Geometry and topology of submanifolds 10, differential geometry in honor of prof. S. S. Chern [Shiing-Shen Chern], Peking university, China, 29 aug - 3 sept 1999 ; TU Berlin, Germany, 26 - 28 nov 1999

n = M"

0 0 0 0 0 2X

XT 0 0

X

e.

and

X2 = jj=dx - V2d2, Xi = di(i--

and its dual frame 1 / 1 • —z=dxi H j=dx2, u)2 = —j=dx 2 , u' = dxt (i = 3 , . v2 2v2 \/2

95 Then the matrix 6 of connection forms u] = w'(VAj) equals

where F = 5 7 7 ( ^ ^ 2 , • • ■, fa-dx?). Hence, the sections [q, u(em,..., e 2 ,1)], E; = ±1, where q projects to the orthonormal frame \{X\ + X2), ^{Xi — X2), X3, ...,Xn are parallel. In particular, the Lie algebra t) of the holonomy group is contained in the image under A of the stabilizer of all u(em,.. ■, £2,1)- This image equals n, where the matrices of so(n) are written with respect to the basis ei + e2, e.\ — e2, e 3 ,..., e„. Now fix a point x G U. By the Ambrose-Singer theorem all linear maps (d8 + 8 A 6){da, 82)(x), a = 3 , . . . , n are contained in I). One calculates (d8 + eA8)(da,d2)

/ 0 0 $ \ = 0 0 0 V 0 2$ T 0 /

where 2\f2 dxadx3' dxadxn Consequently, for the generic choice of / we obtain I) = n and the space of parallel spinors is spanned by the sections [q, u(£m,... ,e2,1)]. For instance, all [q, u(em,.. ■, £2,1)] are pure parallel spinors. □

5

The case of signature (m,m) or (m + l,m)

Here we consider n-dimensional pseudo-Riemannian manifolds of signature (m,m) or (m + 1, m). Let (•, ' ) m m and {•, ■) be those scalar products on E 2m and K2m+1, respectively, +1 /m+l,m which satisfy «,-={-; ^iisodd (12) J 1 1 11 j is even for all 1 < j < n. From (2) and (4) we see that the restriction of $ m , m to Cm,m is an isomorphism onto E(m) and the restriction of $ m +i, m to Cm+i,m is an isomorphism onto R(m)©R(m). In particular, the spinor representations Am_m and A ro+ i >m are real. Therefore, we consider A ^ m and AJ^ +lm in this section. Elements of these spinor modules are called real spinors. Real pure spinors are exactly the real and imaginary parts of those pure spinors in the complexified spinor modules which have real index m. We introduce another basis of Rm'm which is more convenient for some of the following calculations. We define ft = e 2i -i + e2i,

fr = e2i-i - e2i,

(1 < i < m)

96 and calculate $mAf?)

=

E®...®E®(°Q

*«,«.(/,"")

=

E®...®E®[

^\®T®..

\

" }®T®...®T.

(13)

Next we assign to each real spinor in AmiTO or A m + i j m a certain m-form. Let (•, •) be the standard scalar product of M2"*. We define by b(u, v) = (u, eie 3 e 5 . . . e 2 m _ l • v) a Spin+ (TO, m)-invariant bilinear form b on A m j m which satisfies -v) + (-l)mb(X-u,v)

b(u,X

= 0

m m

for all X € R ' , u, v G A m , m and a by b'(u, v) = (u, eie 3 e 5 . . . e 2 m + i • v) a Spin+(m

+ 1, m)-invariant bilinear bilinear form V on A m + i ? m which satisfies + {-l)m+lb'{X

b'{u, X-v) for all X € Rm+1'm,u,v and b'u, by

-u,v) = 0

e A m + i ? m . For u G A m i m and u' e A m + 1 , m we define m-forms bu

bu{Xx,...,Xm)

—}Y^s&i(J'KXo{i)Xrr(2)---Xa(m)u,u)

:= a€5m

6^,(X1,...,Xm)

:=

— Y, m

'

sgna-b'(X^l)X
*€Sm

P r o p o s i t i o n 5.1 1. If u € A m j m is a real pure spinor then bu is decomposable. gously, if u g A m +i, m is a real pure spinor then b'u is decomposable.

Analo­

2. We have 6«(1,1,..,1) = *u(l,l,...,l) = ( _ 1 )

2

(°"l _ ff2) A (CT3 - (T4) A . . . A (<X2m-l - 02m),

where o\, U2, • ■ ■, °V» w a dwaZ ftasis o/ ei, ei,...,

e„.

Proof. It follows from the definition of 6 and B ' that 6 ( u ( e m , . . . ,£\),u{5m,... b'(u(Em, ...,ei),u(Sm,

,5\)) yt 0 ...,6i))

^ 0

iff

Ej = — 5j for all j = 1 , . . . , m

iff

£j■ = -<5,- for all j = 1, . . . , m .

By (13) f o r / , . e { / + , . . . , / + , / r , . - - , / ^ } A ■... • / < m « ( l , 1 , . . . , 1) = c u ( - l , - 1 , . . . , - 1 ) c # 0 holds if and only if {/«,-■-,/i m } = {/i~> • • ■ >/m}- This proves assertion 2. and assertion 2. implies assertion 1. □

97 Remark 5.1 In the following we only list results for signature (m + l , m ) . One obtains analogous facts for signature (m, m) if one omits the last basis vector, the last coordinate etc. Next we will compute the isotropy group H' C Spin(m + 1, m) of u(l, 1 , . . ., 1). Since - 1 ^ H' the map X\H' : H' —> X(H') is an isomorphism. We denote X(H') C SO(m+ l , m ) by H. Note that H and H' are not connected, since e.g. —t\&i € H' but —eie2 ^ S p m + ( m + 1, m). Denote by H+ and H'+ the intersections H n 5 0 + (m + 1 , m) and ff' n Spin+ (m + 1 , m), respectively. P r o p o s i t i o n 5.2 We ftaue

/ ^ B 0 (^T)- 1 ff = <■ V o y

X\

f

0 I

;

2BT(AT)~l + 2A-1B + yTy = o 2A-1X + YT = 0 det 4 = ± 1

and

eH

detA = l

ro+1 witt respect to the basis f. ff+ w t/ie semi-direct product of SL(m, K) and the nilpotent group

/i,/r,-,/;,ewi o/R ''

N :

B -Em 0 Em 0 -2XT

X \ 0

B e R(m), BT + B + 2 X X T = 0, X €

(14)

1J

/n particular, H+ is connected. Proof. An element h' € Spin(m + l , m ) preserves the line C « ( l , 1 , . . . , 1) if and only if A(ft') preserves the optical structure defined by C « ( l , 1 , . . . , 1), i.e. if it preserves Uuci,...,i)Hence we have h e H if and only if h e SO(m + l , m ) , ft preserves the optical structure defined by Cti(l, ! , . . . , ! ) and it preserves ft^n ^ up to sign. Thus, h G H if and only if h g 5 0 ( m + 1, m), /i(span{/ 1 +,..., / + } ) = s p a n j / ^ , . . . , / + } and h ■ b'u(h x) ±il (i,...,i) n holds, where h ■ 6„( lr .. n = b'^ ^ if and only if h £ i ? + . Corollary 5.1 A» : spin(m + \,m) cally onto (A X,t>' =

{

0 \ 0

B -AT -2xT

—> so(m + l , m ) maps the Lie algebra (/ of H'

x \ 0 }\x€M.n, 0 /

A,B eu(m),

trA = 0, Br =

isomorphi-

-B]

C so(m + l , m ) toftere so(m + l , m ) e 2 m + 1 o/R"* 1 "™.

w described

with

respect

to the

basis /{*",...,/+, / x , . . .

,fm,

98 Corollary 5.2 The Lie algebra t}' of the isotropy group of u{\, 1,..., 1) e A m+ i >m with re­ spect to the Spin(m + \,m)-action is spanned as a subalgebra o/spin(m + l,m) = spanR{ejej | 1 < i < j < 2m + 1} by

ftff ft ft

(l
/i + e 2 m +i

(1 < * < m)

eie2 - e2i-ie2i

(1 < i < m).

Now let (M, g) be a pseudo-Riemannian spin manifold of signature (m + 1, m) or (m, m) with Spin-structure Q. Here we consider the real spinor bundle S associated to Q with fibre Am>m = R2™ or A m + i i m = E2™. Sections in 5 are called real spinors. If (M, g) is space and time oriented the bilinear forms 6 and V on Am?m and A m + i j m define bilinear forms on S. Proposition 5.3 Let (M,g) be a pseudo-Riemannian spin manifold of signature ( m + l , m ) or (m,m) which admits a real pure parallel spinor field ip G T(5). Then M is a foliation with m-dimensional totally isotropic leaves such that for the distribution U c TM which is tangential to the leaves the following holds. 1. U is parallel. 2. The restriction of the Riemannian curvature tensor R of (M, g) to U satisfies t?R{X,Y)\u forallX,Y

=Q

€X{M).

S. The image of the Ricci map is contained in U. If, in addition, {M,g) is space and time oriented then 4- there exists a nowhere vanishing parallel decomposable m-form u on M with u(Xi,.. ■, Xm) = 0 if there is an i e { 1 , . . . , m} with Xi £ U. Vice versa, if (M, g) is a simply-connected pseudo-Riemannian spin manifold of signature (m, m) or (ro+1, m) which is a foliation with m-dimensional totally isotropic leaves satisfying Conditions 1. and 2. then (M,g) admits a real pure parallel spinor field. Proof. This theorem is a corollary of Theorem 4.1. Since ip has real index m, the associ­ ated almost optical structure degenerates to an m-dimensional distribution U C TM. The existence of an m-form satisfying Condition 4- follows from Proposition 5.1. □ The next aim is to find metrics of signature (m-fl, m) and (m, m) with real pure parallel spinors. In the special case of dimension 7 this problem was solved by R.L.Bryant. He considers real isotropic (with respect to {•, •) A.)) spinors on pseudo-Riemannian manifolds of signature (4,3). In [K 98] it was shown that these spinors are exactly the real pure ones. In [Br 98] R. L. Bryant proved that any metric of signature (4,3) with a real isotropic parallel spinor can be written in a certain normal form with respect to special local coordinates.

99 Moreover, a generic metric defined on an open set U C E7 satisfying this normal form admits exactly one real isotropic parallel spinor (up to constant multiples) and its holonomy group is isomorphic to the identity component of the stabilizer of a real isotropic spinor. Furthermore, the generic such metric is not Ricci-flat. We can copy Bryant's proof to obtain the following generalization. Theorem 5.1 Let (M,h) be a pseudo-Riemannian spin manifold of signature (m + l,m) with a real pure parallel spinor. Then we can introduce local coordinates (x, y, z), x = (x\,..., xm), y = (y1,..., ym), z € R around each point of M such that m

h=

m

-dzi _ 4 £

dXiaY _ 4 £ aydvV

(15)

M=l

i=\

where g^ are functions depending on x, y and z and satisfying m da ^ " a ~ ^ = °for k = 1,2,...,m,

g{j = gj{ for i,j = 1,2,... ,m.

(16)

Conversely, if we choose functions g^ on a connected open set U C M2m+1 satisfying (16) and define a metric h on U by (15) then (U, h) is spin and admits a real pure parallel spinor. For the generic choice of gij this metric has exactly one real pure parallel spinor (up to constant multiples) and the holonomy group is isomorphic to the image under A of the identity component of the stabilizer of a real pure spinor. The Ricci curvature of h equals R.

0

w

d

'9ki

£> d29u

, ^

,

d2gkl

dgpl 8gk

,

k

.

In particular, h is not necessarily Ricci-flat. As already mentioned for signature (m, m) analogous statements are true where we have to omit in (15) and (17) the terms containing the z-coordinate. We recall the main points of the proof . We will use the Einstein rule for summation. Let P be the bundle of positively oriented orthonormal frames of (M, h). P is isomorphic to the bundle P' of those positively oriented frames for which / h=

0 -2Em

V

0

-1Em 0

0 \ 0 .

0

-l)

Denote by P* the bundle of coframes of P'. Let ip be a real pure parallel spinor on (M,h). This spinor defines ff-reductions PB of P, P'H of P' and P^ of P*. Let (w+, w_, ui2m+1) be a local section in P^, where io+ = {u)\,..., w™) and w_ = {u\,..., w"). Since Vsip = 0 the structure equations can be written {du+,duj.,dw2m+l)T

= -

0

-$T

0

A(a; + ,cj_,u; 2m+1 ) T ,

(18)

100 where $ and & are [m x m)-matrices of 1-forms with tr<3> = 0 and ^ = — \t T and 0 is a vector of m 1-forms. According to Proposition 5.3 tl> defines an m-dimensional distribution U. The annihilator Ann({/-L) C T*M of C/x is invariant under the action of H. In fact, H+ acts as SL(m,R) on Ann([/-L). If as above (u+,uj-,u2m+1) is a local section in P# then Ann([/-L) is spanned by aji,...,aj™. By (18) the system {wl,...,w™} is integrable. Since, furthermore, CJ = wl A ... A w™ is a parallel and, hence, a closed m-form we can choose functions yl,...,ym in a neighborhood of x € M such that span{wl,..., w™} = spanfiij/ 1 ,..., cfr/"1} and w = dj/1 A ... A dym. Since J7+ acts as SL(m, K) on Ann(f/-L) it acts transitively on bases of Ann(£/X) with determinant u. Hence, there exists a section (tD+,ui_,cj2m+1) in P^ with tildew- = dy := (rfj/i, • ■ • ,dym). Consequently, the bundle Pi, := {K,o;_,w 2 m + 1 ) G P*H | to. = A/} defined in a neighborhood of a; is a (local) reduction of P* to the kernel of the projection of H+ to SL(m,R), i.e. to iV. Now we consider a local section (to+,dy,to2m+l) in P^. By (18) dw2m+1 = —2©T A dy holds. Hence we can choose a function z defined in a neighborhood of x £ M such that oj 2m+1 = dz+padya for some functions p i , . . . ,p m . By (14) we can find an element v £ N such that ^•K,d 2 /,o; 2 m + 1 ) T = (a;;,d2/,d2)T. Hence, we can define the bundle P^ := { {co+,dy,to2m+l) e P*H | 0J2m+1 = dz) on this neighborhood. P^ t is a (local) reduction of P£, with structure group ^m

0 0

B

0

Em 0

0

B e K ( m ) , B T + B = o l C AT

1

where k — (m — l)(m — 2)/2. Now let (w+,dy,dz) be a local section in P£k. From (18) we know 0 = ddy = $ T A dy1" 0 = ddz = 20 T A and 0 are contained in span(d)/ 1 ,..., dym}. Again by (18) we obtain dco+ = dy A Q for a certain (m x m)-matrix Q of 1-forms. Consequently, we can choose functions X\,..., xm such that w+ = dx + g ■ dy for x := (a;i,..., xm) and a (m x m)-matrix g of functions depending on a:, y and z. More­ over, multiplying (co+,dy,dz)T by a suitable element of M* C TV we obtain a coframe (r;+, rf, Tfcm+i) = ( ^ + 3 • dj/, dy, dz) such that p is symmetric. We conclude h = - 4 (jy^T/r + ... + i % m ) - ??|m+1,

101 where r)+ = (rjt,...,

rfim) and r) = ( ^ ,. . . , % m ) and, hence, h = —dz2 — Adxidy' — 4gijdy'dyj,

(19)

where g = (0jj)<j=i,...,mNow let ft, = <7jj, (i,j = l,...,m) be functions on a connected open set U in R 2 m + 1 endowed with coordinates (x,y,z), x — (x\,... ,xm), y = (yi,...,ym) and define a metric h by (19). Then we can calculate the matrix & of connection coefficients with respect to the coframe (??+, rf, %™+i) = (dx + g ■ dy, dy, dz) and its dual frame A+ A,"

= =

dxt -gaidxa+dyi

i = 1,... , m i = l,...,m

We obtain

fe+(VX+))ij=1,...,m (•?T(vx;))y=1,...,m (fc.+l(V^))Fl,..,m

K+(V&))i=1,.. (77-(vaz))i=1,..

(^(vx-))tJ=:,...,;

(vr(vx-))t^..., (Thm+liVXj

))j = 1 ,..

%m+l(V9z)

$ *T e \ o -$ o , 0

-20T

0 /

where dgki, S—dVk axj .^iii

(* ' 9a;„

©i

=

-

d9ki n

aj

i

dxa

9gM

dyj

gg J

* ' )dyk

(20)

dgki, -^-dy . dz k

Hence, the values of # are contained in i) if and only if (16) is satisfied. If this is the case then [g, u ( l , 1 , . . . , 1)], is a parallel pure spinor if the projection of q is the orthonormal frame \{Xt + AT), | ( A + - AT), • ■ •, | ( A + + X-), ±(X+ - A " ) , dz. In particular, the holonomy group of the metric h defined by (15) and (16) is contained in H. Moreover, since (U, h) is space and time oriented it must be contained in H+. Now we will prove that for a generic choice of g = (
R =

for(fl(.,.)*+)) V(rfem+l(.R(',-)A+))

0 0

— RQ

-2Re

)x-))

(V7(R(., )XJ)) (r)2rn+l(R(-

tf(R(-,-)dz) r,-(R(-,-)dz) V2m+l{R{- , -)dz)

102 where Rt = d$ + $ A *, #* = d# + $ A * - # A $ T - 20 A G T , Re =rf©+ * A 6. We may assume that 0 G 17. By the Ambrose-Singer holonomy theorem it suffices to prove that all matrices of the form R(X, Y), X,Y £ T0U, generate f) as a Lie algebra. Using (20) we obtain d2gici d2giCi •R* ij = » » dxi A dyk + dz A dyk + iptj fi

ei =

R

A dyk + 9{ 0
= 13

fl—5

+

a

)dxi

a

A

a3

d2gki dxidxa

^

2

,dgaidgkj dgajdgki d gkj d2gki al a} ^ dz dxa dz dxa dzdxa dzdxa , d2gki d2gkj^ + + dz^y--dz^)dzAdyk ^ where tptj, &i, ipij £ spa,n{dyk A dyj | k, I = 1,..., m}. Hence, if we choose 1

*(

fly = 2C»XkXl

!

+

2

2CijZ '

•

■

l 3 =

'

,

'''''m

where Cy and cf,' are constants satisfying cy = c,,, cfj = cj* = c*' and the trace condition ckk\ = 0, then

holds for all i = 1,..., m and we have in 0 G Z7

/ ci

o

o\

R(dx,,dyk)

=

0 -(<7<)T 0 , \ 0 0 0 /

R{dz,dyk)

=

/ 0 0 Ck\ 0 0 0 , \0 -2CJ 0 /

where Clk = (cjj) u=1 ,.., m ;

where Ck = (cku ...

,ckm)T.

Hence, if the constants cy and d[] are taken sufficiently generically, the space {R(X, Y) | X, y G Tof7} contains all matrices of the form (A

0 X\ 0 -^T 0 V 0 -2XT 0 /

103 with tr A = 0. The assertion now follows from the fact that these matrices generate the Lie algebra fj. Moreover, it follows, that there is an open, dense condition on the 2-jet of the functions g^ which ensures that the corresponding metric will have holonomy H+. Equation (17) can be verified by direct calculation. □

References [Bm 81]

H. Baum, Spin-Strukturen und Dirac-Operatoren liber pseudoriemannschen Mannigfaltigkeiten, Teubner Leipzig 1981.

[Bm 98]

H. Baum, Twistor spinors on Lorentzian symmetric spaces, Sfb 288 Preprint No. 313 (1998).

[Br 98]

R.L.Bryant, (4,3)-metrics with a parallel null spinor, personal communication (1998).

[C 38]

E. Cartan, Legons sur la theorie des spineurs, Hermann, Paris, 1938.

[EK 62]

J. Ehlers, W. Kundt, Gravitation, an introduction to current research, L. Witten (et.), New York, Wiley, 1962. I. Kath, G^/2yStructures on pseudo-Riemannian manifolds, J. Geometry and Physics

[K 98]

27 (1998), 155-177. [Ko 94]

W. Kopczyiiski, Pure Spinors in Odd Dimensions Preprint ESI 140 (1994).

[KoT 92]

W. Kopczyiiski, A.Trautmann, Simple spinors and real structures, J. Math. Phys.

[LM 89]

H.B. Lawson, M.-L. Michelsohn, Spin geometry Princeton Univ. Press 1989.

[N 96]

P. Nurowski, Optical geometries and related structures, J. Geometry and Physics 18 (1996), 335-348. I. Robinson, A. Trautman, Optical geometry, in New theories in physics, Procs. of the XI. Warsaw Symposium on Elementary Particle Physics, Kasimierz 23-27 May 1988, ed. by Z. Ajduk, World Scientific, 1989.

33, No.2 (1992), 550-559.

[RT 87]

[S 74]

R. Schimming, Riemannsche RMume mit ebenfrontiger und mit ebener Symmetric, Math. Nachr. 59 (1974), 129-162.

[T 85]

A. Trautman, Optical structures in relativistic theories, Societe Mathematique de France, Asterisque, hors serie (1985), 401-420.

