of the vector space structure on (B^+1 n GA)/H enables us to introduce the scalar product on itself, induced by a scalar product on RN<> . It is usedjn the following procedure, generating step by step a basis of G^-invariants V. It starts setting V\ = B\ and for any w e T*TAHm+1 considering its orbit Orb(w) = Orb\(w). In the (s + l)-th step start from the basis Vs and an element ws = £(as)o.. .o£(a 2 )(w). Consider Bss+1(Orbs(w)), which is by Proposition 2 a fcs-dimensional affine subspace of R Ws for some ks+i < N^+1. Further, consider the orthogonal complement ~Vf+i in the vector space R ^ to V s + i = (££[+! n GA)/H*+1, where H°+1 corresponds to the normal subgroup H in B^+i n GA from Proposition 2. The new G^-invariants are

301

obtained as the components of the unique point Ps+i given by the intersection of Bs+1(Orbs(ws)) with the affine subspace of RN» with the modelling vector space V f coming through the origin. For almost every GA-orbit in the sense of density, the maximal dimension Ks+i is achieved and so it suffices to select only TV*"1"1 — Ks+\ components forming the basis of the additional GAinvariants from the js+l-th step. Finally, Natural T-functions are obtained as the components of Is+i corresponding to those of Ps+i. To express them in formulas, select a linear basis of V s + i formed by + s+1 [ J O + V 1 + 1 ] H . - + 1 > " - [ J o V f + n ^ - Denote by Ort 8 ([j 0 s +V s + i]^ + 1 ) the orthogonal complement to the sequence obtained from this basis omitting the i-th element. Then for any w e T*TAKm+1 we have

Ia+1(w) ~ P.+i(wt)

= Bss+1(ws) + C°+1lf0+1
=§^u*+1

(6)

if we put u\+l = b'o + Vl+i]ff»+ 1 a n ( l Ds+1 — det(u* +1 ) while D°+1 denotes ( - l ) i + 1 ((Bss+1(ws),ui+1), Ort? + 1 (u* + 1 )). In these formulas, we use the vector form of the notation and by ( , ) we denote the scalar product. If we add I}+i, • • •, /(,+!

s+

to Vs, we obtain a new basis of natural T-functions

A

defined on T*T , namely Vs+1 = Vs U {7} + 1 ,.. .,I?'+1~K'+l}. This generating alghoritm is finished if in the (s + 2)-th step ks+2 > N%+i is satisfied. It means that the excessive coordinates can be annihilated by the action of BsJ-^nGA. Clearly, s < r - 1. In the case of the (s + 2)-th step, we come out from ws+i obtained as follows. We use the uniquely determined element as+i(ws) of ~Vs+i = (-8^+1^1 GA)/Hss+l) to obtain Ps+1 and so the element ws e T*TARm+1 is after the (s + l)-th step transformed onto w s +i = £(as+i(ws), ws). This alghoritm yields the classification theorem for natural T-functions defined on T*TA. Theorem 1 Let A = D£/7 he a Weil algebra of width k, d i m M = m > k + 1. Let ZQX : TAM —> T£M be the embedding from Proposition 1, C be a basis of A and B0 a basis o/Der(D£). Further, let B be a basis of functions defined on T*TAM constructed from operators AM ; B» from 1 applying the operation ~. Then all natural T-functions fM : T*TAM —> R are of the form h{Lu{c)TM,

IM.h, ^M;2i • • • ' ^M*2

' ^M;r^ • • • ^M;r

)

CO

for any smooth function h of a suitable type, natural functions Ihx selected directly from B and natural functions I^.s obtained in the s-th step of the reccurent procedure.