Ines Kath Institut fur Mathematik der Humboldt-Universitat Berlin Sitz: Rudower Ch. 25, 10099 Berlin [email protected]

Received January 31, 2000

104

Geometry and Topology of Submanifolds X eds. W. H. Chen et al. (pp. 104-112) © 2000 World Scientific Publishing Co.

Homogeneous geodesies in homogeneous Riemannian manifolds — examples

OLDRICH KOWALSKI, STANA NIKCEVIC AND ZDENEK VLASEK ABSTRACT. In [8] the first author and J.Szenthe proved, for a general homogeneous Rie­ mannian manifold, some existence theorems on geodesies which are orbits of one-parameter groups of isometries. The aim of the present paper is to provide examples showing that the results from [8] are optimal in some sense. KEYWORDS: Riemannian manifold, Homogeneous space, Geodesies as orbits, fc-symmetric space. 1991 MATHEMATICS SUBJECT CLASSIFICATION: 53C20, 53C22, 53C30

1. Introduction. A connected Riemannian manifold (M, (,)) is said to be homogeneous if its full isometry group I(M) acts transitively on M. Then M can always be written in the form M = G/H, where G C I(M) is a connected Lie group acting transitively and effectively on M and H is the isotropy subgroup at some point o e M. In general, we have more than one choice for the group G. A geodesic 7(f) through the origin o is said to be homogeneous (with respect to G) if it is an orbit of a one-parameter subgroup of G. This means that (1)

7(r)=exp(rX)(o),

T 6 K,

where X is a nonzero vector from the corresponding Lie algebra g. Obviously, if J(T) is homogeneous with respect to some isometry group G, then it is homogeneous also with respect to any enlarged group G1 of isometries (but the converse is not true, in general). There exist special homogeneous Riemannian manifolds, so-called g.o. spaces, on which every geodesic is homogeneous with respect to the largest connected group of isometries. For example, all symmetric spaces are g.o. spaces and, more generally, all naturally reduc­ tive spaces (see [4]) are g.o. spaces. Yet, the class of all g.o. spaces is broader than that of naturally reductive ones. The first example was given by A.Kaplan in 1983, [3]. Since that time, an extensive research has been done in this direction. We mention especially [10] where all those g.o. spaces are classified up to dimension 6 which are in no way naturally reductive. Also in [10], many references are given to both naturally reductive spaces and g.o. spaces. From the recent time we mention the joint paper [7] of the first two authors and that of C.Gordon [1]. The first and the third authors have been supported by the grants GA CR 201/99/0265 and CEZ J13/98113200007, the second author was supported by the grant DAAD and SFS Project #0401.

105 A natural related problem is the following one: consider a general homogeneous Rie­ mannian manifold (M, (,)) = G/H. How many homogeneous geodesies (if any) can be found in such a space? V.V.Kajzer [2] proved that there is at least one homogeneous geo­ desic on a Lie group with a left-invariant metric. Then the first author and J.Szenthe [8] generalized this result (using some ideas from [2]) to the general homogeneous Riemannian manifolds. The main result from [8] can be summarized in two theorems: T h e o r e m A. Let (M, (,}) = G/H be a homogeneous Riemannian manifold (i.e., G acts on M transitively and effectively as a group of isometries). Then G/H admits at least one homogeneous geodesic through the origin o 6 M. T h e o r e m B . //, in addition, the group G is semi-simple, then M = G/H admits mutually orthogonal homogeneous geodesies through the origin o, where m = d i m M .

m

Now, some natural question arise. For the simplicity, a finite family of geodesies through o is said to be linearly independent if the corresponding initial tangent vectors are linearly independent. Wc put the following problems: P r o b l e m 1. Let (M, (,)) = G/H be a homogeneous Riemannian manifold where G denotes the largest connected group of isometries and d i m M > 3. Does M always admit more than one homogeneous geodesic, up to a reparametrization? P r o b l e m 2. Suppose that (M, (,)) = G/H admits m = d i m M linearly independent homogeneous geodesies through o. Does it admit m mutually orthogonal homogeneous geodesies? In this paper we show that the answers to both problems are negative. From our examples we shall also see that a solvable Lie group with a left-invariant metric can admit m mutually orthogonal homogeneous geodesies, whereas there is a homogeneous space {G/H, (,)) satisfying [g, g] = g (thus G is "opposite to solvable" but not necessarily semisimple) which admits m linearly independent homogeneous geodesies but not m orthogonal ones. 2. T e c h n i c a l i t i e s . Let, as before, (M, (,)) = G/H be a homogeneous Riemannian manifold with a given origin o and let g and fj denote the Lie algebras of G and H, respectively. Then G/H is a reductive space (cf.[4]) in the sense that there exists an Ad(i7)-invariant direct sum decomposition (2)

g = m+h

where m C g is a linear subspace. (See [8, Proposition 1] for a rigorous proof.) There is a natural identification (canonical isomorphism) of m C g s TeG with the tangent space T0M via the projection 7r : G —> G/H = M . Using this natural identification and the scalar product (, )„ on T0M, we obtain a scalar product B on m, which is obviously Ad (H) -invariant. A nonzero vector X e g for which the trajectory (1) is a geodesic curve is called a geodesic vector. We shall use the following result (cf.[10, p.194], and also [1] and [7]).

106 Lemma 1.1. Under the previous assumptions and notations, a nonzero vector X £ g is geodesic if and only if (3)

B([X,Z]a,Xn)

=0

holds for every Z e n . (Here m written as subscript indicates the m-component of a vector from g with respect to the decomposition (2).) Now, the problem of finding all homogeneous geodesies for a given space (M, (,)) is reduced to the following steps: (a) We calculate the connected component G of the full isometry group I(M), or at least the corresponding Lie algebra g. (b) We find a decomposition of the form (2). (c) We choose a convenient basis {E\,..., Em} of m and a basis {Fi,..., Fr} of h. Then we look for the geodesic vectors in the form m

X = ^xiEi

(4)

•=i

r 1

+

£^Fj.

j=i

The equation (3) gives a system of m quadratic equations for the variables xl,a^ if we substitute Z = Ei for i = 1 , . . . , m. (These equations can be linearly dependent in general; see [7] for the explicit formula.) (d) We determine for which values of x1,...,xm and a1,.. .ar the algebraic equations obtained in (c) are satisfied. Such sets of values for which x1,..., xm are not all equal to zero define geodesic vectors. Let us remark that (M, (,)) is a g.o. space if, and only if, for every nonzero m-tuplet (x1,.. ., xm) there is at least one r-tuplet ( o 1 , . . . ,a r ) satisfying all basic quadratic equa­ tions. This is not the case in general. The following proposition is obvious: Proposition 2.1. A finite family (71, • • •, 7t>) of homogeneous geodesies through o € M is orthogonal, or linearly independent, respectively, if the m-components of the corresponding geodesic vectors are orthogonal, or linearly independent, respectively. 3. Three-dimensional examples. We start with the following result (see [11, p.321]). Proposition 3.1. Let (G, {,)) be a 3-dimensional non-unimodular Lie group equipped with a left-invariant Riemannian metric and let 0 be the corresponding Lie algebra. Then there is an orthonormal basis (X, Y, Z) of g such that the multiplication table of g has the form (5)

[X, Y] = ctY + PZ, [X, Z] = -yY + SZ, [Y, Z] = 0,

107 where a, 0,7,5 are real numbers such that a + 5 ^ 0 and aj + 06 = 0. The basis (X, Y, Z) also diagonalizes the Ricci form and the principal Ricci curvatures are given by the expres­ sions r{X) = ~a2-52-\{{5

(6)

r{Y)

= -a{a

+

+

1)\

6) + ±tf-p2),

r(Z) = -S(a +

S)+1-(p2-7').

It is obvious that the principal Ricci curvatures (6) are distinct in general. Now we are going to prove P r o p o s i t i o n 3.2. Let a,/3,7, <S be such that all Ricci eigenvalues are distinct. Denote D = (/3 + j)2 — 4a<5. Then, up to a reparametrization, the space (G, {,)) admits a) just one homogeneous geodesic through a point if D < 0; b) just two homogeneous geodesies through a point if D = 0; they are mutually orthog­ onal; c) just three homogeneous geodesies through a point if D > 0; they are linearly indepen­ dent but never mutually orthogonal. Proof. First, we see that G itself acts on (G, (,)) from the left as the maximal connected group of isometries (because each isotropy group of 1(G) is finite). Hence, each homoge­ neous geodesic in (G, {,)) is generated by a vector U e g of the form U = aX + bY + cZ. The condition (3) then leads to the system of equations (7)

ab2 + Sc2 + {$ + -y)bc = 0, a(ab + 0c) = 0, a(~/b + <5c) = 0.

Obviously, X is a geodesic vector. Further, D is the discriminant of the first equation (7) and hence we have either zero, or one, or two additional geodesic vectors according to the sign of D. If D = 0, the additional geodesic vector is of the form U = bY + cZ, and hence orthogonal to X. If D > 0, then we get two additional geodesic vectors of the form b\Y' + c\Z and 6 2 ^ + 02^- Then the orthogonality condition would imply a + 6 = 0, which is forbidden by Proposition 3.1. □ Let us notice that a non-unimodular Lie group from Proposition 3.1 is always solvable. Now, consider a solvable Lie algebra with the orthonormal basis (X, Y, Z) satisfying the multiplication table (8)

[X, Y] = 0, [X, Z] = aX,

[Y, Z] = -aY,

a # 0.

The corresponding simply connected group G is unimodular and it is usually denoted as £7(1,1) (cf.[ll]). We have a one-parameter system of invariant metrics on G. Two of the Ricci eigenvalues are zero. Nevertheless, each isotropy subgroup is finite (see [6, p.19] for more details). An easy calculation shows that the only geodesic vectors are multiples of X + Y, X — Y and Z, which form an orthogonal triplet. Hence there are just three

108 homogeneous geodesies through each point and they are mutually orthogonal. This family of examples is interesting for two reasons: 1) These are the only generalized symmetric spaces in dimension 3 which are not locally symmetric (see [6]). 2) These are the only nonsymmetric homogeneous Riemannian manifolds which can be isometrically immersed (not embedded) as hypersurfaces in a space of constant negative curvature (see [12], [13]). It means, a.o., that there are no such examples in higher dimen­ sions. T.Takahashi calls these spaces "B-spaces". Remark. V.V.Kajzer in [2] studies in detail 3-dimensional Lie groups with left-invariant metrics using also the Milnor's paper [11]. But he is occupied with the problem of existence or non-existence of conjugate points on homogeneous geodesies. 4. Four-dimensional example. Next, we shall study a 4-dimensional example which has some more interesting prop­ erties. For example, it admits infinitely many quadruplets of linearly independent homo­ geneous geodesies but no orthogonal quadruplet. The underlying manifold is R4 [x, y, u, v] with the Riemannian metric (9)

g=(-x

+ v t f + 7 + T ) du2 +{x + s/x2 +y2 + l) dv2 - 2y du dv

+y

~{l + y2)dx2 + (1 + x2) dy2 - 2xy dx dy - x2 + y2

where A > 0.

The space (R4, g) can be written as a homogeneous space G/H where G is the 5-dimensional group of equiaffine transformations of a Euclidean space and H is the subgroup of all ro­ tations of the plane around the origin. (See [6, p.136 and 139-140] for more details.) For the simplicity we choose A = 1. Then there exists a reductive decomposition g = m + h, an orthonormal basis (Xi, Yj, X2, Y2) of m and a generator B of f) such that the following multiplication table holds (cf.[5, p.19]):

(10)

[XX,YX] = 0, [XUX2] = - X i , [XUY2] = Xu [YUX2] = Y1, [Yl,Y2] = Xu [X3,Y2] = -2B, [B, Xx] = - y , , [B, Yi] = Xu [B, X2] = 2Y2, [B, Y2] =

-2X2.

Obviously we have [g, g] = g. By a routine but lengthy calculation one can prove that G is the maximal connected group of isometries of (R 4 ,^) (see [5, Theorem 13.5]). Hence, each geodesic vector must be an element of g; let us say U = aX\ + 6Yj + cX2 + dY2 + aB. From the condition (3) we obtain the following system of quadratic equations: (11) (12) (13) (14)

a(c-d) = ab, c(a+b) = a{a + b), a(a + b) = 2ac, {a-b)(a + b) = 2ad .

We shall analyze the possible solutions for (a, b, c, d) ^ (0,0,0,0).

109 P r o p o s i t i o n 4 . 1 . Every nontrivial solution of (ll)-(H) is either of the form (A) 6 = (2c 2 - a2)/a, d = 2c(a 2 - c2)/a2, a = c, where a / 0 and c / 0 are arbitrary, or of the form (B) a = b = a = 0 and c, d are arbitrary, (c, d) ^ (0,0), or of the form (C) b = —a, d = c, o = 0, where a / 0 and c are arbitrary. Proof. Suppose first that a + b / 0 and a / 0. From (12) we obtain c = a =£ 0 and (13) implies a ^ 0, 6 = (2c 2 — a2)/a. From (14) we get d = (a 2 — b2)/2c. Now, replacing 6 by its previous expression, we get d = 2c(a 2 — c2)/a2. We check easily that this solves (11)-(14). Suppose now a + b / 0 and a = 0. From (13) and (14) we get a = 0, 6 = 0, which is a contradiction. Next, suppose a + b = 0 and a / 0. From (13) and (14) we get c = d = 0. Due to (11) we obtain 6 = 0 and consequently a = 0. We get only a trivial solution. Finally, suppose a + b = 0 and a = 0. From (11) we obtain o(c — d) = 0. We have two possibilities and hence the cases (B) and (C) arise. D We shall need more preliminary results. For the sake of brevity, vectors from g will be said to be linearly independent or orthogonal if their m-components are linearly independent or orthogonal, respectively. P r o p o s i t i o n 4.2. At most two geodesic vectors of type (A) are mutually

orthogonal.

Proof. Let us introduce a new parameter t = c/a. Then each quadruplet (a, 6, c, d) of type (A) can be written in the form (a, a(2t 2 — 1), at, 2at(l — t2)), t ^ 0. In particular, it is a scalar multiple of a vector (15)

e t = (1, It2 - 1, t, 2t{\ - t2)), t G R \ {0} .

Assume t h a t t ^ 0 is fixed. Let e x = ( l , 2a;2 - 1, x, 2a;(l — x2)) be another vector of this type, x / t, and check the orthogonality condition (et,ex) = 0. We get for x the cubic equation (16)

4 x ( l - a; 2 )t(l - t2) + {2x2 - l)(2i 2 - 1) + xt + 1 = 0 .

First, we eliminate the special case t = ± 1 . Then (16) has only one nonzero solution, namely i = - j o n = | , respectively. So an orthogonal triplet of geodesic vectors of type (A) cannot arise in this way. Let now t ^ ± 1 . Then the equation (16) is cubic and it decomposes in the form (17)

{2xt + l)(2x2t2

- 2x2 + xt - 2t2 + 2) = 0 .

The corresponding roots are the following:

1

- t ± v / 1 6 ( f 2 - l ) 2 + t2

110 Hence, all three roots are real ones. Now, let us denote by Xij (1 < i < j < 3) the scalar product ( e I f , e x . ) . First, we calculate 1 4- 2t2 *

1 2 =

32t2(f-l)3(t

+

i)3-^where

xl2 = 10t4 - 17t2 + 8 + (t - 2 t 3 ) \ / l 6 ( t 2 - l ) 2 + t2 Put L = 10t4 - 17t2 + 8, P = (2t3 - t ) \ / l 6 ( t 2 - l ) 2 + t 2 . Then il2 = L - P. On the other hand, we get L2 - P 2 = -32(2(t 2 - l ) 2 + t2){t - l) 3 (t + 1 ) 3 / 0. Hence x12 + 0 and x\2 / 0. In the same way we show that x\% / 0. Finally, we get x23 = 2(t+i)2(t-i)i We conclude that an orthonormal triplet of type (A) cannot exist. □

>

"•

Proposition 4.3. Three geodesic vectors of type (B), or of type (C), respectively, are always linearly dependent. Proof. Obvious. We now formulate the basic result about our 4-dimensional example (R4,
&2 = (a2, -a2, c 2 , c 2 ) , e 3 = (0, 0, c 3 , d3) , e 4 = (0, 0, C4, <£j)

and these vectors are linearly dependent, which is a contradiction.

111

(ii) ei is of type (A) and e 2 ,e 3 of type (B), e.g., ei = (1, 2t2 - 1, t, 2t(l - t2)), t 4- 0 , e 2 = (0, 0, c 2 , d2) , e 3 = (0, 0, c3, d3) . Here 02^3 — c3d2 7^ 0 and the orthogonality conditions (ei, 62) = (e\, 63) = 0 imply t = 0, a contradiction. (iii) ei is of type (A) and e 2 ,e 3 are of type (C), e.g., ei = (1, 2t2-l,

t, 2 t ( l - t 2 ) ) ,

e2 = (02, -02, c2, c2) , e3 = (as, -a3, c3, c3) where 02C3 — o3C2 ^ 0, t / 0. The orthogonality conditions (ei, 62) = (ei, e3) = 0 mean a2(-2t2 + 2) + c2(3t - 2t3) = 0 , a 3 (-2< 2 + 2) + c 3 (3t-2i 3 ) = 0 , i.e., t2 = 1 and 2<2 = 3, which is a contradiction. (iv) ei, e2 are of type (A), e 3 is of type (B) and e 4 is of type (C), i.e., ei = (1, 2a; 2 - 1, x, 2x(\-x2)) e2 = (l,2y2-l,y,2y(l-y2))

, ,

e 3 = (0, 0, c3, d3) , e 4 = (1, - 1 , c4, c4) where x ^ y, x ^ 0, y =/= 0 and (c 3 ,d 3 ) / (0,0). The condition (e 3 ,e 4 ) = 0 implies either C3 = -d3 ^ 0 or c4 = 0. In the first case, (ei,e3) = 0 and (e2, e3) = 0 imply {x, y} = {l/\/2, -1/V2} and (ei, e4) = (e 2 , e4) = 0 imply 1 4- -J2 c4 = 0, 1 - v/2 c4 = 0, which is a contradiction. In the second case, (ei,e 4 ) = (e2,e 4 ) = 0 imply {x,y} = {1,-1}. Then (ei,e2) = 1 / 0 , which is a contradiction. D Let us remark that orthogonal triplets of homogeneous geodesies always exist on (R4, g). This is left to the reader as an easy exercise. Remark. The family of spaces defined by (9) has some remarkable properties: a) These are the only generalized symmetric spaces of dimension 4 which are not locally symmetric. More specifically, they are all 3-symmetric (see [5], [6]). b) These are the only 4-dimensional homogeneous Riemannian manifolds which admit a self-dual or anti-self-dual homogeneous structure of class T2. (See [9] for the result and [14] for the general theory of homogeneous structures.)

112 REFERENCES 1. C.S.Gordon, Homogeneous Riemannian Manifolds Whose Geodesies are Orbits, Prog. Nonlinear Diff. Eq. Appl. 20 (edited by S.Gindikin) (1996), 155-174. 2. V.V.Kajzer, Conjugate points of left-invariant metrics on Lie groups, Sov.Math. 34 (1990), 32-44; transl.from Izv. Vyssh. Uchebn. Zaved. Mat. 3 4 2 (1990), 27-37. 3. A.Kaplan, On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), 35-42. 4. S.Kobayashi and K.Nomizu, Foundations of Differential Geometry 77, Interscience Publishers, New York, 1969. 5. O.Kowalski, Classification of generalized symmetric Riemannian spaces of dimension n < 5 (Lecture Note, 61 pages), Rozpravy CSAV, Rada MPV 85 (1975), no. 8. 6. O.Kowalski, Generalized Symmetric Spaces, Lecture Notes in Math., vol. 805, Springer-Verlag, Berlin - Heidelberg - New York, 1980. 7. O.Kowalski and S.Z.Nikcevic, On geodesic graphs of Riemannian g.o. spaces, Arch. Math. 73 (1999), 223-234. 8. O.Kowalski and J.Szenthe, On the Existence of Homogeneous Geodesies in Homogeneous Riemannian Manifolds, to appear in Geom. Dedicata. 9. O.Kowalski and F.Tricerri, Riemannian manifolds of dimension n < 4 admitting a homogeneous structure of class T 2 , Conf. Sem. Matem. Bari 222 (1987), 1-24. 10. O.Kowalski and L.Vanhecke, Riemannian manifolds with homogeneous geodesies, Boll. Un. Mat. Ital. B(7) 5 (1991), 189-246. 11. J.Milnor, Curvatures of left-invariant metrics on Lie groups, Adv. in Math. 2 1 (1976), 293-329. 12. T.Takahashi, Homogeneous hypersurfaces in spaces of constant curvature, J. Math. Soc. Japan 22 (1970), 395-410. 13. T.Takahashi, An isometric immersion of a homogeneous Riemannian manifold of dimension 3 in the hyperbolic space, J. Math. Soc. Japan 23 (1971), 650-661. 14. F.Tricerri and L.Vanhecke, Homogeneous structures on Riemannian manifolds, London Math. Soc. Lecture Note Series 8 3 , Cambridge Univ. Press, Cambridge, 1983.

O.KOWALSKI,

Z.VLASEK

FACULTY O F MATHEMATICS AND PHYSICS C H A R L E S U N I V E R S I T Y , S O K O L O V S K A 83 186 75 P R A H A , C Z E C H R E P U B L I C

e-mail:

[email protected]

STANA NIKCEVIC MATHEMATICAL INSTITUTE,

SANU

K N E Z MIHAILOVA 35, P . P . 367 11 0 0 1 B E O G R A D , Y U G O S L A V I A

e-mail:

[email protected] R E C E I V E D F E B R U A R Y 2, 2 0 0 0

G e o m e t r y and T o p o l o g y of Submanifolds X eds. W . H. Chen et al. (pp. 1 1 3 - 1 3 5 ) © 2000 World Scientific Publishing Co.

113

TWISTORIAL CONSTRUCTION OF SPACELIKE SURFACES IN LORENTZIAN 4-MANIFOLDS. Felipe L e i t n e r ' ( B e r l i n )

Abstract We investigate the twistor space of a Lorentzian 4-manifold with natural almost optical structures and its induced CR-structures. The twistor spaces of the Lorentzian space forms Ml, 4 S and H* are explicitly discussed. The given twistor construction is applied to surface theory in Lorentzian 4-spaces. Spacelike immersed surfaces in a Lorentzian 4-space that have special geometric properties like null-umbilicity or lightlike mean curvature correspond to holomorphic curves in the Lorentzian twistor space. For the Lorentzian space forms R : , Sf and H4 those surfaces are explicitly constructed and classified. Keywords: Lorentzian twistor space, optical structures, spacelike immersed surfaces, Gauss lift. 1991 MSC: 53A10, 5SC50, 53C42.

0

Introduction

T h e aim of this paper is t h e investigation of spacelike surfaces in Lorentzian 4-manifolds with the aid of a so-called twistor construction in 4-dimensional Lorentzian geometry. T h e idea of the draft, t h a t is presented here, comes from Riemannian twistor theory and its well known application to the theory of surfaces in Riemannian four-spaces. We give a short description of t h e Riemannian construction. Let ( M 4 , g ) b e an oriented 4dimensional Riemannian manifold. T h e tangent space at every point of t h e manifold M is iso­ metric to the Euclidean 4-space R 4 . T h e set of complex structures on R 4 can be identified with the homogenous space G £ ( 4 , R ) / G £ ( 2 , C ) . T h e r e are two kinds of orthogonal complex structures on K4: A+ :={Je A.

50(4)|

:= {J e SO(4)\

where J0 := (

l

0

3 A e 5 0 ( 4 ) with J = A~l ( J° 0 Jo 0

3 A e 5 0 ( 4 ) with J = A " 1 ( J°

° ) A} Jo 0 ) -Jo

_°

and A},

) G fll(2, M). Both sets A+ and A_ of orthogonal complex structures are

naturally identified with the homogenous space 5 0 ( 4 ) / ( 7 ( 2 ) , which is t h e 2-sphere 5 2 . We choose the set A- and define t h e twistor space A-{M) of M 4 due to Atiyah/Hitchin/Singer [AHS78] as the associated fiber bundle A-{M)

:= SO(M)

x 5 0 ( 4 ) A.

= SO(M)

x s 0 ( 4 ) Sa

"Supported by the SFB 288 of the DFG. This paper is in final form and no version of it will be submitted for publication elsewhere.

114 over M4 consisting of orthogonal complex structures on TM. Thereby, SO(M) denotes the bundle of positive oriented orthonormal frames on M. The twistor space A-(M) admits two natural almost complex structures J\ and J2. The first almost complex structure J\ is integrable if and only if the Riemannian 4-manifold M4 is self-dual. The second complex structure J2 is never integrable. Let us consider a conformal immersion / : N2 —► M4 of a Riemannian surface JV2 into the Rieman­ nian 4-space M4. Every oriented 2-plane V2 in the oriented Euclidean four-space K4 gives naturally rise to an orthogonal complex structure on R4, which is the rotation around the angle 5 in positive direction on V and negative direction on VL and which is an element of the set A-. The image df (TnN) of the tangent space at every point n 6 N is an oriented 2-plane in the Euclidean vector space Tf(n)M. Hence, the 2-plane df(TnN) in Tj^M corresponds uniquely to an element Jn in the fiber of the twistor space A-(M) over the point f{n). This gives rise to a natural lift of the immersion / to the twistor space A-(M): If.

N

-> A- (M) ,

n

>->

Jn

if '

/ : N ->

A-(M) ■!■

M

J. Eells and S. Salamon studied this 'Gauss' lift in [ES85] and proved the following relation for minimal surfaces. T h e o r e m Conformally immersed minimal surfaces in a Riemannian 4-space M4 correspond bijectively to non-vertical J2-holomorphic curves in the twistor space A-(M) over M4. In particular, the lift of a conformally immersed surface to the twistor space is horizontal iff the immersed surface is superminimal. This means that in the case of a self-dual Riemannian 4-space M4, superminimal surfaces can be constructed by horizontal holomorphic curves in the twistor space, which is a complex manifold. Those constructions of superminimal surfaces have been done by Th. Friedrich in [Fri84] and [Fri97]. On this way, a well known result is obtained, which says that a superminimal surface in the Euclidean 4-space has locally the form C -> z H>

Cxcai (z,f(z))

4

,

where / is a holomorphic function. Using the twistor space P 3 (C) of the sphere S4, R. Bryant proved in [Bry82] the global result that every Riemannian surface admits a conformal superminimal immersion into S4. The twistor method has also applications in semi-Riemannian geometry. G.R. Jensen and M. Rigoli constructed in [JR90] the reflector space of a neutral 4-space and applied this construction to the theory of immersed neutral surfaces. The idea of twistor theory in context of Lorentzian geometry is as follows. The twistor space 2(M4) of an oriented Lorentzian 4-manifold M4 is defined to be the bundle of null directions in the tangent space TM over M (comp. [Nur96]): Z(M4) := SO(M4) xSO(3,i) P, where P is the space of null directions in the Minkowski space K4. Instead of almost complex structures, the twistor space Z(Mf) admits natural almost optical structures, which are related to

115 CR-structures. Let us consider a conformal immersion / : N2 —> Mf of a Riemannian surface N2 into an oriented Lorentzian 4-space Mf. In every point f(n) S Mj* of the immersed surface, there exists an ordered pair of normal null directions on the tangent space df(TnN) to the surface in M\. By choosing one of these normal null directions, we obtain a natural lift of the immersion / to the bundle of null directions Z(M\),

Z(Mf) V / 2 f : N -»

I ■ Mf

Similar as in 4-dimensional Riemannian geometry, spacelike surfaces in M* with special geometric properties can be characterized and constructed by holomorphic curves in the almost optical manifold Z[M\). The paper is organized in two parts. The first part is concerned with the twistor space of a Lorentzian 4-space and its natural almost optical structures. Special attention is given to underlying CR-manifolds (Theorem 1.3). The twistor spaces of the Minkowski space Rj, the pseudosphere S\ and the pseudohyperbolic space H\ are discussed. In the second part, we apply the Lorentzian twistor construction to surface theory. The second fundamental form II of an isometrically spacelike immersed surface (JV2, h) is decomposed to / / = h ® H+ + h ® H- + L+ + L_, where H+ and H- are the lightlike parts of the mean curvature. The vanishing of components in this decomposition is related to the holomorphicity of the Gauss lift of the immersion (Proposition 2.1). We prove that non-vertical holomorphic curves in the twistor space over a Lorentzian 4-space Mj project to null-umbilic (L_ = 0) surfaces and to surfaces that have mean curvature of vanishing length in M\ (Theorem 2.1). For the Lorentzian space forms Kf, S\ and Hf, we construct explicitly null-umbilic surfaces, which have in addition mean curvature of vanishing length.

1

Lorentzian twistor space and optical geometry

1.1

Optical geometry and CR-geometry

We recall in this section some basic facts about CR-geometry, optical geometry and the relation between them. Optical geometry was first introduced in relativistic theories by A. Trautman in [Tra85] (see also [Nur96]). We start with the definition of CR-structures: Definition 1.1 Let M 2 n _ 1 , n > 1, be a (2n - 1)-dimensional C°°-manifold. 1. A pair [H, J), where H C TM is a smooth distribution of codimension 1 on M and J : V. —> ti is a smooth bundle isomorphism on H with J2 = -id, is called almost CR-structure on M2n~l.

116 2. An almost CR-structure (H, J) on M2n of J onW vanishes, i. e

1

is said to be integrable iff the Nijenhuis tensor N

[JX, Y] + [X, JY] € r(W) VA", Y £ r(W) and N{X, Y) := J([X, JY] + [JX, Y]) - [JX, JY] + [X, Y] = 0 VA, Y e T(W). /n iftj's case uie coH (M, W, J) a CR-manifold. 3. A C°°-map f between two almost CR-manifolds (M, H, J) and (M,i-L,J) is called a CR-map if and only if df(H) C H and df o J = J odf. Example The natural CR-structure on the unit sphere bundle of an oriented Riemannian 3-manifold Let (N3,h) be a 3-dimensional oriented Riemannian manifold. The unit sphere bundle of iV3 is given by S2(TN)

:= SO(N) x s o ( 3 ) S 2 = SO(N) xso{3)

50(3)/50(2) C TN,

where SO(N) denotes the 50(3)-principal fiber bundle of orthonormal frames on N. Let it : S2(TN) -> N denote the natural projection. The Levi-Civita connection of (N3,h) decomposes the tangent bundle TS2 (TN) into a horizontal and a vertical part: TS2{TN)

= TVS2{TN)

0

THS2(TN).

On S2(TN), it exists a natural smooth distribution ^ s 2 ( r j v ) c TS2(TN) in a point I € S2(TN) by H$\TN)

= w -i( ( W )-L) =

TVS2(TN)

®

of codimension 1 given

(THS2(TN)nn:1{m-L)).

Let Js denote the standard SO(3)-invariant complex structure on S2. It exists a natural complex structure on the distribution % s ( TiY ', which is pointwise defined by jf(™) =

ff,-loJio7r,

+ [S]-1ojS2oW,

where [s] denotes the identification of a fiber with S2 by an orthonormal frame s and w~l o Jt ow, is the horizontal lift of the orthogonal complex structure J; on (Rl)1- C TwmN, which is the rotation around the angle § in positive direction. This CR-structure (HS'2(TN\ JS'2
C°°-manifold.

1. A triple O = (/C, C, J) consisting of subbundles K. and C in TM such that K C C,

dim/C = 1, dim£ = In — 1

and an almost complex structure J : LjK —> C/K, J2 = —id, on the quotient bundle LjK., is called an almost optical structure on M2n.

117 2. An almost optical structure O = (/C, £, J) on M2n is said to be integrable iff the following two conditions are satisfied: (A) [r(£),T(£)] C T(C)

and

kt*J = J V/c G r(/C),

where $ denotes the flow of the field k to the time t. This condition means that locally in every point m 6 M the almost optical structure O may be pushed down to an almost CR-structure ("Hm, Jm) on a locally induced quotient manifold C/ m /~ K , where Um C M is a suitable neighborhood of m 6 M and two points «i,«2 £ Um ore ^-related iff both belong to the same integral curve of the distribution K\vm. (B) The locally induced CR-structures (Wm, Jm) are integrable in every point m S M. 3. A C°°-map f : M —► M between two almost optical manifolds (M, K., £, J) and (M, it, C, J) is called an optical map iff df(K.)C)C,

df(C)GC

and dfoJ

= jodf.

It is instantly clear from the definition that optical structures are closely related to CR-structures. We express this by the following facts: 1. An almost CR-manifold (N2n~l,H, J) gives rise to a canonical almost optical structure on the manifold M := E x N. This almost optical structure on M is defined by ICM:=TM,

CM :=TR®rl

and

JM .=* J : CM/K.M =* rl -> CM/K.M =* H.

2. If an almost optical structure (/C, C, J) on M satisfies condition A, the locally induced almost CR-structure on Um/~K, m 6 M, is given by Hm := fr,(£)

and

J™ := fr, o J o jt'1,

where ff : Um -> C/ m /~ K is the natural projection. The open neighborhood Um C M of m € M may be chosen in such a way that (Um,IC\um,C\um,J\um) is optically equivalent to R x Umj^ with the induced almost optical structure given as above. 3. Let jV 2 " - 1 be a submanifold of codimension 1 in the almost optical manifold (M 2 n , K,, C, J) such that TN(B>C\N

=TM\N. N

It follows that 'H := TNnC\x is a distribution in TN with codimension 1. The distribution HN is naturally identified with the restricted bundle C/IC\N and J induces an almost complex structure JN on HN. The pair (~HN,JN) is a naturally induced almost CR-structure on N2n~l. We say that (N,HN', JN) is an almost CR-submanifold of the almost optical manifold (M,IC,CtJ). If (/C, C, J) satisfies condition A, the almost CR-structure (HN ,JN) in m S N C M is locally equivalent to the naturally induced almost CR-structure (Hm,Jm) on the quotient manifold Um/~K in the point {m} € f/m/~K. 4. An 1-dimensional distribution £ on a manifold M is called regular if there is a smooth differentiable structure on the quotient set Mj ~ K such that ff : M —► M / ~ K is a C 00 submersion. If (M, K, C, J) is an optical manifold with regular distribution K, the quotient manifold M/~K admits globally a natural CR-structure.

118 After we have defined CR- and optical manifolds, we want to consider mappings between them and define for this an appropriate notion of holomorphicity. Definition 1.3 Let (Q,J®) denote an almost complex manifold, (N,MN,JN) manifold and (M,KM\CM,JM) an almost optical manifold.

an almost CR-

1. A C°°-mop / : Q -> JV is called holomorphic if df(TQ) C HN

and

df o J<3 = JN o df.

2. A C°°-map g : Q -» M is called holomorphic if dg(TQ)c£M,

dg(TQ)nKM

=0

and w o dg o J « = JM ot o dg,

where H : C —► LjK is the natural projection. 3. A C°°-map h: N -» M is called holomorphic if dh(nN)cCM,

dh{HN)nlCM

= 0

and fr o dho JN = J M o jr o df.

With the same notations as in the definition, we have Proposition 1.1 1. If the mappings f : Q -> JV and h : N —► M are holomorphic, the map g := ho f ; Q -> M is also holomorphic. 2. let (KM,CM,JM) on M satisfy condition A and let k € T(KM) be a vector field on M. If the flow (f>\ of k is defined on] — e, e[xU, where e > 0 and U C M is an open subset, and if g : Q —► M is holomorphic, then the map 9t ■= 4>kt ° g\9-'(u) ■ 9 - 1 (E0 C Q -)• M is holomorphic for every t £ (—e, e). The proof of this is obvious. The second part of the proposition says that the deformation of a holomorphic map by an arbitrary smooth flow along the distribution K is still holomorphic. 1.2

Twistor space of a Lorentzian manifold M*

The twistor space of an oriented Lorentzian 4-manifold M* := (M4,g) can be defined as the fiber bundle of null directions in the tangent space TMf (comp. [Nur96]). We give in this section an equivalent definition of the Lorentzian twistor space as bundle of positive projective spinors on JWj*. The twistor space admits natural almost optical structures. The integrability, the conformal invariance and the underlying CR-hypersurfaces of these optical structures are investigated. At the end of this section, we discuss explicitly the twistor space of the Minkowski space E.*, the pseudosphere S* and the pseudohyperbolic space H*. We start with description of the fiber type of the twistor bundle. Let W.\ := (E 4 , <, >|) denote the Minkowski space, ( u i , . . . ,«4) the standard basis of I 4 with < « i , « i > 4 = —1 and let P : = { B - t i e P J ( l ) : « ^ 0 , ?= 0} denote the set of null directions in K4. The space P is a submanifold of the projective space P 3 (K). There are several characterizations of the space P. For the first, P may be written as homogeneous

119 space, since t h e Lorentzian group 5 0 ( 3 , 1 ) acts in a natural way transitively on P. Let H be the isotropy group of t h e natural 5 0 ( 3 , l)-action in o := R • (ui + u2) 6 P. It holds P = 50(3,

l)/H.

Let Eij := CiXji - ejX{j, where t{ :=< Uj,Uj > | and Xy 6 M ( 4 , R ) is t h e matrix, whose entry in the i-th row and j - t h column is 1 and elsewhere 0. T h e set {Eij : i,j = 1 , . . . , 4, i < j} forms a basis of the Lie algebra o(3,1). T h e Lie algebra t) of t h e isotropy group H is t h e n given by 1) = Span{El2t

El3 + E23, Eu

+ #24, EM}

and o(3,1) = m © f), where m = Span{E\3 - E23, Eu - £24} is t h e complement to rj in o(3,1). This decomposition of t h e Lie algebra o(3,1) is not reductive. T h e space P of null directions in R | is also naturally identified with t h e set of positive projective spinors. For this let A + = C 2 denote t h e standard 5 p m ( 3 , l ) - m o d u l of positive spinors (comp. [Baum81], chap. 1.4). A spinor 0 ^ t ) € A + induces t h e R-linear m a p V :

»i x

->• A + , ^-> x ■ v

where x ■ v is the Clifford product. T h e kernel kerv of this mapping is a null direction in Bf and the m a p 1:

P(A+) [v]

-> t-y

P kerv,

v S [v]

is bijective and 5 0 ( 3 , l)-equivariant. T h e n a t u r a l complex structure J P ' A + ' on P ( A + ) = P : ( C ) = P is invariant by the natural 5 0 ( 3 , l)-action and is given on m S T0P by J ^ + ) ( E 1 3 - E23) = -(Eu

- E2i)

and

J P ( A + > ( E 1 4 - E2i)

= E13 -

E23.

Every unit timelike vector T £ R j , < X , T > } = —1, gives an identification of t h e space of null directions P and t h e 2-sphere S2^^-) = 5 2 in the orthogonal complement TL = R 3 : 1:

P R(T + 5)

=* S >->■

2

^ ) S

.

The twistor space of a 4-dimensional oriented Lorentzian manifold Mf : = ( M 4 , g ) is defined to be the positive projective spinor bundle Z(M?)

:= P(S+)

= SO(Mf)

xso(3il) P(A+).

Let -K : Z(M) - + M b e t h e natural projection. We may interpret this bundle also as the bundle of null directions in TMf, Z(M)

= SO(M)

x s o ( 3 i l ) P = SO(M)

x S 0 G M ) 50(3,1)/H.

T h e tangent space TZ(M) decomposes with respect to t h e Levi-Civita connection of M\ vertical and a horizontal part:

TZ(M)

=

TVZ(M)®THZ{M).

in a

120 On the twistor space Z(M), there are given two natural almost optical structures 0+ = (£, £, J+) and 0~ = ()C,C,J~). We define them pointwise as follows. Let [tp] 6 Z(M) be an arbitrary positive projective spinor. For a suitable orthonormal basis s = ( s i , . . . , S4) S SO(M) we may write [V>] = [s,K(«i +«2)]- To [i/>] G Z(M), it corresponds the orthogonal optical structure OM = (KMf L W , j W ) on the tangent space T ^ j A f that is defined by # M = K(si + s 2 ),

L^ 1 = 5pon{si + s2,

JW (a3 + JfW) = s 4 + ^

W

and

s3,s4},

M (s 4 + ^ W ) = -«3 + tfW ■

The almost optical structure 0+ = (K, C, J+) is given in Ty^Z(M) 1

1

by

p A

O+j = jr." o OM o „-» - [s]- o J ( +) o [s], where TT7 * O OW OTT» denotes the horizontal lift of the optical structure ©M in Tw( W ) M to T ^ 2 (M). The almost optical structure C~ is given in T^Z(M) by O - = jr,-1 o OM o „•„ + [s]" 1 o J p ( A +) o [a]. Theorem 1.1 (comp. [Nur96] and [Lei98]) Let M4 be an oriented 4-dimensional Lorentzian man­ ifold and let Z(M4) be its twistor space. 1. The almost optical structure 0+ is integrable in p e Z(M) iff p is a principal null direction inTM4 (comp. [0'N95]). 2. The almost optical structure C + on Z(M4)

is integrable iff M4 is conformally flat.

3. The almost optical structure 0~ doesn't satisfy condition A and is never integrable. We want to consider the conformal invariance of the twistor space Z{M4) with its almost optical structures. For this, let g — exp(2p)g, where p : M4 —> R is a smooth function, be a conformally equivalent metric to g on M4. The twistor spaces Z(M4,g) and Z(M4,g) are naturally identified, since the null directions in TM4 are conformally invariant. Theorem 1.2 Let M4 := (M4,g) be an oriented Lorentzian 4-manifold and let g = exp(2p)g be a conformally equivalent metric to g. The corresponding almost optical structures 0+ and 0+ on Z(M) are equal. Proof: The distribution /C is the lightlike geodesic spray of the Lorentzian manifold M4. The lightlike geodesic spray is conformally invariant. It follows that the optical flag K C C C TZ(M) is conformally invariant. It remains to show that the almost complex structure J + and J + on the screen space C/K are equal. For this purpose, let s = («i, S2, «3,S4) : U C M ->

SO(M,g)

be a local frame and denote by s = exp(—p)s the conformally changed frame in SO(M,g). In the trivialization U x 50(3,1)/H of Z(M) induced by s, the horizontal lift sf of st to U x 50(3,1)/H is given by «? = »i + 9(V Si (*i + «a), s3)El3~E23H

+ S (V S i ( Sl + «,),

s,)El4~E24H.