302

2

Natural operators T -> TT*TA

In this section, we are going to classify all natural operators AM : TM —> TT*TAM lifting vector fields on a manifold M to T*TAM. It follows from the very beginning of the paper that they are identified with natural T-functions / M : T*TTAM -> R gM = / M ° ^ M o s T /i M linear on fibers of the vector bundle T*{T*TAM) -> T*TAM. Therefore we are going to discuss base natural T-functions fM : T*TD®AM^> R from 7 for the Weil algebra D A and their linearity. In what follows, every base T-function / M will be assigned either the zero operator or a natural operator JM • TM —> TT*TAM. We shall need the coordinate expressions. Let yla denote the canonical coordinates on TARm+2 and qfdyi define the additional coordinates on T*TARm+2, l m > k + 2 = width(A) + 2. Further, let z a = dy^ and pfdyi + sfdz^ determine the additional coordinates on x*TTARm+2. Then the coordinate expressions of tTARm+2 o s~\Rm+2 from the very beginning of Section 1 yields sf = qf. Further, every base operator T R m + 2 -> TT*TARm+2 is identified with a function T*TTARm+2, which is of the form ^a(Jo + 1 ^.!/a.9f )P? - Q?Uro+1X,vlq?)£ %

where Y£ = dy

a

(8)

and Qf = dqf denote the additional coordinates on

In the first step, we investigate A] : T*TTARm+2 -> R and sort them into four classes. Among them we select those which are properly linear, construct Aj : TRm+2 -> TT*TARm+2 and give their geometrical description. If the multiindex 7 does not contain k + 1, write AJ to distinct it from AJ : T*TARm+1 -> R. Let T denote the flow operator associated to the tangent bundle and V : T —> TT denote the vertical lift. Further, let E —> M be a vector bundle and f : E —> E be & vector bundle morphism over the identity on M. The vertical tangent bundle VE is canonicaly identified with E y-M E. Let us denote by Wf the map ids x / in this identification. Hence Wf is a vertical vector field on E. We state the following 1 proved in 10 . Lemma 1 Let Y : TM —> TTM be a linear vector field, 6 . Then there is a unique vector field Y : T*M —> TT*M satisfying YotMosJ^

— Y. Moreover,

Y is dual to Y. Particularly, for the flow prolongation Y = TX of a vector field X it holds Y = T*X. Let us sort A] : TRm+2

-> TTTARm+2

into four classes, investigate the

303

linearity of AJ and construct Aj. (i) kf = T o A^ for j ^ k + 1. It follows from ?? that such an operator is properly linear and A^ = T* o Aj. Computing its coordinate form, one can easily verify that the map lmm defined before 4 yields the value pP for hr, . (ii) As for Ay®+V the map lmm returns the value pf +1 and A fc+1 = 0. (Hi) A? + l = VoAj for j / k + 1. The map lmm yields the value q1? in this case and the operation ~ yields the zero operator. ~3k+l

^r;

(iv) A f ^ 1 = W(L(T0)). It follows from ?? that A fe+1 = W(L{c)) = W*(L(c)) = -W(L*{c)), where L*(c) denotes the dual map to L(c). In coordinates, the linearity appears in zla and the map lmm yields qf.+1 for Af+j 1 . The following Theorem gives the main result of the section. Theorem 2 Let A be_a Weil algebra of width k, d i m M > k + 2, C be a linear basis of A and T> a linear basis ofVerA. Further, let AD be absolute natural operators T —> TTA associated to D &T>. Then all natural operators AM •• TM -* TT*TAM lifting vector fields on M to T*TAM are of the form AM(X,w)

= gcM(X,w)T*

o

(L(c)MTAX)

+ gD-,M(X, w)T* o AM + gdM(X, w) W*{L{d)).

(9)

A

for some natural T-functions gM, gD,M o,nd gM defined on T*T M and corresponding to two sets of c, d forming C and a set D forming V. Further, X is a vector field on M and w is an element ofT*TAM. Proof: We investigate all natural T-functions fM : T*TTAM -> R. Taking into account their coordinate forms, the formula 8 implies

j=»L

A_«L

dp? m=#=(» Wo

dz

«

A

IK=P7=O)

d(

(10)

lT

Let PM • TM —-> M denote the tangent bundle projection and 0M the zero vector field on M. Let SM • TT*M —> T*TM denote the natural equivalence by Modugno and Stefani. Then the space STAM(0T,TAM) C T*TTAM is inj4 variant in respect to the action of T*TT -morphisms and naturaly equivalent to T*TAM. Therefore natural T-functions fM • T*TTAM -> R constant on fibers of the natural bundle pT*TAM o s~\M : T*TTAM -> T*TAM (in coordinates independent on z%a and pf) can be identified with natural T-functions defined on T*TAM. Particularly, every V o A is identified with A for a natural operator A : T -> TTA.

304

In what follows, we prove that no base natural function defined on T*T^®A except those obtained in the initial step of the reccurent procedure described before Theorem 1 generates a non-zero natural operator T —> TT*TA. In order to show it, consider natural functions generated in the (s + l)-th step of the procedure, i.e. Is+1. Define Is+1't by Is+1,t+1(w) — £(at+i)o . . . o 1 1 s+1 £(a2)(w) for 1 < t < s. Clearly, /*+ ;»+ = J , i.e. the originally given natural function. _ _ _ Further, put ./«+i;t+i = Dt+i tfjs+ut+i^ b y j»+i;t+i denote t h e /_ t h component of p+1 TT*TA. It completes the proof, n We have the following corollary for natural transformations T*TAM —> T*TAM Corollary 1 Let A be a Weil algebra of width k, d i m M > k + 1. Then all natural transformations OM '• T*TAM —> T*TAM satisfying QT^M ° aM(T*TAM) = T*TAM are of the form T*s0 o pr 2 o gdM(X,w)W*(L(d)) A A for any natural equivalence T M —> T M induced by a Weil algebra automorphism so : A —> A and d G A. Proof: The only thing to discuss is the verification of the assertion for d i m M — k + 1. It is given by the fact that we are not interested in nonabsolute operators in Theorem 1 and 2 which enables to do all proofs for dim M decreased by one. n In the following assertion we give some result concerning the composition of natural operators T —> TT*TA with some natural transformations on rprrt^rpA

Proposition 3 Let A be a Weil algebra of width k, d i m M > k + 2. Then every natural operator AM : TM —> TT*TAM is of the form OM ° ^Tfy for a natural transformation OM • TT*TAM -> TT*TAM over idT,TAM. Proof: We are going to verify the assertion for each of three sets of the base natural operators from 7 and prove that the coefficient natural Tfunctions from 7 are obtained as the composition of a natural T-function defined on TT*TA with the flow operator T*TA. As for the first set, we have

305

T*LM{C)TM — TL*M(c) o T*T$, where L*M(c) denotes the dual linear map to LM(C). Let D € VerA be tangent to a one-parameter subgroup "f(t) in AutA. Then T* o KDM = ^ | o ( T * 7 ( t ) ) = £ | o (7 _ 1 (*)* °P°T*TA), where * indicates the pull-back, p denotes the tangent bundle projection. The fact that Jj, (7 _ 1 (i))* is a natural transformation in question is directly verified. The assertion is obvious for the third set of base natural operators. For the coefficient natural T-functions, consider a vector field X on M and w € T*TAM. Then we have LM{c)T^X(w) =< pT.TAM o TL{c)*{T*T^X(w)), TqTAM{T*T$X{w)) > for the tangent and cotangent bundle projections p and q while <, > denotes the tensor contraction. The fact that qt^o T£Y = Y and the definition of W(L(c)) : T -> VTTAM

yield Lui^fT^Xiw) = A for A = W{LM{c)) o STAM{T*T^X{W)), where s denotes the natural equivalence by Modugno and Stefani. Further, we have / = I opT.TAM(T*Tj^X(w)) which verifies the assertion for the second set of base coefficients. It completes the proof, o 3