121 For the conformal change of the Levi-Civita connection holds (comp. [Bes87]) VXY

= VXY + dp(X)Y + dp(Y)X - g(X, Y)grad(p).

For the horizontal lift sf of S, with respect to the connection V, we have s" = exp(-p) • (sf

+

{g({si +

S2)(p)-si,s3)-g(s3(p)-si,s1+S2)]—^——H JP

+

_ JP

[9((si + S2)(p) -Si,s 4 ) -g(s*(p) -Si,si +s2)\-^-^

-H)

and we see that J+(Sf + K) =sf

+ /C = J + ( s f + K)

and

J+(s? + K.) = - s f + K, = J + ( s f + /C),

i.e. J + and J + are equal

□

The proof also shows that the almost optical structure 0~ isn't conformally invariant. We are now interested in the underlying CR-hypersurfaces of (Z(Mf), 0 + ) . Let T g F(TM), g(T,T) — — 1, be a timelike unit vector field on Mf. The choice of such a field T is equivalent to a 50(3)-reduction SOT(M) of the frame bundle SO(M) over M. The twistor bundle may then be written as Z(M) = SOT(M)

x s o ( 3 ) P S SOT(M)

x s o ( 3 ) S2.

The twistor fiber ir~l(m) over a point m £ M\ is identified via Tm 6 TmM with the sphere S2{T^) C T^, i.e. we can think of the twistor bundle as the bundle of 2-dimensional unit spheres, which are orthogonal to the timelike vector field T e T(TM). Let us consider an oriented spacelike hypersurface P 3 in Mf, i.e. the restriction g\p is positive definite. There is an unique timelike unit normal field T on P such that (T,S2,s$,sn) is positive oriented on Mf if (s2,S3>«4) £ SO(P). The field T induces a 50(3)-reduction of the restricted frame bundle SO(M)\p and we have a natural identification *:Z(M)\P?iSO(P)xSo{3)S2 of the restricted twistor bundle and the unit sphere bundle over P 3 . The restriction Z(M)\p submanifold of codimension 1 in Z(M) and obviously, it holds T(Z(M)\P)nlC

is a

= {0}.

It follows that the almost optical structure 0+ induces an almost CR-structure (H, J + ) on Z(M)\p (comp. section 1.1). Theorem 1.3 Let M\ be an oriented A-dimensional Lorentzian manifold and let P3 be an oriented spacelike hypersurface of Mf. 1. The almost CR-structure (W, J + ) on the restricted twistor bundle Z(M)\p grate.

is always inte­

2. The CR-structure (M,J+) on Z(M)\P and the natural CR-structure (HS^TP\ J s 2 ( T P >) on the unit sphere bundle S2(TP) are equivalent under the identification $ if and only if P 3 is a totally umbilic hypersurface in Mf.

122 Proof: First, we compare the CR-structure (•H s2 ( rp ), J^C" 1 )) and the almost CR-structure (H,J+) on Z(M)\P =? S2(TP). To every point n € P C M, it exists an open neighborhood U C M and a local frame s = (si,s2, sit s4) : U C M -> ^©(Af) such that T = «i on {7nP and (S2,53, S4) : (7 n P -> SO(P) is an orthonormal frame. An arbitrary point p € Z(M)\pnu may be written as [s • A, B(«i + U2)] for some A G 50(3). We have *(p) = * (R(«i + (s ■ A)2)) = (s ■ A) 2 6 S 2 (TP) and it holds ■K,{UP) = ( R T ^ , © B ( s x W • A)2)^ = Span{( S7r(p) • A)3, (sn(f) ■ A)4} = n, (Hf$P)) It follows ¥ , ( « „ ) = Wf^™' for every p S Z(M)\p. of Z(M)\P and S 2 (TP):

■

A local frame s as above induces trivializations

-Z(M)| P r w

=

S2(TiV)|pnJ7

(P n U) x 5 0 ( 3 , l ) / f f

=

(P n [/) x SO(3)/50(2)

The horizontal lifts of Sj, i = 2,3,4, with respect to the Levi-Civita connection on P are given in the trivialization of S2(TP)\unp by *f = Si ~ g(VSis2,s3)E23SO{2)

-

g{VSis2,s4)E24SO{2).

1

Let sf denote the horizontal lift of Sj, i = 2,3,4, with respect to the Levi-Civita connection on M\. It holds t" = ^,{s^)+g(VS3sus3)E23SO{2)

+

g{VS3sl,s4)E2ASO(2),

i f = * . ( « ? ) +9{VS3si,s4)E23SO(2)

+

g(VSisl,s4)E24SO(2)

and the almost complex structure J * by

+

on % is given onHs

J+(t») = i f - 2g(yS3s1,s4)E23SO{2)

+ ( 9 (V„si,s 3 ) -

J + ( t f ) = - i f + 2g(VS3sus4)E24SO(2) J+(E2;iSO{2)) = E24SO(2),

and

frp' in T 5 2 (TP) via the identification

+ {g(VS3sus3) +

J (£24SO(2)) =

ff(V,4si,s4))^S,0(2), -g(VSisus4))E23SO{2), -E23SO(2).

The calculation of the corresponding Nijenhuis tensor shows that the almost CR-structure ("H, J + ) on Z(M)\P s S 2 ( r P ) is integrable. Moreover, the above formulas show that J + and Js ( r p ' on Hs ' r p ' are identical iff ff(V„si,S3) = p(Vj 4 si,44)

and

g(V S3 *i,s 4 ) = 0

for every orthonormal basis {s2,s3,s4) on P . But this condition just means that the second fundamental form II of P 3 in Mf looks like II = g® H, where H is the mean curvature, i.e. the hypersurface P 3 in Mf is totally umbilic □ The almost CR-structure on the restriction Z(M)\p of the twistor bundle to a spacelike oriented hypersurface P that is induced by the almost optical structure 0~ on Z(M) is never integrable.

123 Remark 1.1 The twistor bundle Z(M?) on Mf can also be identified with the bundle P(S-) of negative projective spinors. The naturally induced almost optical structures on P(5_) are antiholomorphic to 0 + and 0~ on Z(M) = P(S+) (comp. [Lei98]) and the results of this section carry over to the bundle P(S-). Examples A. The twistor space Z(R\) of the Minkowski space Mj with optical structure 0 + Let Rj be the flat Minkowski space. The twistor space Z(K.\) of Rf is given as R\ X P(A + ) (x,[v])

a o

R4 x P (x,kerv) = (x,R(ui + S)) = (x,R[A-(ui+u2)})

s R< x 50(3,1)/H «• {x,AH)

s R< X S2 <-> (x, S)

Let it* denote the standard basis in TXW.\ = Rj and let uf = ( u i , . . . , 144) • A be the orthonormal basis transformed by A S S0(3,1). The optical structure 0 + on Z(S.\) is given in [ip] = {x, AH) G Z(R\) by % ] = « « + «£),

CM = Span{ut + u^u^u^},

J+ (u^ + £ M ) = uf + % ]

and

J+\Tv

m

^

=-JP{A+\

J+ (uf + ICW) = - u ^ + /C M .

+

The twistor space (Z(Rj), 0 ) is optical diffeomorphic to R x (R3 x 5 2 ) with the optical structure (TR,TR®ff s ' 2 < TK3 >,J 5 ' 2 < TK3 )), where (R3 x 5 2 ,ff 5 2 ( T R )}JS2{T«. )J i s t n e u n i t S ph e r e bundle over the Euclidean space R3 with the natural induced CR-structure (comp. section 1.1). B. The twistor space Z(S\) of the pseudosphere S\ with optical structure 0 + The hypersurface S\ := { i £ i f : <x,x>\=

1}

in Rf with the induced metric is an oriented Lorentzian 4-space of constant sectional curvature 1 and is called the 4-dimensional pseudosphere (comp. [0'N83]). The pseudosphere 5f is a symmetric space: S± = 50(4, l)/50(3,1), where we choose the embedding 1: 50(3,1) (Aij)

^> H4

50(4,1) / Au

0 Au

\

0 V -4(1

0

1 0 Aij

)

of the isotropy group 50(3,1). The twistor space Z(Sf) 50(4,1)/H and is diffeomorphic to R x S2(TS3) where S2(TS3)

is given as homogenous space by

= R x 50(4)/50(2) a R x 5 3 x 5 2 ,

s* 50(4)/50(2) is the unit sphere bundle over the sphere 5 3 S 50(4)/50(3).

The optical structure 0 + = (/C, £ , J + ) on Z(S\) is S0(4, l)-equivariant and is given as follows. The Lie algebra o(4,1) = Span{Eij : 1 < i < j < 5} is decomposed by o(4,l) = m® f)ffib,

124 where o(3,1) = m © t) and b = Span{Ei2, E23, E24, E<&}. The subspace b is Ad(H)-invariant there exists an >td(ff)-equivariant optical structure (S, I, J) on 6 defined by I := M(-Ei2 + E23),

l:=Span{E24,Ev,}®l,

KE24 + t) = E25 + t,

and

and

l(E2& + i) = -E24 + t.

The twistor bundle Z{S\) splits into a horizontal and a vertical part, where the horizontal bundle is given by THSO(4, \)IH = 50(4,1) xAd{H)

b.

The distributions K and C of the optical structure C + are defined by K = 50(4,1) xAd{H)

t

C = (TvSO(4,1)/H)

and

9 (50(4,1) xAd{H)

().

The complex structure J + : £//C -» £//C is given on the vertical part by J+| r vSO(4,I)/H — - / P ' A + ^ and on the horizontal part by

j + p , ( + e]) = [;i,J(( + e)], ,4 e50(4,i), lei The distribution /C is regular on Z(Sf) and the underlying CR-manifold of (2(5f), 0 + ) is equivalent to the unit sphere bundle of 5 3 with natural CR-structure. We describe this CR-structure on 50(4)/50(2) ^ S2(TS3). Let {£„ } be the standard basis in o(4), t:

50(2) /" cos ( ( sint

•->

— sin t \ cost J

, ^

50(4) eX

, T-, P*B34

the embedding of 50(2) and let o := Span{E\$, En, E23, £74} be a subspace of o(4). The subspace 0 is Ad(50(2))-invariant and the linear map J : a —► 0 given by l(E13)

= Eu,

J(E 2 3) = E24,

J(£ , i4) = - ^ i 3

and

J(Eu)

=-E23

is an ^4d(50(2))-equivariant complex structure on a. The canonical CR-structure on 50(4)/50(2) is given by «

<

»=50(4)xMSO(2))0

and

^

^

^

^

o g a

C. The twistor space Z(Hf) of the pseudohyperbolic space Hf with optical structure 0+ The hypersurface Hj := {1 e »j : < x , i >2= - 1 } in R2 with the induced metric is an oriented Lorentzian 4-manifold of constant sectional curvature - 1 and is called the 4-dimensional pseudohyperbolic space. The pseudohyperbolic space H* is a symmetric space: H* S= 50(3,2)/50(3,1), where we choose the embedding t:

50(3,1)

->

50(3,2)

A

-

( i °A)

125 of the isotropy group. The twistor space Z{B.\) is given as homogenous space by 50(3,2)/ff and is diffeomorphic to S 1 x S2(TH3)

= 5 1 x 50 0 (3, l)/50(2) s S 1 x R3 x S 2 ,

where S2{TH3) = 50„(3, l)/50(2) is the unit sphere bundle over the hyperbolic space tf3 S SO„(3, l)/50(3). The optical structure 0 + = (/C,£, J + ) on 2(ffj) is 50(3,2)-equivariant. We describe 0 + as follows. The Lie algebra of SO(3,2) splits into o(3,2) = m © t) © b, where the subspace b = Span{Ei2, E13, Eu, E15} is >lrf(fl")-invariant. The subspaces t := R(£v2 + £13),

1 := Span{El2 +

EWtEu,El6}

of b and the complex structure J : l/t —> l/t given by J(Ei 4 + t) = £15 + e and

J(£i5 + t) = -B14 + t

form together an ^4
/C := 50(3,2) x „ r f W e, ■7*Vso<s.2)//r

s

P{A+)

-J

and

© (50(3,2) xAd(H)

l),

+

J ([A,J + e]) = [Jl,JZ(J + t)],

H

where [.A,Z + t] &

Cr\T SO(3,2)/H.

The locally induced CR-structures of O^ are equivalent to the natural CR-structure on the unit sphere bundle of H3. We describe this CR-structure on 50 0 (3, l ) / 5 0 ( 2 ) a S2{TH3). Let 1: 50(2)

->

S0„(3,l)

A - (i 2) be the embedding of S0(2) in S0 0 (3,1). The pair (a, J ) defined by a := Span{Ew, Eu, E23, E2i} C o(3,1), J(Ei 3 ) = Eu,

l(Eu)

= -£13,

l(E2Z)

= #24

and

is an Ad(50(2))-equivariant CR-structure on Span{Ei3,Eu,E2i, structure ( « s 2 ( ™ 3 ) , JS2(TH3)} i s t h e n g i v e n b y

^

2 2.1

=^ , l ) x ^ .

and

J ^ )

= -£23

E24, E34} C o(3,l). The CR-

• «[AB]

^

[ A

«

M a e a

Surface theory and Lorentzian twistor construction Spacelike i m m e r s e d surfaces

We study the second fundamental form of a spacelike immersed surface in a Lorentzian 4-space.

126 Let Mf := (M4,g) be an oriented Lorentzian 4-space and let / : N2 -»• Mf be an isometric immersion of an oriented Riemannian 2-manifold (N2, h). A local Darboux frame on N2 is a map s = (su...

,s4):UcN->SO(M)\N

such that («3,S4) is locally a positive oriented orthonormal frame in TN. Let TN1- denote the normal bundle of the immersion / in TMf. The metric g on TM induces a metric h1 of signature (1,1) on TN-1-. The normal bundle splits into a positive and a negative line bundle of lightlike normal vectors, TN1- = TN1 ® TN1, where these line bundles over N are locally defined by TN± = R( S l + a2)

and TN1 = R(si - s2)

with respect to a Darboux frame s. The second fundamental form of the immersion / is defined to be the normal part of the Levi-Civita connection V on Mf, IT{X, Y) = NVXY,

X, Y S r(TJV).

Let hfj := e a 9(V J!j .s. ; ',s a ), a € {1,2}, i,j & {3,4}, denote the components of II with respect to a Darboux frame s. The mean curvature vector H of the isometric immersion / is given by H

= \trII

= \gi>h?jsa = H+ + H-.

where H- e TN1 and H+ € T7V|. Let t := ~"j|*< € C ® T*M. It is II

=

:=

fle[(5(-ftj3+/ij4

- his + h2u) + i{hlM + h2u)) -tot] («i + s2)

/j®fl"

+

ft®H+

+ Re [(i(/i»3 - /4 4 - h23 + h2u) + i ( - 4 , + h2A)) ■ t o t] ® (s2 + /i®ff_ + I + + L_,

where £_ € Sym2{TN) ® TN1 and L + G Sym2(TN) in the normal null line bundles.

Sl)

® TAT^ are symmetric 2-forms with values

Definition 2.1 Let f : N2 -> Mf be a spacelike immersion of an oriented surface, i.e. the induced metric h := f*g is positive definite on N2. We call the spacelike immersion f null-stationary *» H- = 0 null-umbilic «■ L_ = 0 isotropic -S4- H- = L_ = 0 stationary ** H = 0 totally umbilic •» L+ = L_ = 0. Remark 2.1 J. / / we change the orientation on N2, the null line bundles TN1 switch. A null-stationary (null-umbilic) surface satisfies then H+ = 0 (L+ = 0).

and TN1

2. Let us consider a spacelike immersion f : JV2 —> Mf and let g := exp(2p)g be a conformally equivalent metric to g on M4. Denote by II the second fundamental form of the isometric immersion f : (N,f*g) —¥ (M,g). The comparison of the covariant derivatives gives VXY

= VXY + dp(X)Y + dp(Y)X - g(X, Y)gradp

127 and this yields II = II - g® Afgradp = f*g ® [exp(-2p) • (H - Mgradp)] + L+ + L~. This shows that the vanishing of the components L+ and L_ is invariant under conformal change of the metric g on M4. In particular, the property of a spacelike immersion to be totally umbilic is conformally invariant, whereas the stationary condition isn't conformally invariant. 2.2

Holomorphic Gauss lifts of spacelike immersed surfaces

We define now the Gauss lift of a conformally spacelike immersed surface to the twistor space. Geometric properties of the conformal immersion are related to the holomorphicity of its Gauss lift. Let (N2, JN) be a Riemannian surface and let / : N2 -> M4 be a conformal spacelike immersion into an oriented Lorentzian 4-mamfold, i.e. the complex structure JN is orthogonal with respect to the induced metric f*g on N. The positive null line bundle TNj; exists on N and the space TnN+ is for every n G N a null direction in TM4, i.e. an element of the twistor space Z{M). We define the Gauss lift 7/ of / into the twistor space Z(M) by jf.N n

-> Z(M) H-> TnN±

11 / f:N-*M

'

Z(M) 4

•

Definition 2.2 Let f : N2 —> M4 be a conformal spacelike immersion of a Riemannian surface into an oriented Lorentzian 4-space. The immersion f is called 0 + -holomorphic if jf : (N,JN) N

0~-holomorphic if~/f : (N,J ) 2

—> (Z(M),0+) -> (Z(M),0~)

is holomorphic

or

is holomorphic.

4

The conformal immersion f : N —> M is called horizontal if the lift 7/ to Z(M) is horizontal that means dyf(TnN) C Tf{n)Z(M) for all ne N. That a conformal spacelike immersion / : N2 -¥ M4 is horizontal means that the parallel displacement of a positive normal null vector along an arbitrary curve on N2 remains a positive normal null vector. Let us consider the differential of the lift jf. The horizontal part d-yf is simply the horizontal lift of df. For the vertical part d-yY holds *l) = ~ " 1 3 ~ a , a ( g i 3 - E23)H + - ^ - ^ [ E u

-

E24)H,

where UHJ :— g(VMsi,Sj) are the connections forms of the Levi-Civita connection on M4 with respect to a local Darboux frame s. From this formula we obtain by a direct calculation Proposition 2.1 Let f : N2 —> M4 be a conformal immersion. The following relations hold: 1. f is 0~ -holomorphic e> H- = 0 «* f is null-stationary 2. f is (D+-holomorphic O £_ = 0 •*=*■ f is null-umbilic

128 3. f is horizontal <=> H_ = L_ = 0 «• f is isotropic Remark 2.2 We could say that isotropic surfaces in Mf are super-null-stationary. This would be analogous to the notation for surfaces in a Riemannian 4-manifold, that have a horizontal Gauss lift to the Riemannian twistor space. Those surfaces satisfy a stronger condition then minimality and are called superminimal. 2.3

Twistorial construction of spacelike surfaces

The Proposition 2.1 of the previous section relates geometric properties of a conformally and space­ like immersed surface to the holomorphicity of its Gauss lift. In the following, the reconstruction of null-stationary and null-umbilic surfaces in Lorentzian 4-spaces from holomorphic curves in the Lorentzian twistor space is established. We give a local classification of null-umbilic surfaces in conformally flat Lorentzian 4-spaces. In particular, we classify isotropic surfaces in the Lorentzian space forms «[, Si and H\. Theorem 2.1 Let (AT2, JN) manifold.

be a Riemannian surface and let Mf be an oriented Lorentzian 4-

1. If 1 : (N2,JN) -» {Z(M),0~) is a holomorphic map into the twistor space over Mi such that d"f / 0 on N and 7 is non-vertical that means dy(TnN) <£ T^n^Z(M) for alln £ N then the projection f := 7T o 7 : AT2 -> Mi is a conformal spacelike immersion with H- = 0. In particular, there is a bijective corre­ spondence between 0~-holomorphic non-vertical curves in Z(Mi) and conformally immersed null-stationary surfaces in Mi2. There is a bijective correspondence between 0 + -holomorphic non-vertical curves in and conformally immersed surfaces in Mi with L_ = 0.

Z(Mf)

Proof: We prove only the first statement, since the proof of the second statement works the same. Let 7 : (N, JN) —> (Z(M),0~) be a non-vertical holomorphic curve with d-y / 0 on N. Obviously, the projected map / := n o 7 is an immersion. It is dj(TN) C C and it follows n, (df(TnN))

C L^"' N

Moreover, it is 7r o d~f o J PTy(n) °dfoj£>

Vn € N. = J~ a i o tfy, which means that

= /?(">

0 prrf{n)

0 df

. TnN

_> Ll(n)/K7(»)

Vn

g

N<

where pr 7 ( n ) : L 7 ' n ) -> L'r(n)/i{"i'(») ; s the natural projection. J"<(n) is orthogonal with respect to Qf(n) a n < l therefore the immersion / : N2 -► Mi is conformal. The Gauss lift 7/ of the immersion / = 7r o 7 is equal to the original map 7 and we can apply Proposition 2.1 to obtain that / is a null-stationary immersion □ Parts of this theorem can also be found in [Bob98]. Similar, it can be proved that there is a bijective correspondence between conformally immersed totally umbilic surfaces in a Riemannian 3-manifold P 3 and non-vertical holomorphic curves in the unit sphere bundle (S2(TP),ns^TP1,Js2(TP1)

(comp. [Lei99]).

129

One way to obtain holomorphic curves in an almost optical manifold is to construct holomorphic curves in a CR-hypersurface. For example, if P3 C Mf is an oriented spacelike hypersurface in a Lorentzian 4-manifold then the unit sphere bundle S2(TP) with CR-structure (H, J+) is a CRhypersurface in (Z(M), 0+) (Theorem 1.3). A non-vertical holomorphic curve in (S2(TP),M, J+) is then a non-vertical holomorphic curve in ( 2 ( M ) , C + ) and projects to a null-umbilic immersed surface in Mf. The existence of a holomorphic curve in an integrable optical manifold gives rise to a whole family of holomorphic curves. This is the idea to Corollary 2.1 Let Mf be an oriented conformally flat Lorentzian 4-manifold and let k 6 T(K) be a vector field in the distribution K. on the twistor space Z(Mf). If f : N2 -> Mf is a conformally immersed Riemannian surface with L- = 0 then the map ft:=no^aJf:N2^Mf, where ffi denotes the flow of the field k, is at least locally for small t's a conformally immersed surface in Mf with L_ = 0. Proof: If / : N2 —> Mf is a conformally immersed surface with L- = 0 then the Gauss lift N

V:(N,J

)^(Z(M),0+)

is holomorphic. Since Mf is conformally flat, 0 + on Z(M) is integrable and therefore
is locally and for small t's a non-vertical holomorphic map (Proposition 1.1), which projects to a null-umbilic immersed surface in Mf □ Corollary 2.1 may be interpreted as follows. Let N2 be an oriented spacelike surface in Mf. The normal bundle TN1- over N2 in Mf decomposes to the bundle of positive and negative normal null directions on the surface N2 in Mf. Every positive normal null vector on N2 defines a geodesic that intersects the surface N2. We call such a geodesic through N2 positive normal null. The distribution K on Z(M) is the lightlike geodesic spray of Mf. Hence, Corollary 2.1 says that a smooth bijective deformation of a null-umbilic surface N2 along its positive normal null geodesies is also null-umbilic. In case that the integral curves to fc € T(/C) on Z(Mf) are complete, the map ft = TT O ffi o 7^ in Corollary 2.1 is defined for every i e l . But it may happen that the holomorphic curve <j>\ o 7^ in Z(Mf) isn't any more a non-vertical curve for some t £ R, i.e the deformation ft isn't in general an immersion for every t S K. Corollary 2.2 Let Mf be conformally flat. Every null-umbilic surface N2 in Mf is locally a deformation of a totally umbilic surface N2 in Mf along the positive normal null geodesies on N2. Proof: Since Mf is conformally flat, there exists a hypersphere segment P ^ C Mf in every point m 6 Mf, i.e. a totally umbilic hypersurface. Let n G N2 be arbitrary and let P3 be a hypersphere segment in n. Then exists an open neighborhood U C N2 of n S iV2 such that every positive nor­ mal null geodesic through U intersects the hypersurface P%. Moreover, we can find a vector field fc,

130 which is tangential to the positive normal null geodesies through U, and a neighborhood U C U of n such that N2 := 0*, (U) C P 3 for some t\ > 0 and such that the map $ : U -> <$((!) is a diffeomorphism for every t 6 [0,ti]. Since N2 is null-umbilic, it follows that N2 is totally umbilic in P 3 . This implies that N2 is even totally umbilic in Mf and the open set U C N2 is a deformation of iv"2D There is no analogous version of Corollary 2.1 for null-stationary surfaces, sine the almost optical structure 0~ on Z(M) doesn't satisfy condition A. However, we will prove at the end of this section an analogous version of Corollary 2.1 and 2.2 for isotropic surfaces in case that Mf has constant sectional curvature. Examples A. Null-umbilic surfaces and isotropic surfaces in the Minkowski space Kj Let us consider the flat Minkowski space S.\. The closed totally umbilic spacelike hypersurfaces in M.\ are isometric embeddings of the Euclidean or hyperbolic 3-space. We choose the totally geodesic embedding P 3 = R3 ^ R*. (2/1,2/2,3/3) >">• (0,2/1.3/2,2/3) The closed totally umbilic surfaces in the Euclidean 3-space R3 are isometric embeddings of the 2-sphere or the Euclidean plane. The embedding j:

S2 M- m\ (2/1,2/2,3/3) >-+ (0,3/1,3/2,3/3),

2/i + 2/1 + 3/1 = *

is totally umbilic. The null vector field

is positive normal to the surface j(S2) C Rf. The positive normal null geodesies intersecting the sphere j(S2) are given by 7(a,bfi) (*) = (<> to> *6> tc)>

te&,

a2 + b2 + c2 = 1.

Every deformation of the surface j(S2) C JR* along the positive normal null geodesies has the form

h-

S2

-+ Kf,

3/= (2/1,3/2,2/3)

>-► (A,Ayi,Aj/2,Aj/3)

where A(y) is a smooth function on S2. If A ^ 0 is a positive or negative function then the map j \ is a conformal embedding. By Corollary 2.1, it follows that those conformal embeddings of S2 are null-umbilic surfaces in R*. Since the Gauss lift of the embedding j isn't horizontal, the embedding jx has no horizontal Gauss lift for any function A ^ 0 and the surface jx(S2) is nowhere isotropic in U*. In case that X(y) = 0 for some y e S2, the map jx isn't an immersion. The isometric embedding i:

(E2,<,>)

->

(Ki,<,>?)

(21,22)

i-4

(0,0,2i,22)

131 of the Euclidean plan is totally geodesic. T h e positive normal null geodesies intersecting the plane j(R 2 ) are given by 7(a,l>) (<) = (*.*> a. &) >

t£WL.

Every surface t h a t is a deformation of j(R 2 ) C R* along t h e positive normal null geodesies on »(R 2 ) has t h e form

R2

->

(zi,Z 2 )

H>

i\ :

RJ, (\(zi,Z2),\(zi,Z2),Zl,Z2)

where A(zi,Z2) is a smooth function on R 2 . T h e embedding i^ is isometric and i,\(R 2 ) is a nullumbilic surface in Rj for every function A on R 2 . If we calculate t h e second fundamental form of i\ into R\, we obtain / / = < , >

2

® ( A

l 2

A ) . ^

+

L+.

T h a t means an embedding of t h e form i\ is even isotropic. Let / : JV2 —> Rj b e an arbitrary conformal null-umbilic immersion of a connected Riemannian surface N2. It follows from Corollary 2.2 t h a t t h e immersion / has locally t h e form j \ or i\ up to an isometry of R\ (comp. [Elg96]). Furthermore, the subset of N2, where / is isotropic, is open and closed, i.e. t h e immersion / is globally isotropic or globally only null-umbilic. In case t h a t the immersion / is only null-umbilic, it can b e locally deformated along t h e normal null geodesies to an unique point in R*. T h e fact t h a t every complete lightlike geodesic in M.\ intersects t h e hyperplane P3 C t j in a single point gives rise to an uniquely denned smooth m a p /' :

N2 n

-> P 3 S R 3 , >-¥ rn

where rn is t h e intersection point of t h e positive normal null geodesic through / ( n ) with t h e hyperplane P 3 C K*. If / is isotropic t h e n the map / ' is an immersion. In case t h a t / is only null-umbilic, it may h a p p e n t h a t / ' has singularities. Since t h e inverse image of a singularity of / ' is closed, it follows t h a t / ' has to b e already constant on N2. Because two null geodesies in Rj have at most one intersection point, we can find another totally geodesic hypersurface P' ~ R 3 C Rj such t h a t t h e unique deformation of / into P' = K3 is an immersion. Hence, we obtain in any case a smooth deformation / ' : N2 —> R 3 of t h e null-umbilic immersion / , which is by Corollary 2.1 conformal and totally umbilic. Moreover, t h e immersion / is isotropic iff t h e induced immersion / ' into R 3 is totally geodesic. In this case t h e induced metric / ' * ( < , >*) on N2 is flat, whereas if / ' isn't totally geodesic the metric / ' * ( < , >*) on N2 has positive constant sectional curvature. We can conclude that in the class of compact Riemannian surfaces only t h e sphere S2 admits a conformal null-umbilic immersion into R\. Such an immersion has the form j \ : S2 —► R\ u p to an isometry of R*. Moreover, since t h e only complete flat Riemannian 2-manifold t h a t admits a totally geodesic immersion into R 3 is the plan R 2 , it follows t h a t there exists no isotropic conformal immersion of a compact Riemannian surface into R*. T h e property that a surface is null-umbilic is independent of t h e conformal class of the ambient Lorentzian 4-space. Hence, the immersions of t h e form i\ and j \ describe locally every null-umbilic surface in a conformally flat Lorentzian 4-space.

132 B. Null-umbilic surfaces and isotropic surfaces in the pseudosphere Sf Let S 4 be the pseudoshere. The image of the isometric embedding i:

S3 C E4 (S/l.y2iJ/3.J/4)

*-> 5 4 C Rf I"*- (0,1/1,J/2,J/3it/4)

is a totally geodesic spacelike hypersurface in Sf. Every closed totally umbilic surface in S3 is up to an isometry of S3 the image of a conformal embedding of the form S3 C K4. >-> (c, Vl - c2 • j/i, Vl - c2 • 3/2, Vl - c2 ■ y3),

]-■

(l/i,J/2,3/3) 2

2

c€B, |c|
3

Only the surface i(S ) := j°(5 ) C S is totally geodesic. Every deformation of the embedding j c : S2 "-> S 4 along its positive normal null geodesies has the form

S2

Jl:

^

Sf,

i-+ (A, A Vl - c2 + c, (Vl - c2 - Ac)j/)

V= (j/i.Jte.Jfe)

where A : S2 -> B is an arbitrary smooth function. The map jjj : 5 2 -> 5 4 is a conformal embedding if c = 0 or A ^ ' 1~c on S 2 . Those embedded surfaces are null-umbilic in Sf. The embedding

h ■■=%■■ S2 ^ y

->

SfcRl

(A,A,j/),

||y|| = l

is isometric for every smooth function A on S2. Let sp :

St\{y

6 S4 :

M

= 1}

A

1 4 \H 0 3

I/= (j/l,!/2,J/3,J/4,J/5)

!">

Trjj(S/l,!/2,J/4,y5)

l+fxll 3 ^ 1 ' 1 2 ' ^ ' 2 ~ 1 ' : E 3 ' 1 4 )

^

x = (xi,x2,a;3>^4)

denote the stereographic projection in (0,0,1,0,0) 6 S 4 . We set p = In t , iLiia and A := exp(-p)Ao sp. It holds ap_1otx:

B2

<-> S 4 ,

(zi,z 2 ) >-► 1+z I +z a (A, A, z ' + ^~ 1 ,^i,z 2 ) = (A, A,j/3,1/4,3/5), yl + y \ + yi = ^ i.e. the stereographic projection sp maps the surface «A(S 2 ) in S 4 to the surface ij;(B2) in R4. We can use this to calculate the second fundamental form IT of the isometric embedding i\ : S2 •—> Sf. It holds //

=

<, >s* ®[exp(-2p){H - tfgrad(p))] + L+

This proves that i\ : S2 <-} Sf is an isotropic embedding into Sf. The embeddings of the form Jl, c ^ 0, are never isotropic. If A is a spherical function to the eigenvalue —2 of the Laplace operator A* on S 2 , the surface J^tS 2 ) C Sf is stationary and null-umbilic.

133 Let / : N2 —► Sf b e an arbitrary conformal null-umbilic immersion. Again, t h e immersion / can b e deformated along its normal null geodesies t o a conformal totally umbilic immersion / ' : N2 -> S3. T h e immersion / is isotropic iff the induced immersion / ' is totally geodesic. T h e induced metric / ' * ( < ! >Sl) on N2 has positive constant sectional curvature. We can conclude t h a t in the class of compact Riemannian surfaces only t h e sphere S2 admits a conformal null-umbilic immersion into S*. T h e sphere S2 admits even isotropic embeddings into Sf. C. Null-umbilic surfaces and isotropic surfaces in the pseudohyperbolic space Hf Let us consider the pseudohyperbolic 4-space H\ : = { i £ l ^ : < i , i > ^ = - 1 } C K | . T h e isometric embedding i:

H3 C «.\ (VI>V2,3/3,J/4)

<-+ Hf C B | >-» (yi,0,y2,y3,y4),

j/i > 0, \\y\\ = - l

is a spacelike totally geodesic hypersurface in Hf. U p t o an isometry of H3, every closed totally umbilic surface in H3 has one of the following forms: j:

S2

->

(21, 22, 23) i:

2

H (zi,z2,z3)

H3,

!-> (c, V c 2 - 1 • Zi, V c 2 - 1 - 2 2 , Vc 2 - 1 • 23), -> 1-4

T h e deformations of t h e surfaces j{S2) geodesies look as follows:

s2

hh:

H

(zi,22,2 3 )

C Hi a n d i{H2) C ffj* along its positive normal null

^ Hi,

z = (zi,z2,z3) 2

C>1

H3. (zi,0,z2,z3)

->

>-> (A, AVc 2 - 1 + C, ( V C 2 - 1 + Ac) • z),

c > l

Hi,

*-» (21, A, A, 22, z 3 )

where A(z) is a smooth function on S2 resp. H2. The m a p J'A is a conformal null-umbilic embedding for every function A ^ — ' - 1 on S2 a n d t h e map i\ is an isometric null-umbilic embedding for every function A o n H2. For t h e second fundamental form of t h e isometric embedding i\, we calculate

/ Z ^ > - . [ A - A - 2 A ] . I ( A + _^.)

+

,+,

i.e. t h e embeddings of the form i\ are isotropic, whereas t h e embeddings j \ are only null-umbilic. T h e only compact Riemannian surface t h a t admits a conformal null-umbilic immersion into Hi is t h e sphere S2 and there is no isotropic immersion of a compact Riemannian surface into Hi. T h e discussion of t h e isotropic surfaces in t h e space forms K.\, Si a n d Hi proves t h e following analogous result t o Corollary 2.1 and 2.2: T h e o r e m 2 . 2 Let Mi be an oriented Lorentzian 1. Every smooth deformation of an isotropic geodesies of N2 remains isotropic.

4-manifold surface N

2

with constant

sectional

in Mi along the positive

curvature. normal

null

134 2. Every isotropic surface in Mf is locally a smooth deformation of a totally geodesic surface N2 in M* along the positive normal null geodesies on N2. Remark 2.3 The property of a surface to be null-stationary isn't a conformal invariant of the ambient Lorentzian manifold. In particular, a stereographic projection doesn't map every nullstationary surface again into such one.

References [AHS78] M.F. Atiyah, N.J. Hitchin and I.M. Singer. Self-duality in Four-dimensional geometry. Proc. R.S. London A362 (1978), 425-461.

Riemannian

[Baum81] Helga Baum. Spin-Strukturen und Dirac-Operatoren iiber pseudo-Riemannschen nigfaltigkeiten. Nummer 41 in Teubner-Texte zur Mathemtik. Teubner, 1981.

Man-

[Bes87] Arthur L. Besse. Einstein manifolds. Band 10 der Reihe Ergebnisse der Mathematik und ihrer Grenzgebiete, 3.Folge. Springer, 1987. [Bob98] M. Bobienski. Master thesis (Polish). Warszawa, 1998. [Bry82] R.L. Bryant. Conformal and minimal immersions of compact surfaces into the ^-sphere. Joun. Diff. Geom. 17(1982), 455-473. [ES85] J. Eells, S. Salamon. Twistorial constructions of harmonic maps of surfaces into fourmanifolds. Ann. Scuola Norm. Sup. Pisa 12(1985), 589-640. [Elg96] R. Elghanmi. Spacelike surfaces in Lorentzian manifolds. Joun. Diff. Geom. and its Appl. 6(1996), 199-218. [Fri81] Thomas Friedrich. Self-duality of Riemannian manifolds and connections. In Teubner-Text Bd. 34. Teubner-Verlag Leipzig, 1981. [Fri84] Thomas Friedrich. On surfaces in Four-Spaces. Ann. Glob. Analysis and Geometry Vol. 2, No. 3(1984), pp. 257-287. [Fri97] Th. Friedrich. On superminimal surfaces. Archivum Mathematicum vol. 33(1997), pp. 4156. [Jac90] Howard Jacobowitz. An introduction to CR structures. Volume 32 of the series Mathemat­ ical Surveys and Monographs. Amer. Math. Soc, 1990. [JR90] G.R. Jensen, M. Rigoli. Neutral surfaces in Neutral Four-spaces. Le Matematiche Vol. XLV(1990) -Fasc. II, pp. 407-443. [Lei98] F. Leitner. The twistor space of a Lorentzian manifold. Preprint No. 314 of the Sonderforschungsbereich 288. Berlin, 1998. http://www-sfb288.math.tu-berlin.de [Lei99] F. Leitner. Twistorial Constructions of spacelike surfaces in Lorentzian ^-manj/oW«. SfB 288 Preprint No. 370. Berlin, 1999. [0'N83] Barret O'Neill. Semi-Riemannian Press, 1983.

geometry. Pure and Applied Mathematics. Akademic

135 [0'N95] Barret O'Neill. The geometry of Kerr black holes. A K Peters, Wellesley. Massachusetts, 1995. [Nur96] Pawel Nurowski. Optical geometries and related structures. J. Geometry and Physics 18(1996), p. 335-348. [Tra85] Andrzej Trautmann. Optical structures in relativistic theories. Societe Mathematique de France, Asterisque, hors serie, 1985, p. 401-420. Felipe Leitner Institut fur Reine Mathematik Humboldt-Universitat zu Berlin Sitz: Ziegelstr. 13a 10099 Berlin, Germany email: [email protected] RECEIVED JANUARY 31, 2000

136

Geometry and Topology of Submanifolds X eds. W. H. Chen et al. (pp. 136-144) © 2000 World Scientific Publishing Co.

Affine differential geometry and partial differential equations of fourth order A n - M i n Li*

Fang Jia

KEYWORDS: classification of Blaschke hypersurfaces, Monge-Ampere equations, affine Bernstein problem 2000 MATHEMATICS SUBJECT CLASSIFICATION: 53A15, 53C40, 58J05, 58J60, 35G25 It is well known that differential geometry has been an extremely rich source of interesting problems in partial differential equations. In affine differential geometry the study of affine curvature functions leads to the study of nonlinear partial differential equations of second order or fourth order. Let An+1 be the unimodular affine space of dimension n + l, and M be a locally strongly convex hypersurface in An+1 with affine principal curvatures Ai, A2, ■ ■., A„. Set L r

= ( j ^ '

]C

A

'i'--V

(r = 0 , l , . . . , n ) .

1<«1< —<«,

Wc call Li the affine mean curvature and Ln the affine Gauss-Kronecker curvature. The simplest hypersurfaces in affine differential geometry are affine hyperspheres. The study of parabolic affine hyperspheres is equivalent to the study of the convex solutions of the equation (see [L-S-Z-2])

The study of elliptic and hyperbolic affine hyperspheres is, by a Legendre transformation, equivalent to the study of the equation (2)

det(^-) = (*„)—.

Unlike in the case of the Euclidean hyperspheres, there are many equiaffinely non-equivalent affine hyperspheres. In fact, given any convex domain fl and any function <j> e C°°(9fi), one can construct a hypersphere from the solution of a boundary value problem of the equation (1) or (2). The classification of complete, locally strongly convex affine hyperspheres is a fundamental and important problem, which was solved through works of many geometers during the last decades. We mention E. Calabi, A.V. Pogorelov, L. Nirenberg, S.T. Yau, S.Y. Cheng, T. Sasaki, S. Gigena, A.-M. Li and others. "partially supported by a NSFC grant, a NSFC-DFG cooperation project and a Qiu Shi grant This paper is in final version and no form will appear elsewhere

137 A more general interesting problem is the global classification of hypersurfaces with Lr = const. In case when M is compact, one can derive a Minkowski-type integral formula for an affine hypersurface using a method of S.S. Chern. By means of Minkowski integral formulas the first author proved that if Lr = const everywhere on M for some fixed r (1 < r < n) then M must be an ellipsoid. Under stronger assumptions this result was first stated in [SU], and extended in [SI-3], [SI-4], [HS-S]. In case when M is complete and noncompact, the study of hypersurfaces with Lr = const leads to the study of a complicated nonlinear PDE of fourth order. All elliptic and hyperbolic affine hyperspheres have constant affine mean curvature L\ ^ 0. So far the classification of affine-complete, locally strongly convex hypersurfaces with L\ = const ^ 0 is open. In recent years great progress in the study of hypersurfaces with L\ = 0 or Ln = const has been made. In this article we give a short survey on the recent developments of the affine Bernstein problem and the classification of complete hypersurfaces with constant Gauss-Kronecker curvature.