Natural T-functions on TAT* and natural transformations rpArp*

rp*rr
In this section, we are going to find all natural T-functions defined on TAT*. For A = D j , we obtain them immediately since there is the natural equivalence T[T* —> T*T[ by Cantrijn, Crampin, Sarlet and Saunders, 2 . However, there is not a natural equivalence of this kind for a general Weil algebra since it is proved in 3 that it does no exist for A = D£ whenever k > 2. We start from the construction of a TAT*-invariant map A*M : TAT* M —> A determined by a natural operator AM '• TM —> TTAM. Its components yield the set of natural T-functions defined on TAT*M. Their definition is given by the following formula AMX{w)

= TAev(w, KM o AMX O TAqM(w)) A

A

(12)

for the canonical isomorphism KM '• TT M —*• T TM, the cotangent bundle projection qM, w G TAT*M and the tensor evaluation ev. Let us verify the correctness of the definition, i.e. the naturality condition. It reads A*MX(w) = A*N{Tip o X o ip-1) o TAT*ip{w) for any local diffeomorphism tp : M -> N'. It holds A*N(TtpoXotp-^oTAT*oip(w) = TAev(TAT* T*M, 7 and jArj — KM O .AM-X" ° jA{qMl)Then the last expression equals to T ^ e v f j ^ T V o -y),TAT
306

jAev(f,rj) = TAev(jAi,jAr]) = TAev(w,nM o AMX o jA(qM o 7)) = A A T ev(w, KM ° AMX O T qM{w)). It proves our claim, n In the following investigations we shall be interested only in Weil bundles associated to monomial Weil algebras, i.e. those A = D £ / J for which J is generated only by monomials. For such Weil algebras, we have a linear basis formed by monomials and the so called zero section 1: A —> D£ introduces in the simple way the coordinates on TAM. Consider natural operators A? and (A^ )*. Denote by F? a natural function which is an a-th component of A^* such that a, f3 and fj, satisfy /3+/J, = exit is easy to verify that Fj* does not depend on the choice of a and f3 and so it is well defined. We prove the assertion giving the classification of all natural T-functions fM • TAT*M -» R. Proposition 4 Let A be a monomial Weil algebra of width k and m = d i m M > k. Then every natural function fM '• TAT*M —> R is of the form h(FM ) for all couples (n,j) corresponding to base absolute natural operators TM —> TTAM and any smooth function h of a suitable type. Moreover, for m > k + 1 every natural T-function fM '• TAT*M —• R is of the form /I(A(C)M'?MI -^M,?) for anV s e * °f c forming a basis of A. Proof: Let m> k. By general theory, we are searching for all GJ^t"1-invariant functions defined on (TAT*)oRrn+1. Therefore we investigate the the action of G^t,1! on (TAT*)oRm+1 It will be necessary to introduce coordinates on (TAT*)0Rm+1. Let xi be the canonical coordinates on R m + 1 and let pt % defined by pidx be the additional ones on T * R m + 1 . The so called zero section 1 : A —> D£, mentioned before the present Proposition introduces the coordinates xza,pf on TAT*Hm+1. Then the transformation laws for the action of G ^ on (TAT*)0Rm+1 are of the form

< = <.../^L\ • • • *'«..

P? = ~<...A • • • *W~ 7 (13)

for all decompositions c*i,..., as of a and 7 1 , . . . , 7« of 7 for any 7 C a. Consider operators A^ : T R m + 1 - • TTARm+1 from 1. We can extend the formula 12 also to non-natural operators hfj, which enables the unified notation for all of A^. Clearly, it holds Ff = Ff for natural A^. Taking into account 3 and the property of A being monomial we obtain their coordinate forms as follows d

"

=

(a-7^7-^)!^-^" 7 ' 6

» =a - *

W

By general lemmas, , Chapter VI, every natural function must be of the form h{x%ap1) for all combinations of admissible multiindices a, 7 and a