1

Affine Bernstein problem

The affine maximal hypersurfaces are extremals of the interior variation of the affinely invariant volume, the corresponding Euler-Lagrange equation is L\ = 0. Historically the hypersurfaces with L\ = 0 were called "affine minima! hypersurfaces". Calabi (see [CA1]) suggested to call locally strongly convex hypersurfaces with L\ — 0 "affine maximal hypersurfaces" because of the following result: (Calabi) Let x, x* : fi —► An+l be two graphs defined on a compact domain fi C n A by locally strongly convex functions / , / # , namely xn+i = f(x\,...,xn) and xn+\ = f*(xi,..., xn), respectively. Suppose that / = / * and J £ = 4^-, for i = 1 , . . . , n on dQ. Denote by Li,Lf, respectively, the affine mean curvatures of x and x#, and by dV, dV* their equi-affinely invariant volume elements, respectively. If L\ = 0 on Cl, then

[dV> f

Jn

dV*,

J(i

and equality holds if and only if / = / * on fi. Let x : M —> An+1 convex function

be an affine maximal hypersurface, given as a graph of a strictly Zn+l = / ( Z i , . . . , £ „ ) .

Then the Blaschke metric is given by G = det

(3)

97 ^dxjdx

-l/(n+2)

fl2

^ ^

d X i d X i

-

The affine conormal vector field U can be identified with (4)

U

det

a2/ xi- 1 ^ 2 ) / of dxidxj

dxi' " ''

df dxn'

138 The formula AU = —nL\U implies that x is an affine maximal hypersurface if and only if / satisfies the following PDE det

(5)

52/ dXidxj,

-l/(n+2) :

0,

where A denotes the Laplacian with respect to the Blaschke metric, which is defined by

All parabolic affine hyperspheres have vanishing affine mean curvature, so there are many examples with L\ = 0. The elliptic paraboloid

z»+i = gK*i)2 + • • • + K) 2 ]. (zi,..., xn) e An. is the only example of complete, locally strongly convex hypersurfaces with L\ = 0. To study affine maximal surfaces, Terng ([T]) derived an affine Weierstrass representation. Given a triple of functions U = (Ui (u, v), U2(u, v), U3(u, v)) defined on D, C R2 satisfying the following two conditions: (1) Ui, i = 1,2,3, are harmonic with respect to the canonical metric of R2;

(2)det(f,f,t/)>0ina

Then one can construct an affine maximal surface x : fi —> A3 as follows: x= ["'"

[U,Uv]du-[U,Uu]dv,

where f2 is a simply connected domain and (uo,vo), (u,v) G fi. From the point of view of local differential geometry this formula gives all affine maximal surfaces. About complete affine maximal surfaces there are two open conjectures, due to Chern and Calabi, both are called the "affine Bernstein problem" (see [CH], [CA-1]): (Chern) Let X3 = f(x\,x2) be a strictly convex function defined for all (xi,x2) € A2. If M = (xi,Xi,f(xi,X2)) is an affine maximal surface, then M must be an elliptic paraboloid. (Calabi) A locally strongly convex affine-complete surface x : M —> A3 with Li = 0 is an elliptic paraboloid. These conjectures were generalized to higher dimensions (see [L-S-Z-2], [CA-1], [SI-2]). Note that there are several notions of completeness in affine geometry. The two versions of the affine Bernstein problem assume different completeness conditions. Calabi's conjecture assumes affine completeness, that is, completeness of the Blaschke metric on M; Chern's conjecture assumes Euclidean completeness, that is, completeness of the Riemannian metric on M induced from a Euclidean metric in An+1. Generally, the affine completeness and the Euclidean completeness are not equivalent (see [SCH], [NO]). Some partial solutions to Chern's conjecture have been made by several authors. Calabi proved that an affine maximal surface M is an elliptic paraboloid if it is both Euclidean

139 complete and affine complete (see [CA-1]). The first author (see [L-1]) proved that an affine maximal surface M is an elliptic paraboloid if there exists a real constant N > 0 such that |Ai • A2I < N everywhere. For the higher dimensional affine Bernstein problem Pogorelov (see [P-2]) proved the following result: Let x : M —> An+l be given by a strictly convex function xn+i = f(xi,...,xn). If / satisfies d2f(x)

—> y^^CjjdxjdXj, *,j

as x —► 00, where the ci3- are constants, then M is an elliptic paraboloid. Recently we solved Chern's conjecture in dimension n = 2 and n = 3 (see [L-J-l]). Precisely, we have proved the following Theorem A: Let xn+i = f{x\,...,xn) be a strictly convex function defined for all \X\, • • - , Xn ) e An. If M = {(xi,..., xn, f{xi,..., xn))} is an affine maximal hypersurface, then in the case n = 2 or n = 3, M must be an elliptic paraboloid. We indicate basic ideas of the proof. Denote

*.=

« p

By a direct calculation we get A$ >

(1 - S)n ^ $ 2 2(n - ■\)*— 1) ^ $ 22

n2 - (1 - 2S)n - 2 . 2 ( n -- l 1) ) ^ 2(n ^ 2

2

'' P»

2

( n-- l1) ) (n - 2)) (n n +fa-2 fa-2^ $ , 2J ^ 2 8n 2(n - 1) p where 5 > 0 is a small number. The key point of the proof of the theorem is to estimate $. Suppose that x : M —> An+1 is an affine maximal hypersurface , given as a graph of a strictly convex function f(x\,..., xn) defined for all (xi,... ,xn) £ An. For a fixed point p £ M we may assume that the plane x n+i = 0 is parallel to the tangent hyperplane of M at p and p has the coordinates (0,..., 0). For any C > 0 the set Mc := {x e M 11„+1 = / ( x i , . . . ,i„) < C} is compact. We consider the following function (6)

^ e x pl

{

^

C-/

+

* / r - / ) p2 ^ ^ + ^ ^ 2 J }v ( ( JC

(/ + <*)

V

(/ + d)

where m and d are positive constants, and (7)

* = exp{(C - / ) * } .

F attains its supremum at some interior point p € Mc- Then we derive the following estimate In JS-J + 2 In C + a (8) $<_^=/

C-f

140 under a restriction (2 - (" %^n ^ - "vt^-i) 2 ) > 0, where o is a constant independent of C. This restriction can be satisfied only for n = 2 or n = 3. We let C —> oo to obtain $ = 0

on

M.

This implies that, if f(xi,x2, ■ ■ ■ ,xn) is a global convex solution of the fourth order PDE (5), then f(x\,x2,.. ■, xn) must be a global convex solution of the PDE det I - — - — I = const \OXiOXj J

for n = 2 or n = 3. It follows from a result of Pogorelov (see [P-1]) that / must be a quadratic polynomial. D Trudinger and X.J. Wang also solved Chern's conjecture for dimension n = 2 (see [T-W]), using another PDE method. They proved Chern's conjecture by three steps, starting with another version of equation (5), which can be written as -(n+l)/(»+2) ^

(9)

£E7*

&

dxkdxi

det(dxidxj

= 0,

where (£/*') denotes the cofactor matrix of ( a ^X.)■ Step 1. Show that the Bernstein problem can be reduced to a problem of a priori estimate for solutions of the equation (9). Step 2. Apply the recent Holder estimate of Caffarelli and Gutierrez [C-G] for the linearized Monge-Ampere equation to obtain the Bernstein property under a restriction of uniform strict convexity. Step 3. Establish a modulus of convexity estimate for solutions of equation (9) in two dimensions. So far Calabi's conjecture is open. Some partial solutions to this conjecture have been made. The first author ([L-1]) proved that a locally strongly convex, affine complete, affine maximal surface is an elliptic paraboloid, if the Gauss map g omits 4 or more directions in general position. Martinez and Milan ([M-M]) proved that a locally strongly convex, affine complete, affine maximal surface M is an elliptic paraboloid, if there is a constant N > 0 such that the affine Gauss-Kronecker curvature satisfies I-L2I < N everywhere on M.

2

Classification of complete hypersurfaces with con­ stant affine Gauss-Kronecker curvature

Consider a locally strongly convex hypersurface x : M —> An+1. Denote by G its Blaschke metric and by B its Weingarten form. We call x of hyperbolic (elliptic) type if B is negative (positive) definite, respectively. If B is negative definite we define B := —B, then B is a Riemannian metric on M, called the Weingarten metric. The affine Gauss-Kronecker curvature is given by _ det(ff) "~det(G)-

141 In case when M is of elliptic type, the Ricci tensor Ric(G) of the metric G is positively bounded below. Thus (M, G) must be compact. But then Ln = const implies that M must be an ellipsoid (see [L-S-Z-2]). Now we consider the case that M is of hyperbolic type. The classification of complete hypersurfaces of hyperbolic type with constant affine Gauss-Kronecker curvature is an extension of the classification of complete hyperbolic affine hyperspheres. Suppose that M is locally given as a graph by X n + i — J{Xi, . . . , Xn).

Denote by U and Y the affine conormal vector field and the affine normal vector field, respectively. In terms of the coordinates xi}.. .,xn and the function / , the condition Ln = const is a complicated PDE of fourth order. By considering the affine conormal immersion U :M — ► i? n+1 Li-Simon-Chen could split this PDE of fourth order into two linked MongeAmpere equations. The key point is the following fact Proposition: Let x : M —> An+1 be an affine hypersurface. Then Ln = const if and only if both U(M) and Y(M) are affine hyperspheres. To sketch the method of Li-Simon-Chen, consider a locally strongly convex hypersurface of hyperbolic type x : M —> An+1. Suppose that Ln = \ and that the Weingarten metric B defines a complete Riemannian metric on M. Then, both U(M) and Y(M), are complete affine hyperspheres. Moreover, Y(M) is a graph of a convex function /* : Rn —> R, and we can consider x : M -> An+1 as graph x

n+\

=

/(xli ■ ■ ■ :

x

n)

of some function / : Q. —> R defined on some domain H of R™. Introduce the Legendre transformation dxC

and denote by f! the Legendre transformation domain of / , i.e. u : il —> R and

« = {(&.---,UI6 = |p(zi,...,a:„)eft}. Then (see [L-S-CH]) the Legendre domain ft of / is a bounded convex domain and the Legendre function u of / satisfies

where u* is the Legendre transformation function of /*, which is the solution of the PDE fl2)

v

'

1 det(H-) = (-„•)-- 2 in ft,

' u* = o on an.

142 On the other hand, for any bounded C°° convex domain fi and any function (/> e Coc(3f2) (see [L-S-CH]), there is a unique solution « € C 00 (n) of the equations (13),(14). (13)

( det(^-) = (-«*)-»-2 ™ " . 1 u = on 9f2,

(14)

J det(^%) = (-«•)-- 2 in fl, 1 u* = 0 on dQ.

From the solution « of (13)-(14) Li-Simon-Chen construct a hypersurface M as the graph of the Legendre transformation function / of u: du

f(xi,...,xn)

= Y2^aF ~u> ^

€ i l

Then M has constant affine Gauss-Kronecker curvature Ln = 1 and it is Euclidean complete (i.e. it is closed with respect to the Euclidean topology of Rn+l). The following is an interesting problem: when is the Blaschke metric of the constructed hypersurface complete? Li-Simon-Chen proved that in the case when f2 is the ball Si(0), M is complete with respect to the Blaschke metric. We believe that this result is true for any smooth convex domain fi. In [L-S-Z-l] Li-Simon-Zhao further studied the prescribed Gauss-Kronecker curvature problem. They extended the global investigations of [L-S-CH] to a hypersurface with Ln ^ const. Li-Simon-Zhao proved that: Given a C°° bounded convex domain fi and functions tj> £ C°°(<9fi) and Sn G C°°(fl) with Sn > 0, there is a hyperbolic equiaffine hypersurface M in An+1 which has the following properties: 1) its equiaffine Gauss-Kronecker curvature is the given function Sn; 2) M is both Euclidean complete and Weingarten-complete; 3) if H is the ball {(£i, ■■■,(„) G -R"| £ ^ < 1}, then M is also complete with respect to the Blaschke metric. Liao ([LIA]) gave a local classification and a global classification of locally strongly convex surfaces with L2 = 0. The key point is the following fact: Proposition: Let x : M —> A3 be a locally strongly convex surface then x(M) has affine Gauss-Kronecker curvature zero if and only if U(M) is of the Gauss curvature zero. Liao proved the following global result: Theorem: Let x : M —>• Az be a Euclidean complete convex surface with L2 = 0. If its affine conormal immersion U : M —> R3 is Euclidean complete, then x(M) is either an elliptic paraboloid or the surface defined by x\ — x\ + C2 cosh2 X3 = 0.

143

References [BLA]

W. Blaschke, Vorlesungen fiber Differentialgeometrie, vol. II. Berlin, Springer, 1923.

[CA-1]

E. Calabi, Hypersurfaces with maximal affinely invariant area, Amer. J. Math. 104(1984), 91-126.

[CA-2]

, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Joergens, Mich. Math. J. 5(1958), 105-126.

[C-G]

L.A. Caffarelli and C.E. Gutierrez, Properties of the solutions of the linearized MongeAmpere equations, Amer. J. Math. 119(1997), 423-465.

[C-Y]

S.Y. Cheng and S.T. Yau, Complete affine hypersurfaces, Part I. The completeness of affine metrics, Commun. Pure and Appl. Math. 39(1986), 839-866.

[CH]

S.S. Chern, Affine minimal hypersurfaces, Proc. Japan-U.S. Semin., Tokyo 1977, 17-30 (1978).

[GIG]

S. Gigena, On a conjecture of E. Calabi, Geom. Dedicata 11(1981), 387-396.

[HS-S]

C.C. Hsiung and J.K. Shahin, Affine differential geometry of closed hypersurfaces, Proc. London Math. Soc. 17(1967), 715-735.

[L-l]

A.-M. Li, Affine completeness and Euclidean completeness, Lecture Notes Math. Springer 1481(1991), 116-126.

[L-2]

, Some theorems in affine differential geometry, Acta Mathematica Sinica, New Series, 5(1989), 345-354.

[L-3]

, Affine maximal surfaces and harmonic functions, Lect. Notes Math. Springer 1369(1989), 142-151.

[L-4] [L-5]

, A characterization of ellipsoids, Results in Math. 20(1991), 657-659. , Calabi conjecture on hyperbolic affine hyperspheres (2), Math. Ann. 293(1992), 485-493.

[LIA]

H.-K. Liao, Surfaces with affine Gauss-Kronecker curvature zero, to appear.

[L-J-1]

A-M. Li and F. Jia, Affine Bernstein problem on affine maximal hypersurfaces, Technische Universitat Berlin, Preprint Reihe Mathematik No.656/1999.

[L-S-CH] A-M. Li, U. Simon and B.-H. Chen, A two-step Monge-Ampere procedure for solving a fourth order PDE for affine hypersurfaces with constant curvature, J. reine ang. Math. Vol. 487(1997), 179-200. [L-S-Z-l] A-M. Li, U. Simon and G.-S. Zhao, Hypersurfaces with prescribed affine Gauss-Kronecker curvature, Geom. Dedicata, to appear. [L-S-Z-2]

, Global Affine Differential Geometry of Hypersurfaces, Walter de Gruyter, BerlinNew York, 1993.

[M-M]

A. Martinez and F. Milan, On the affine Bernstein problem, Geom. Dedicata 37(1991), 295-302.

144 [NIR]

L. Nirenberg, Monge-Ampere equations and some associated problems in geometry, Proc. ICM Vancouver 1974; vol. 2(1975), 275-279.

[NO]

K. Nomizu, On completeness in affine differential geometry, Geom. Dedicata 20(1986), 43-49.

[N-P-S]

K. Nomizu, U. Pinkall and U. Simon (eds.), AfEne Differential Geometry, Proc. Conf. Math. Forschungsinstitut Oberwolfach, Febr. 1991, TU Berlin 1991.

[P-l]

A.V. Pogorelov, On the improper convex affine hyperspheres, Geom. Dedicata 1(1972), 33-46.

[P-2]

, Complete affine minimal hypersurfaces, Soviet Math. Dokl. 38(1989), 217-219.

[SAS]

T. Sasaki, Hyperbolic affine hyperspheres, Nagoya Math. J. 77(1980), 107-123.

[SCH]

R. Schneider, Zur affinen Differentialgeometrie im Grojien, I. Math. Z. 101(1967), 375406.

[SI-1 ]

U. Simon, Hypersurfaces in equiaffine differential geometry, Geom. Dedicata 17(1984), 157-168.

[SI-2]

-(ed.), Affine Differential Geometry, Proc. Conf. Math. Forschungsinstitut Oberwolfach, Nov. 1986, TU Berlin 1988.

[SI-3]

, Integralformeln zur Kennzeichnung von Hyperfldchen durch Krummungen und Stiitzabstand, Dissertation FU Berlin 1965.

[SI-4]

, Minkowskische Integralformeln und ihre Anwendungen in der Differentialgeome­ trie im Grossen, Math. Annalen 173(1967), 307-321.

[SU]

W. Suss, Zur relativen Differentialgeometrie V: Uber Eihyperfldchen im Rn+1, Tohoku Math. J. 31(1929), 202-209.

[T]

CX. Terng, Affine minimal surfaces, Ann. Math. Stud. 103(1983), 207-216.

[T-W]

N.S. Trudinger and X.J. Wang, The Bernstein problem for affine maximal hypersurfaces, The Australian National University, Mathematics Research Report No. 99. 020.

[YAU-S] S.T. Yau and R. Schoen, Differential Geometry, Chinese. Science Press, Beijing, China, 1998. A.-M. Li and F. Jia: Department of Mathematics, Sichuan University Chengdu, PRC. Received February 2, 2000

Geometry and Topology of Submanifolds X eds. W. H. Chen et al. (pp. 145-153) © 2000 World Scientific Publishing Co.

145

SUBMANIFOLDS WITH POINTWISE PLANAR NORMAL SECTIONS IN A COMPLEX PROJECTIVE SPACE

SHI-JIE LI

ABSTRACT. In this article, we investigate submanifolds of a complex projective space C P m with pointwise 2- or 3-planar normal sections via the first standard imbedding of C F m . Key Words: Pointwise planar normal section, the first standard imbedding of €Pm, projective space. Mathematics Subject Classification 1991: 53C40.

1.

complex

INTRODUCTION

Let M be an n-dimensional submanifold of an m-dimensional Euclidean space E m . For a point p in M and a unit vector x tangent to M at p, the vector x and the normal space Tjj-M to M at p determine an (m - n + 1)-dimensional affine subspace E(p,x) in E" 1 through p. The intersection of M and E(p, x) gives rise to a curve 7 in a neighborhood of p which is called the normal section of M at p in the direction of x. In general the normal section 7 is a twisted space curve in E(p, x). A submanifold M is said to have pointwise fc-planar (2 < k < m — n) normal sections if each normal section 7 at p satisfies 7' A 7 " A ■ • ■ 7 f c + 1 = 0 at p for each p in M. The study of submanifolds with pointwise planar normal sections was initiated by B. Y. Chen in the early 1980's in his fundamental paper [2]. In particular, Chen obtained in [2] a necessary and sufficient condition for a submanifold of E m to have pointwise 2-planar normal sections. Using this and the other results of Chen in [3], Chen and the author classified in [4] surfaces with pointwise 2-planar normal sections in E m . More precisely, they proved that such a surface must be a surface lying locally in a E 3 of E m , or an open portion of the product surface of two planar circles lying in a E 4 of E m , or an open portion of a Veronese surface lying in a E 5 of E" 1 . In this article, we investigate the submanifolds with pointwise 2- or 3-planar normal sections in a complex projective space via the first standard imbedding of the complex projective space and find the situation is quite different from the previous cases. In fact, we prove the following. The author is supported by NSFC (Project 19771039) and PNSF (Project 960179) of Guangdong Province, China. This paper is in final form and no version of it will be submitted for publication elsewhere.

146

Theorem 1. Let M be an n-dimensional submanifold of a complex projective m-space QP m (4) of constant holomorphic sectional curvature 4. Then M has pointwise 2-planar normal sections via the first standard imbedding of €Fm(4) if and only if M is totally geodesic in CP m (4) , i.e., M is an open portion of RP"(1) or QPr(4) in CP m (4) with r = n/2. Theorem 2. Let M be an n-dimensional Kaehler submanifold of a complex projective m-space OP m (4) of constant holomorphic sectional curvature 4. Then M has pointwise 3-planar normal sections via the first standard imbedding o/CP m (4) if and only if M is an open portion of a totally geodesic QPr(4) in QPm(4) with r — n/2. 2 . SUBMANIFOLDS OF A SUBMANIFOLD OF A RlEMANNIAN MANIFOLD

Let M be a submanifold of a submanifold N of a Riemannian manifold K. Denote by V , V" and V the Levi-Civita connections on M, N and K, respectively. Denote by D' and D the normal connections of M in N and K, respectively, and by D" the normal connection of N in K. Letters x, y, z will be some vector fields tangent to M, u a vector field normal to M and tangent to N, and £ a vector field normal to N. Then we have (2-1)

V:y = Vxy + o'(x,y),

(2.2)

Vxy = Vxy + o(x, y) = v£y + a"(x, y),

where a' and a are the second fundamental forms of M in N and K, respectively, a" the one of N in K. (2.3)

V > = -A'ux + D'xu,

(2.4)

Vxu = -Aux + Dxu = Vxu + a"(x, u),

(2.5)

Vxt = -A€x + Dxt = -A'{x + D'&

where A' and A are the Weingarten maps of M in N and K, respectively, A" the one of N in K. Then from (2.1) and (2.2), we have (2.6)

o-(x,y) = o-'(x,y) + a"(x,y).

From (2.3) and (2.4), we have (2-7)

Au = A'u,

(2.8)

Dxu = D'xu + a"(x,u).

147 From (2.5), we have (2.9)

D x £ - D'H = AKx -

A'lx.

We define the first covariant differentiations V
(2.10)

(2.11)

(2.12)

(Vxo-)(y,z)

= Dxa(y, A ~
(V x a')(y, z) = D'xa'(y, z) - o'(Vxy,

(V x
V'xz),

z) - a'(y,

Vxz),

z) - o>'(V'iv, z) - a"(y,

V^z).

In a similar way, we can define the second covariant differentiations VVa, VVcr' and VVcr" of a, a' and a", respectively, (cf. [1] for more details). Applying (2.6) and (2.8) to (2.10), we have (V x
'

'

z) + a"(x, a'(y, z)) + Dxa"(y,

~ °'{V'xy, z) - a"{Vxy,

z)

z) - a'(y, Vxz)

- a"(y,

Vxz),

and then we obtain, by using (2.11) and (2.12), (V I cr)( 2/ , z) = ( V , * ' ) ( y , z) + {Dxa"(y, [

'

'

z) - D'y'(y,

z)} + (Vxa")(y,

+ {o-"(x, a'(y, z)) + o»(y, a'(z, x)) + a"(z, a'(x,

z)

y))}.

For the later use, we need the following basic lemmas: L e m m a A [2]. Let M be an n-dimensional submanifold of an m-dimensional Euclidean space E m . Then M has pointwise 2-planar normal sections if and only if the second fundamental form of M in Wn satisfies the following: (2.15)

(Vxa)(x,x)

Aa{x,x)

= 0,

for any x tangent to M. L e m m a B [5]. Let M be an n-dimensional submanifold of an m-dimensional Euclidean space E m . Then M has pointwise A-planar normal sections if and only if the second fundamental form of M in E m satisfies the following: (2.16)

{ ( V . V ^ j f i , x) + 3
for any x tangent to M.

A (Vx<7)(a;, x) A a(x, x) = 0,

148 3.

T H E F I R S T IMBEDDING O F C P m ( 4 )

Let Q P m be the complex projective space obtained as a quotient space of the unit sphere S m + 1 = {z G C m + 1 \zz* = zz* = 1} by identifying z with \z, where A G C and |A| = 1. Let g be the canonical metric on C P m , that is, the invariant metric such that the fibration ■K : S 2 m + 1 —> C P m is a Riemannian submersion. It is known t h a t C P m with this metric is a complex space form of constant holomorphic sectional curvature 4. Let HM(ra + 1) = {A G gl(m + 1, C ) | A = A*} be the set of all Hermitian matrices of degree (m + 1) with metric g(A, B) = \trace(AB), for any A, B G H M ( m + l ) . It is known t h a t the mapping F : S 2 m + 1 -> H M ( m + l ) given by F(z) = z*z = zlz, z G S 2 m + 1 , induces an immersion F : C P m - + H M ( m + l ) such that: (1) F ( O P m ) = {Ae H M ( m + l)\AA = A, traceA = 1}. (2) F is an equivariant full isometric imbedding into H\M{m + 1) = {A € HM(m + \)\traceA = 1} which is a hypersphere of H M ( m -f 1) centered at ^Vj-^ and with radius y/2m(m + l). F is in fact the first standard imbedding of C P m into H M ( m + 1). In the following we identify C P m with F(CPm). Under this identification, the second fundamental form a" of C P r o in H M ( m + 1) is parallel and satisfies 1

9(a"(x> ?/)' ( 7 " ( ? ; ' w ) ) =29(x' y)9(v, w) + g{x, v)g(y, w) + g{x, w)g{y, v) + 9{Jx, v)g(Jy, w) + g{Jx, w)g(Jy, v),

for any x,y,v,w tangent t o Q P m , where J is the complex structure tensor of C P m (cf. [6,7] for more details). Let M be a submanifold of Q P m of HM(m + 1) via F. Then, for any vector u normal to M and tangent to C P m , we have g([Dxa"(y,z) (3.2)

= g(Vxa"(y, =

- Dy\y,z)],u)

=

z), u) = -g(a"(y,

z),

g(Dxa"(y,z),u) Vxu)

-g(a"(y,z),a"(x,u)),

thus we have by (3.1) and (2.9), for any vector field x tangent to M, (3.3)

Dxa"(Vxx,

x) - D'y'(Vxx,

x) = 0,

and (3.4)

Dxa"(x,

x) - D'y'(x,

x) = 0.

Consequently, for any vector field x tangent to M, we have, from (2.14) and the fact that a" is parallel, (3.5)

(Vxa)(V'xx,x)

= (Vxa')(Vxx,x)

+ 2a"(a'(Vxx,x),x)

+

and

(3.6)

{Vxa){x,x)

= (Vxa'){x,x)

+

3a"(a'(x,x),x).

a"(a'(x,x),V'xx),

149 Further from (3.6), we may compute {VxVxa){x, (VxV^Xx, x) = Dx((Vxa)(x, (3.7)

x)) -

x) as follows.

3{Vxa)(Vxx,x)

= Dx{{\/xa'){x,x)) + 3Dx{a"{a\x,x),x)) =:I + 311 - 3III,

-

3(Vxa){V'xx,x)

where from (2.8) and the Codazzi equation we have / =D'x((X7xa')(x,x)) + '

a"((Vxa')(x,x),x))

=(Va;Va;cr')(a:Ja;) + 3(Vx
and we have, with the help of (2.1), (2.3), (2.11) and the fact VCT" = 0, II =D'x'{a"(a'(x,x),x) ,

.

=Ao"(o,(x,x),x)x

-

+ a"((Vxa')(x, +

+ Aa„(„,{XiX):X)x A

o"(a'(x>x),x)x-

a

A'^l{(T,{XtX)tX)x (Ao'(x,x)X'X)

x), x) + 2a"{a'{V'xx, x),x) + a"{a'{x, x), Vxx)

a"(a'(x,x),a'(x,x)),

and by (3.5), we have (3.10)

/ / / = (Vxa'){Vxx,

x) + 2a"(a'(V'xx, x), x) + a"(a'(x, x), Vxx).

Substituting (3.8)-(3.10) into (3.7), we obtain (VzVxc7)(z, x) =(VxVxa')(x, (3.11)

x) +

-3Al„(a,(XtxU)x

3A
-3a"(A'a,{XtX)x,x)

+

Aa"{{Vxo'){x,x),x) 3a"(a'(x,x),a'(x,x)).

4. PROOFS OF THEOREMS

Proof of Theorem 1. Let M be a submanifold of QP m . If M has pointwise 2-planar normal sections via F, then by Lemma A we have (2.15), which implies that there exist coefficients 0 and 7, not all zero, such that (4.1)

/3{Vxa)(x,x) + ya{x,x) = 0,

for any vector x tangent to M. Applying (2.6) and (3.6), we may deduce the following from (4.1). (4.2)

/3(Vxa')(x,x)+'ya'(x,x)

= Q,

and (4.3)

3/3a"(a'(x,x),x)

+ ya"(x,x) = 0,

150 for any vector x tangent to M. Take the inner product of (4.3) with a"(x, x), where x is a unit vector tangent to M, since we have by (3.1) (4.4)

g(a"{a'(x,x),x),a"(x,x))

(4.5)

= 0,

g(a"(x,x),o"(x,x))

= 4,

then we may obtain (4.6)

7 = 0.

Since /? and 7 are not all zero, applying (4.6) in (4.2) and (4.3), we obtain the following (4.7)

(Vxa'){x,x)

= 0,

and (4.8)

a"(a'(x,x),x)

= 0,

for any unit vector x tangent to M. Since we have by (3.1) g((j"(a'(x,x),x),a"(a'(x,x),x)) (4.9) = g(a'(x, x), a'(x, x))g(x, x) - g{a'(x, x),

Jx)2,

thus (4.8) implies (4.10)

cr'(x, x) = g(a'{x, x),

Jx)Jx,

for any unit vector x tangent to M. Let -75(2: ± y) take the place of x in (4.10), where y is a unit tangent vector of M perpendicular to x, we may obtain, by the linear behavior of a' and g, (4.11)

a'(x,y)

= -{g{a'{y,y),Jy)Jx

+

g(a'(x,x),Jx)Jy}.

If there exists a vector x tangent to M such t h a t a'(x,y) vanishes for any vector y tangent to M , then by (4.10) and (4.11), we have cr'(y,y) = 0, for any vector y tangent to M , which implies t h a t M is totally geodesic in Q P m . If for any vector x tangent to M there is at least a vector y tangent to M such that a'(x, y) ^ 0, then by (4.10) and (4.11), we may find t h a t Jx belongs t o the first normal space of M in C P m , which implies t h a t M is totally real in C P m . Then we have, for any x, y tangent to M ,

,

x

ff(*'(*,

x), Jy) = g(Kx, Jy) = -g(V';jx,

y)

(4.12)

= -g{^xJx,y) = g(Jx,a'(x,y)). Combining (4.10) and (4.12), we may conclude that a'(x,x) = 0 for any x tangent to M, which implies t h a t M is totally geodesic in Q P m again. Conversely, if M is totally geodesic in CIP m , then by (3.6) we may find t h a t (2.15) holds on M and thus M has pointwise 2-planar normal sections via F.

151

Proof of Theorem 2. Let M be a submanifold of CP m . If M has pointwise 3-planar normal sections via F, then by Lemma B we have (2.16) which implies that there exist coefficients a, /? and 7, not all zero, such that (4.13)

a{(VxVxa)(x,x)

+ 3a(x, Aa{x,x)x)}

+ /3(Vxcr)(x, x) + ja{x}x)

= 0,

for any vector x tangent to M. Using (2.6), (3.6) and (3.11), we may deduce the following from (4.13). a{(4a"((Vxa')(x,x),x)-Za"(x,A'ir,(x!x)x)

(4.14)

+ 3a"(x,A„{XtX)x)}

+

3a"(a'(x,x),a'(x,x))

+ 3/3a"(a'{x)x),x)+ya"(x,x)

= Q,

for any vector x tangent to M. Take the inner product of (4.14) with cr"(x, x), where a; is a unit vector tangent to M, then since by (3.1) we have (4.15)

g(a"((Vxa')(x,x),x),a"(x,x))

(4.16)

(4.17)

g{a"{x,A'al(x^x),a"{x,x)) g(a"(
(4.18)

= 0,

=

A\a\x,x)\\

= 2\a'(x,x)\2 +

g(a"(x,A^x)x),a"(x,x))

(4.19)

g(a"(a'(x,x),x),a"(x,x))

(4.20)

g(o"(x,x),a"(x,x))

2g(J(j'(x,x),x)2,

= 4\a(x,x)\2, = Q, = 4,

then we have (4.21)

3a{-4\a'(x,x)\2

4- 2\a'(x, x)|2 +

2g(Ja'(x,x),x)2

+ 4|cr(x,x)i 2 } + 47 = 0,

which implies (4.22)

3a{\a'(x, x)\2 + 2\a"(x, x)\2 + g(Ja'(x, x), x)2} + 2 7 = 0.

Then take the inner product of (4.14) with a"{a1(x, x), a'(x, x)). In the same way as above we have g(a"((Vxa')(x, =

x),x), a"(a'(x, x),a'(x, x)))

2g(x,Ja'(x,x))g((Vxa')(x,x),Ja'(x,x)),

152 g{a"{x, A'al(XiX)x), 4

= 2\a'(x,x)\

(4.25)

CT"(
+

2g(A'a,(x^x,Ja'(x,x))g(x,Ja'(x,x)),

g(cr"(a'(x, x), a'{x, x)), a"(a'(x, x),a'(x, x))) = 4\a'(x, g(cr"(x, Aa{X:X)x),

x)\\


= 2\a{x, x)\2\a'{x, x)\2 + 2g{Aa^x>x)x, Ja'(x, x))g(x, Ja'(x, x)),

(4.27)

(4.28)

g{a"(a'(x,x),x),a"(a'(x,x),a'(x,x)))

=0,

g(a"(x, x), cr"(a'(x, x), a'(x, x))) = 2\a'(x, x)\2 + 2g(x, Ja'(x, x))2,

then we have x))g((Vxal)(x,x),Ja'(x,x))

a{4g(x, Ja'(x, x

(x,x) i J° ( ' ))9(xi Jcr'{x, x)) (4.29)

x x

+ 3g(A„,(XtX)x, Ja'(x, x))g(x, Ja'(x, x)) + 3\a"{x,x)\2\a'(x,x)\2 2

+ -y{\a'(x,x)\

+

6\a'(x,x)\4}

+ g(x, Ja'{x, x))2} = 0.

Take the inner product of (4.14) with a"(cr'(a:,a;),a;), we have 9{cr"({Vx
x),x))

= g({Vxa')(x, x),a'(x, x)) + g{{Vxa')(x, x), Jx)g{x, Ja'(x, x)),

(4.31)

(4.32)

(4.33)

(4.34)

(4.35)

g (a"{x, A'a,(xx)x),

a"{a'(x, x),x)) = giA'^^x,

Jx)g{x, Ja'{x, x)),

9{o"{v'{x, x), a'{x, x)),o"{a'{x, x),x)) = 0,

g(a"(x,Acr{x,x)x),cr"{a'(x,x),x))

g(
= g{Aa{XtX)x, Jx)g{x, = \
g{a"(x,x),a"(a'{x,x),x))

= 0,

Ja'{x,x)), Ja\x,x))2,

153 then we have a{4ff((Vx(r')(2;, x),a'(x, x)) + 4g((Vxa')(x,x), (4.36)

- ^9(K'(x,x)x'

Jx)g(x, Ja'(x, x))

Jx

)9(x, Ja'(x, x)) + Zg{A^XtX)x, Jx)g(x, Ja'(x, x))}

+ 3/3{\a'(x, x)\2 - g(x, Ja'{x,x))2}

= 0.

If in addition M satisfies the following condition : g(x, Ja'(x,x)) = 0, for any unit vector x tangent to M, then (4.22), (4.29) and (4.36) become the following, respectively, (4.37) 3«{i<j'(ir, x)\2 + 2\a"(x, x)\2} + 2 7 = 0, (4.38)

\a'{x, x)\2{3a(\a"(x, x)\2 + 2\a'(x, x)\2) + 7} = 0,

(4.39) 4ag{{Vxo-')(x,x),a'{x,x)) + 3P\o'(x,x)\2 = 0, where (4.38) implies that either a'(x,x) = 0 or (4.40) ia{\G"{x,x)\2 + 2\a'{x,x)\2} + i = 0. For the last case, combining (4.40) and (4.37), we obtain acr'(x,x) = 0. Here if a = 0, then 7 = 0 by (4.37) and f3a'(x,x) = 0 by (4.39). Since a, (3 and 7 are not all zero, thus we still have a'(x, x) = 0. Thus M must be totally geodesic in CP m . Conversely, a totally geodesic submanifold in CP m has pointwise 3-planar normal sec­ tions since it has pointwise 2-planar sections according to Theorem 1. Now we may conclude the result above as follows. Proposition 1. Let M be a submanifold of an m-dimensional complex projective space CP m (4). If the following condition holds on M: (4.41) g(x,Ja'{x,x)) = 0, for any vector x tangent to M, then M has pointwise 3-planar normal sections via the first standard imbedding o/CP m (4) if and only if M is totally geodesic in CP m (4). It is clear that (4.41) holds on any Kaehler submanifold in CP m and thus Theorem 2 is just a consequence of Proposition 1. REFERENCES [1] B.Y. Chen, Geometry of Submanifolds, Dekker, New York, 1973. [2] B.Y. Chen, Submanifolds with planar normal sections, Soochow J. Math. 7 (1981), 19-24. [3] B.Y. Chen, Differential geometry of submanifolds with planar normal sections, Ann. Mat. Pura Appl. 130 (1982), 59-66. [4] B.Y. Chen and S.J. Li, Classification of surfaces with pointwise planar normal sections and its appli­ cation to Fomenko's conjecture, J. Geometry 26 (1986), 21-34. [5] S.J. Li, Spherical submanifolds with pointwise 3- or ^-planar normal sections, Yokohama Math. J. 35 (1987), 22-31. [6] K. Sakamoto, Planar geodesic immersions, Tohoku Math. J. 29 (1977), 25-56. [7] S.S. Tai, Minimal imbedding of compact symmetric spaces of rank one, J. Diff. Geom. 2 (1968), 55-66. DEPARTMENT OF MATHEMATICS, SOUTH CHINA NORMAL UNIVERSITY, GUANGZHOU 510631, CHINA

E-mail address: lisjfflscnu.edu.cn RECEIVED O C T O B E R 11,

1999

154

Geometry and Topology of Submanifolds X eds. W. H. Chen et al. (pp. 154-170) © 2000 World Scientific Publishing Co.

Tchebychev hypersurfaces of § n+1 (l) TSASA LUSALA*

Abstract We consider a new class of hypersurfaces in spheres, called Tchebychev hypersur­ faces. This class generalizes the class of isoparametric hypersurfaces with at most two distinct principal curvatures. We give examples and partial classification results, which also give some new insight for isoparametric hypersurfaces. Basic concepts and methods come from Affine Differential Geometry. Keywords: Isoparametric hypersurfaces, Tchebychev hypersurfaces in spheres, conformal structures. 1991 Mathematics Subject Classification: 53 B 25, 53 C 40, 53 C 42, 53 A 15. Acknowledgements: We sincerely thank the geometry research group at the Technische Universitat Berlin for fruitful discussions, particularly with U. Simon, L. Vrancken and M. Wiehe.

1

Introduction

E. Cartan proved that isoparametric hypersurfaces in space forms of constant curvature K admit at most two principal curvatures, except for the case of isoparametric hypersurfaces in spheres (K > 0) [4], [6]. He gave examples in spheres with more than three principal curvatures [5]. Another well known result of Cartan is that isoparametric hypersurfaces in space forms satisfy the so called Cartan formula for the principal curvatures. Recently, I.B. Kim and T. Takahashi characterized isoparametric hypersurfaces in a space form, with no more than two principal curvatures, provided that the type number (the rank of the Weingarten operator) is zero or greater or equal to three [11]. In this paper we introduce a new class of hypersurfaces in spheres which are called Tcheby­ chev hypersurfaces. We extend some of the results mentioned above to this class. A first example of such a result is that Tchebychev hypersurfaces with constant Gaufi-Kronecker curvature (not necessarily isoparametric) satisfy again Cartan's formula, we state some more examples of results at the end of the introduction. The idea to consider this new class of hy­ persurfaces comes from affine differential geometry, in particular from [14]. The basis for our investigation is the following. We call a hypersurface in a sphere with a regular Weingarten "This work is supported by DFG (Graduierten Kolleg "Geometrie und Nichtlineare Analysis" HU-TU Berlin). This paper is in final form and no version of it will be submitted for publication elsewhere.

155 operator S a non-degenerate hypersurface. For such a hypersurface the second fundamen­ tal form I and the third fundamental form n are semi-Riemannian metrics. If V1, V2, V3 denote the Levi-Civita connections of the first, second and third fundamental form, respec­ tively, it is known that the triple (V 1 ,!, V3) is a conjugate triple. We recall this concept, which is well known from affine hypersurface theory below. The symmetric (l,2)-tensor 1C := V 1 — V3 has the Tchebychev form T as its trace, nT(u) : = trace{tu i—> C(u,w)}, and T as its associated Tchebychev vector field, implicitly defined by I(T, it) := T(u). We call a regular hypersurface in a space form of Tchebychev type if the Tchebychev operator T := V 2 T, defined as V2—covariant derivative of T, and the Weingarten operator S satisfy | ( 5 — KS"1) - j - ^ r =: L = fj,-id. An equivalent condition is the following interesting higher order "Codazzi equation" for the traceless part C of C. The covariant derivative V 2 C is totally symmetric exactly for Tchebychev hypersurfaces. This class has further interesting geometric and analytic properties. We prove basic properties, give examples and, under additional assumptions (e.g. constant GauB-Kronecker curvature), partial classifications. In particular, we establish a link with the condition of LB. Kim and T. Takahashi in [11]. As examples of results we explicitly state the following: (i) A non-degenerate (hyper)surface in a sphere satisfies the equation C = 0 if and only if it is an isoparametric hypersurface with at most two distinct principal curvatures (Theorem 3.12). (ii) For dimension n > 3, the condition C = 0 is equivalent to the local symmetry of the first fundamental form metric of the hypersurface (Theorem 3.16). (iii) Deforming the Clifford torus we get non-isoparametric locally symmetric examples of Tchebychev surfaces in S 3 (l) (Example 3.19).