307

smooth function ft of a suitable type. The last expression can be replaced by h{G^x\p1,x)p\) for |a| > 2 and \6\ = r. Let us consider the immersion element i defined after 3. By the map Imm defined in n and recalled before 4, the recent expression is transformed onto h(pj, 0 , . . . , 0,pf), where * indicates the transformed values of p's. It follows from 14 and the fact that dim M > k + 1 that p\ can be annihilated by B^^ except the case i = k + 1 corresponding to a non-absolute natural operator. By Bs£x we denote { j S + V G G'+^jfo = JS«*R™+I}In the following step one can verify by direct computations the invariancy of all Gj in respect to the map Imm using 13 for x = i. It results into the coincidence of C* with pj over TAq^} (i) for the cotangent bundle projection It remains to annihilate all pf corresponding to non-natural Gj. It is done by the action of St(i) = GA, the correponding formula for which reads Pj = ^ i T ^ r - 7 Setting /i = a —/3, we have p*~ = 5j 7 p"~ and the action of GAC\B]^+1 yields p? = P°l~ ^ja-pPr Consider the coordinate expression of G\j\ Then it is easy to verify that Gj is natural if and only if for any multiindex 7 satisfying — | y| < r the relation TJT7^13 G I implies T 7 € / . Let 1 TjT $ I for any 7 C / 1 , Then the recent implication is fulfilled in the trivial way for any admissible 7 < a, which results into the naturality of Gf. It is the contradiction with the assumption of non-naturality of &*, which follows that all p^ corresponding to non-natural G^ can be step by step annihilated by alj starting with the largest rank of /x. It proves the first claim. As for natural T-functions, the proof is very analogous. Consider a linear basis of A formed by monomials ra. We investigate the action of a slight modification of GA, namely the stability subgroup of i and j ^ g^£TT) (for more details see n ) . Then the values of base non-absolute T-functions coincide with Pm+i and by means of the recently mentioned subgroup all excessive pf can be annihilated. It completes the proof. • In what follows, we are going to investigate natural transformations TAT*M —> T*TAM for monomial Weil algebras. For a general Weil algebra A and a linear function / : A —• R, recall the natural transformation (sf)M • TAT*M -+ T*TAM defined by Doupovec, 3 as follows (sf)M(X)(Y)

= f o TA(<

PT*M(X),PTM(KM(Y))

>)

(15) A

for the tangent and cotangent bundle projections p, q, X G T T*M and Y G TTAM such that Y G TTAm{x)TAM. This claim means that (sf)M are natural transformations satisfying qx^M ° ( S / ) M = TAqM- Clearly, without

308

loss of generality we can limit ourselves to the natural transformation of this kind. For a monomial Weil algebra the linear basis of which is formed by 1 and the monomials Ta, put / = xa. Then we obtain the natural transformations (S<*)M for all admissible a including the zero one. As for their coordinate expressions, consider the coordinates Xla on TAM and the additional coordinates P« on T*TAM defined by P?dX%a. Clearly, Xia=xia. Then 15 yields the coordinate expression of sa given by (Pa)] = rt-v1~~1 • In what follows, we give the construction of natural T-functions defined on TAT* by means of natural transformations TAT* -> T*TA. Let A = R x N be of order r and iV be its nilpotent ideal. Kolaf in 5 defined the factor Weil algebra As = A/Ns+1, the underlying Weil algebra of order s < r and the underlying s-th order Weil functor TA". Let (PS)M • TAM —> TA'M denote the projection natural transformation and consider a natural transformation tM : TAT*M ->• T*TA'M XTASM TAM. Clearly, there is a natural injection (P*S)M XidTAaMidTAM • TAT*M -+ T*TA°M XTA,MTAM - • T*TAM which enables to consider the natural transformations of the recent kind as natural transformations TAT*M -> T*TAM. The symbol (p*s)M denotes here the pull-back of (PS)M- The following assertion, which is easy to verify, yields the construction of a natural T-function TAT* -> T*TA Proposition 5 construction of nf Let AM '• TM —> TTAM be an operator such that T(ps)M ° AM • TM -> TTA°M is natural and tM : TAT*M -+ T*TA'M XTA3MTAM be a natural transformation. Then the function /j^" 4 : A T T*M -» R defined by fMA(w) =< tM(w), AM ° TAqM(w) > is natural. noindent The following Theorem gives the result for natural transformations. Theorem 3 last Let A = Drk/I be a monomial Weil algebra, M be a manifold satisfying d i m M > k. Then every natural transformation tM '• TAT*M —> T*TAM satisfying qTAM o aM(TAT*M) = TAM is of the form tM(w) = hM(w)(sp)M(w)

(16)

for the base natural transformations {S^M defined after 15 and any natural functions hM : TAT*M -> R. Proof: We use the very similar technique to that applied in the proof of Proposition 8. As for the classification result presented in Proposition 8, the assumption for d i m M can be decreased by one since we consider only natural, i.e. on vector fields independent functions. In what follows, we shall need the transformation laws for P?. They are determined by 3 modified to the case of a monomial Weil algebra. They are of the form ^T

=

a t l ^ i - . < » ^ 7 i ••• S 7»-^T 7 I n t n e ^ r s t

ste

P>

cons

ider a general nat-

309

ural transformation corresponding to s = r. Further, consider natural operators A.?M : TM —> TTAM for maximal multiindices f3. Then we obtain fj^ j from construction of nf. By general lemmas, 6 , Chapter VI, we have Pf = h^x^p^pj for all combinations of admissible multiindices rj and fi. By Theorem 8 and the application of the same annihilation technique as that in its proof we have

fu''=tyFt)*7 = l$(Ft&Ff.

(17)

j

It follows that P- = H^{F^)p0j taking into account the assumption d i m M > k uniquely determining P? from 17. Then P? — Hl3(Ffl)p'j which corresponds to the natural transformation H^{F^)sp. If we subtract the natural transformation H^{F^)sp from t considered in the beginning of the proof, we obtain the situation corresponding to s = r — 1 from construction of nf. We can modify the first step to the following ones and by the gradual subtraction of H/3(FfJ')s0 finally obtain all Pf equal zeros except P° which correspond to the natural transformation HQ(F^)SQ. It completes the proof. • Acknowledgments This contribution was presented at the conference "Differential Geometry Valencia 2001" hold on the occasion of prof. Naveira's 60-th birthday. The author was supported by the grants No. 201/99/0296 and No. 201/99/D007 of the GA CR. He was also partially supported by research projects MSM 261100007 and J22/98 261100009 CZ 204709 References 1. J. R. Alonso, Jet manifold associated to a Weil bundle, Archivum Mathematicum Brno 36, 195 (2000). 2. F. Cantrijn, M. Crampin, W. Sarlet and D. Saunders, The canonical isomorphism between TkT*M and T*TkM, C. R. Acad. Sci. Paris 309, 1509 (1989). 3. M. Doupovec, Natural transformations of the composition of Weil and cotangent functors, to appear in Annates Polonici Mathematici , (). 4. I. Kolaf, On the Natural Operators on Vector Fields, Ann. Global Anal. Geometry 6, 109 (1988).

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5. I. Kolaf, Affine structure on Weil bundles, Nagoya Math. J. 158, 99 (2000). 6. I. Kolaf, P. W. Michor and J. Slovak, Natural Operations in Differential Geometry, Springer - Verlag , (1993). 7. I. Kolaf, Covariant Approach to Natural Transformations of Weil Bundles, Comment. Math. Univ. Carolinae , 99 (1986) 8. I. Kolaf and Z. Radziszewski, Natural Transformations of Second Tangent and Cotangent Bundles, Czechoslovak Math. J. , 274 (1988). 9. M. Modugno and G. Stefani, Some results on second tangent and cotangent spaces, Quaderni dell'Universita di Lecce , (1978). 10. J. Tomas, Natural T-functions on the cotangent bundles of some Weil bundles, Diff. Geometry and Appl., Proc. of the Satellite Conf. of ICM in Berlin, Brno , 293 (1998). 11. J. Tomas, Natural T-functions on the cotangent bundle of a Weil bundle, Czechoslovak Math. J. , to appear ()