2

Preliminaries

Notations and facts. In the following we use concepts which are well known in the affine differential geometry of hypersurfaces and adopt notations used in [22], [15]. Let Mn+l(K) be an (n+l)-dimensional oriented space form of curvature K with n > 2. Let x : Mn —> Mn+1(K) be an immersion of an orientable C°°-manifold Mn of codimension 1 into Mn+1(K), and y be a unit normal vector field along x. Denote by <, > the metric and by V the Levi-Civita connection of Mn+1(K). At any point p e Mn, the shape operator, denoted by 5, is the linear map defined as follows: S : TpMn —> TvMn,

dx{S{u)) := -Vuy.

An immersion is said to be non-degenerate or regular if and only if the corresponding shape operator S is of maximal rank; this is equivalent to the fact that the Gaufi-Kronecker cur­ vature function is everywhere non-vanishing. In the following we suppose that the immersion is non-degenerate. Denote by X(Mn) the C°°-module of vector fields on Mn. Recall the definition of the first, the second, and the

156 third fundamental forms, denoted by I, H, and M, respectively: for all u,v £ X(Mn), l(u,v) = ; W(u,v) = < dy(u),dy(v) > .

-I(u,v)

= < dx(u),dy(v) >;

(2.1)

They are semi-Riemannian metrics on Ma. In fact, the first and the third fundamental form are positive definite. Structure equations. For all u, v £ X(M), the structure equations are given by: Vudx{v) = dx{V\v) + l(u,v)y (GauB equation); dy(v) = —dx(Sv) (Weingarten equation), where V1 is the Levi-Civita connection of I. It is well known that the three fundamental forms are related through S: I(u,v) = I(S-1u,v),

W(u,v) = I(Su,v).

(2.2)

Conjugate connections. i,From the relation (2.2) and the Codazzi equations for S (see Codazzi-S below), it is easy to check that the Levi-Civita connection V 3 of HI satisfies V3u = S^CV^Sv) (see details in [18], p.154), and that the triple (V 1 ,1, V3) is conjugate, in other words, for all u, v, w € X(M): wl(u,v) = l(V1wu,v)+l(u,V3wv). 2

x

(2.3)

3

Hence V = j(V + V ) is the Levi-Civita connection associated to the semi-Riemannian metric I. The difference tensor C, defined by C := ^(V 1 — V 3 ), is a symmetric (1,2)-tensor field to which is associated the totally symmetric cubic form C, defined by C(u,v,w) = I(C(u,v),w). The invariants I, V 1 , V2, V3, C and C are related as follows: 2C = - V 1 ! = V 3 I;

V1 - C = V2 = V 3 + C.

(2.4)

The Tchebychev form T and the Tchebychev vector field T are defined by: nf(u) := tr[v >—> C(u,v)];

I(«,T) := f(u).

The (l,2)-tensor field C, defined by C{u, v) := C(u, v)

^]f(v)u

+ f{u)v + TL{u, v)T\

(2.5)

is traceless, it is the so called the traceless part of C. Proposition 2.1 ([2]). If K := det(S) denotes the Gaufi-Kronecker curvature of a regular hypersurf ace, then rif = -dlnlKf1;

nT = ^grad2In \K\~\

where grad2 is the gradient with respect to the second fundamental form.

(2.6)

157 Relations b e t w e e n t h e intrinsic geometries of I, I , M. P r o p o s i t i o n 2.2. The Riemannian curvature tensors Rl, R2 and R3 associated to I, I and M, respectively, satisfy the following relations: R1{u,v)w-R2(u,v)w

=

(V2uC){w,v)

- (X72vC){u,w)

+C{u,C{w,v)) R3{u,v)w-R2{u,v)w

=

-

(2.7)

C{C{u,w),v)

{V2vC){u,w)-(V2uC){w,v)

(2.8)

+C{u,C{w,v))-C{C(u,w),v) R1(u,v)w-R3(u,v)w 3

R (u,v)w

=

2[(VlC)(w,v)~(V2vC)(u,w)]

=

S-1R\u,v)Sw.

(2.9) (2.10)

Proof. ^From (2.4) and calculating the curvature tensors (see also in [22], p.60), one gets (2.7), (2.8) and (2.9). (2.10) is a consequence of Proposition 3.1, p.155, in [18]. D Integrability conditions. The structure equations imply the following integrability conditions: Codazzi-S:

(ViS)t)

=

(VjS)u;

Codazzi-H:

(ViH)(«,u>)

=

(Vjl)(u,iu);

GauB-V 1 :

R1(u,v)w

=

K[l(w,v)u-I{w,u)v]

J(Su,v)

=

I(u, Sv).

Ricci-equation:

+

I(v,w)Su-I(u,w)Sv;

The equations above can be found in many books e.g. [3] p.134-138, [9] p.47, [22] p.67-68. As consequence of (Gaufi-V 1 ), using (2.10), one has the following equation: GauB-V 3 :

R3{u, v)w = K[l{w, v)S~1u - I(io, u)S'lv]

+ W(v, w)u - TS.(u, w)v.

Remark 2.3. If M " + 1 = S" + 1 (l), the immersion x being non-degenerate, the associated GauB map defines a regular immersion x": Mn —>• S" + 1 (l) such that: I* = m,

I* = I ,

HI* = I;

S* = S- 1 ;

V*1 = V 3 ,

V* 2 = V 2 ;

V*3 = V 1 .

The pair (x, x*) is called a polar pair, and the correspondence x i—> x* is the polarization of the regular immersion x [2]. The formula (GauB-V 3 ) is then the integrability condition (GauB-V*1) for the immersion x". From the foregoing the second fundamental form and its intrinsic geometry are invariant under polarization. The study of such polarization is of geometric interest. Using (2.9), (GauB-V 1 ) and (Gaufi-V 3 ), one has ( V 2 C ) (v,w)

=

(V2vC)(u,w)

+ ^{K[I{w,v)u-l(w,u)v]

+ I(v,w)Su

-H(u, w)Sv ~ K[J(w, v)S~ u - I(w, u)S~ v]-W

(2.11)

(v, w)u + M(u, w)v \.

158

3

Tchebychev hypersurfaces.

Definition 3.1. Let the immersion x : Mn —> Mn+1(K) be regular. The linear operator r defined on Mn by T(«) := V 2 T,

u e X(Mn)

(3.12)

is called the Tchebychev operator. Lemma 3.2. Consider the traceless part C of C. and define the linear operator L on M" by L := ^(-KS-1 + S) - JJ^T. One has: (V2uC)(v, w) - (V2vC)(u, w) = I(v, w)Lu + I(Lu, w)v - I(u, w)Lv - I(Lv, w)u.

(3.13)

Proof. Straight forward calculations using (2.5) and (2.11) give: (V2uC)(v,w)

= {VlC)(u,w) + ^{K[l{w,v)u-l(w,u)v]

+ I(v,w)Su

(3.14)

1

- I ( u , w)Sv - M(v, w)u + l ( u , w)v - K[I{w, v)S~ u - I(u>, u)S~lv]} n {l(w, T(U))V + l(v, w)r(u) — I(iu, T(V))U — l(u, w)r(v)} . n+2 The equation (3.14) and the definition of L establish (3.13).

□

Proposition 3.3. The following statements are equivalent: (i) (V2uC)(v, w) = {V2vC){u, w) for all u,v,w<= X(Mn); (ii) there exists ft € C°°{Mn) such that L = \x- id; (iii) div 2 C = 0, ■where div2 is the divergence taken with respect to the second fundamental form. Proof. Our proof of this proposition is quite similar to the proof of Proposition 2.3, p.209, in [14]. For the convenience of the reader, we give a proof. (iii) = > (ii): Consider a frame (ej)i<j<„. One can see that div2 C = 0 if and only if for any

1
vlc

* = vhvlTl + ^+~fhl div2T'

2

(3 15)

-

2

where we use V = V . the covariant derivative in the direction of the vector field e,-. From (2.11), one has: Vfc* = nV 2 T, + £(I U - l w ) + 1(H - H')KM. The expressions (3.15) and (3.16) imply: n+2 "

' ' n + 2 '"

'" ' ' 2 K M

a

"

2V

(3.16)

159 then LM = %(H - H* — ^ d i v 2 T ) I I h , = ±(trL)I u , where H and H* denote the functions on M defined as follows: H := — trS = —tr,H = tr„ni (mean curvature function) n n

H* := nitrS-^-Ur.^-Ur.I. n n (ii) = > (iii): If L is proportional to the identity, then

Lhl = k-h, /

+ MM) - ~ V 2 T , = \{H -H'n +1

A

J^div2T)Iw.

n +1

So

\{H-

H')IM - I ( - I W + Mhl) = ^ K T ,

- ^ 1 1 * 1 div2 T.

(3.17)

Substituting (3.17) in (3.16), one gets (3.15). (ii) = > (i): If L = fj, ■ id, one can see that the right hand side of (3.13) vanishes. (i) => (ii): If (i) is satisfied, from (3.13), one has HjkLhi ~ Ljk^hi — 0, which gives L/,j = i(trL)I w . □ n+1 Definition 3.4. An immersion of a regular hypersurface x : M" —> M is said to be a Tchebychev hypersurface (or is of Tchebychev type) if and only if one of the statements in Proposition 3.3 is satisfied. Example and Remark 3.5. (i) Regular hypersurfaces for which (7 = 0, particularly those which satisfy C = 0, are trivial examples of Tchebychev hypersurfaces of the Euclidean sphere. (ii) On proper affine spheres the affine cubic form has totally symmetric covariant deriva­ tive ( with respect to any of the three affine connections on a non-degenerate hyper­ surface). [14] contains a systematic study of such symmetry relations for covariant derivatives of cubic forms. From this Liu, Simon and Wang [12], [13] introduced the concept of a Codazzi tensor of order m > 2. Definition 3.4 says that the class of Tchebychev hypersurfaces in space farms can be characterized by "Codazzi equations" for the pair {V 2 ,C}; that means V 2 C is totally symmetric. Proposition 3.6. In spheres of curvature 1, the concept of Tchebychev hypersurfaces is invariant under polarization, in other words, a regular immersion is of Tchebychev type if and only if its Gaufi map defines a regular immersion of Tchebychev type. Proof. If the pair (x, x*) is polar (see Remark 2.3), then the operators L and V corresponding to x and x*, respectively, satisfy L* = —L. □

160 Case of constant Gaufi-Kronecker curvature. Theorem 3.7. Let the immersion x: Mn —► S n+1 (l) be a regular hypersurface without umbilic points. Suppose that the Tchebychev operator r of x is identically zero. Then x is a Tchebychev hypersurface if and only if x has exactly two different principal curvature functions Ai and A2 such that A1A2 = — 1. Proof. Let r = 0. The immersion x is of Tchebychev type if and only if

- 5 " 1 +S = (-H*+H)-id. Each principal curvature is a solution of the following second order polynomial equation: A2 - (-ff* + H)X - 1 = 0.

(3.18)

Since the immersion has no umbilic points, the equation (3.18) has two different solutions Ai and A2 such that A1A2 = —1 and Ai + A2 = — H* + H. Conversely, suppose that r = 0 and x has, at any point, exactly two distinct principal curvatures Ai and A2, such that AiA2 = —1. Using an orthonormal frame of eigenvectors of the shape operator, one can see that — S"1 + S = (Ai + A2) ■ id; the definition of L gives the assertion. □ Lemma 3.8 (Cartan's formula: [6], p. 237). Suppose that a given isoparametric hyper­ surface of dimension n immersed in a space form of curvature K, has I distinct principal curvatures Ai,..., A; with multiplicities mi,...,mi, respectively. Then, for 1 < i < I, one has: rrf

m](K+X,Xj)=Q A; — A;

(3lg)

Particularly, if the immersion has exactly two distinct principal curvatures Ai and A2, then K+ X^^O. Remark 3.9. Theorem 3.7 shows that Cartan's formula can be generalized to Tchebychev hypersurfaces with r = 0. Corollary 3.10. Tchebychev hypersurfaces of constant Gaufi-Kronecker curvature K in S2*(l) are isoparametric ones. Proof. A"=constant implies r = 0, see (2.6) and (3.12). Let Ai and A2 be two distinct principal curvatures, and denote by n\ and 712 the multiplicities of Ai and A2, respectively. From Theorem 3.7, A1A2 = —1. One has: K = A?1 AI?2 = A?'(--!-) n2 = (-l)" 2 A ( 1 ni "" 2) . Ai

Since n\ + ni = 2k — 1 is odd, one of n\ and n2 is even while the other one is odd. So 0 ^ n\ — 712 is odd. Then Ai is constant, so is A2. □

161 Remark 3.11. If 9 is a (0, 2)-tensor on M, then for all u, v, w e X(M), (2.4) implies: (VlJ)(v, w) = (V2u9)(v,w) - 9(C(u, v),w)- 8(v, C(u, w)); (V3J)(v, w) = (Vl9)(v, w) + 8(C(u, v),w) + 9(v, C(u, w)).

(3.20) (3.21)

Theorem 3.12. Let x: M" —> S n+1 (l) be a regular immersion and consider the associated symmetric (l,2)-tensor C. Then the equation C = 0 characterizes isoparametric hypersurfaces with at most two distinct principal curvatures. Proof. Suppose that x is isoparametric with two principal curvatures. From Proposition 2.1, T = 0. Then Lemma 3.8 and Theorem 3.7 imply that x is of Tchebychev type. So - S " 1 + S = {-H* + H) ■ id or - I + HI = (-H* + H)-I. Since H* and H are constant, one has: — V^(Ijj +HIij) = 0., for any frame (ej)i<j<„, and for 1 < i,j, k < n. Using Remark 3.11 for 9 = I and 9 = HI, respectively, one gets: Clkj(lu + 1H„) + Ck(I,-( + % ) = 0.

(3.22)

Choose a I-orthonormal frame (e,-) of principal eigenvectors of the immersion, and denote by Aj the principal curvature to which e< is associated. Then (3.22) becomes: (1 + A?)C{,- + (1 + A?)C£ = 0. If Ai = A3 then C^ = -Cjj... Particularly, C{ki = 0. If Ai / Aj, then from Lemma 3.8, A;A.,- = —1 and then C'kj = — A~2C^. But absolutely, A*, is one of A; and Aj. Assume \k = \ . One has:

which gives C^ = 0. Finally, C = 0. If x has only one principal curvature, that means that x is umbilical, then C£. = —C,* = C*^ = -C\.p which implies C^. = 0. Conversely, if C = 0, x is of Tchebychev type with r = 0, and from Theorem 3.7 x has one or two principal curvatures. Moreover, C = 0 implies V 1 ! = 0, particularly x has constant principal curvatures. □ Corollary 3.13. For any regular and isoparametric hypersurface of the sphere with more than 2 principal curvatures, one has: div 2 <7^0. Following Proposition 3.3, they can not be of Tchebychev type. Corollary 3.14. Let x, x*: Mn —► S" +1 be a polar pair. Then x* o x~x: x{Mn) —► x*(Mn) is a geodesic diffeomorphism if and only if x and x* are isoparametric with at most two principal curvatures. Proof. V 1 and V 3 = V* are projectively equivalent, i.e. the map x* o x _ 1 is geodesic, if and only if there exists a 1 —form a such that Vlv ~ ^*uv = a(u)v + °(v)u,

for all u,v£

X(Mn).

162 As 2C = V1 - V*, we get 2C(u,v,w)

= 2I(C(u,v),w)

= (r(u)I(v,w) + cr(v)I(u,w).

The total symmetry of C gives a(u)J(v, w) — a(w)I(u, v) = 0; as rank(I) = n we get a = 0 by contraction. Thus (7 = 0 and C = 0. n

□

n+1

Theorem 3.15 ([19]). Let x: M —> S (l) be a regular hypersurface immersion of a connected n-dimensional manifold Mn into S" +1 (l) with constant mean curvature. Assume that at least one of the following assertions is satisfied: (i) M" is compact and has non-negative sectional curvature; (ii) Mn has non-negative sectional curvature and the squared norm of the second funda­ mental form is constant. Then the immersion x is of Tchebychev type, more precisely, the tensor field C vanishes on Mn, consequently Mn is of constant Gaufi-Kronecker curvature. Proof. If one of the conditions (i) and (ii) above is satisfied, then the immersion x is umbilic or has exactly two distinct constant principal curvatures. □ Theorem 3.16. Let x: Mn —> S n+1 (l) be a regular hypersurface of dimension n > 3. Then the following statements are equivalent: (i) the immersion is locally symmetric; (ii) the shape operator is parallel with respect to V1; (iii) the (1,2) -tensor field C from (2.4) is identically 0. Proof, (ii) «=> (iii): For any u,v € X(M), one has: (VlS)v = V > - S(Vi«) = S {Vlv ~Viv)

= -2S(C(u,«)).

(3.23)

Since the immersion is non-degenerate, from (3.23), the shape operator is parallel with respect to V 1 if and only if C = 0. (iii) = » (i): Differentiating covariantly (GauB-V1) with respect to the first fundamental form, one has for z,u,v € X(M): -l{VlRl)(u,v)w

= H(w,v)SC{u,z)-1L(w,u)SC(v,z) +C(v, z, w)Su — C(u, z, w)Sv.

So if C = 0, then the immersion is locally symmetric.

(3.24)

163 (i) ==$■ (iii): Suppose that x is locally symmetric. Locally, one can rewrite the equation (3.24): ItfSJC* - *HS'hC*k + C%llhS° - C&KS]

= 0.

(3.25)

Consider a I-orthonormal frame («j)ik-X,j6'jC{k

= 0.

So if s 7^ i and s ^ j , then C^ = 0, and the same argument gives C\k = 0; if s = i, then C\k = — C^.. Since n > 3, there is a third index m such that m / i and m ^ j . And one has: C\k = -C™t and C]k = -C™k, and then C\k = 0. Finally C = 0. □ Remark 3.17. The equivalence (ii) -$=> (iii) and the implication (iii) =$> (i) in Theorem 3.16, hold for any dimension n of the immersion. For n > 3, LB. Kim and T. Takahashi proved in [11] that the condition (i) in Theorem 3.16 is equivalent to the fact that the immersion has at most two principal curvatures. From Theorems 3.12 and 3.16, one can conclude that for the dimension two, regular isoparametric surfaces in S 3 (l) are also locally symmetric. Proposition 3.18. For the particular case n = 2: (i) Compact regular surfaces of S 3 (l) with constant Gaufi-Kronecker curvature are of Tchebychev type. (ii) Only Clifford tori are minimal Tchebychev surfaces with constant Gaufi-Kronecker cur­ vature in S 3 (l). Proof. For (i), such surfaces are spherical or have constant GauB-Kronecker curvature equal to —1; see ([23], p.139) and Theorem 3.7. For (ii), minimal Tchebychev surfaces with constant Gaufi-Kronecker curvature have two principal curvatures —1 and 1. D Example 3.19. In the following, we construct a family of Tchebychev surfaces by deforma­ tion of the Clifford torus. While the Clifford torus is minimal and isoparametric, the rest of the family consists of non-minimal isoparametric and of non-isoparametric surfaces. Consider the Clifford torus x: M2 —> §3(1) t-> R4- That immersion satisfies: (i) < x, x >= 1; < xu, xu >= 1 =< xv, xv >; < xu, xv > = 0; (ii) the unit vector fields y normal to x satisfies: duy = -xv; dvy = -xu; duxv = y = dvxu; (iii) duxu = -x = dvxv; (iv) the immersion has constant principal curvatures - 1 and 1.

164 Take now the map

3

X:

(1), X = -220L,

where ip:

-1,1)

c

C°°-function with values in the open interval (—1,1), depending only on u. Denote by ' the derivative with respect to u. One has: (HV) 2 Xv —•

/l+v1

< -^u,-^u >— (i+[f,i)i + 1|

y/l+V* < Xu, Xv >— , , Jsi i+v;

[xv - tpxu

1+

< Xv, Xv >— 1;

w

PW

Since at any point,

(jliati

7^ 0)

so

the map X defines an

Denote by Y a unit vector field on R 2 normal to the immersion X, Ex = Xu and E2 = Xv tangent vector fields. The vector Y satisfies the following equations: =0;

=0;

=0;

=1.

iFrom Y G sp&n{xu,xv,x,y} = TR 4 , there are a, f3, a, b e R such that Y = ax + j3y + axu + bxv. iFrom the first equation, one finds a = -

so Y = ,

-
Since Y is a unit vector, /? = ± - 1 2 ± V 2 2 V( +»' )[(l-»' ) +V'2]' If 5 denotes the shape operator associated to the immersion X, then: -SX2

=

dvY:

V^V

*^U

- 5 J f i = S „ y = <^ - V j 9 ' - v'f3 +

+ < - i-ip1

{y 1 — ip1-tp2

fx)

x + ln'

2^'V 2 /? (1-V2

\-
(V'V + V'2)/? l-^2

165 So, I(X2,X2)

= \{X2,SX2)

= -\{X2,dvY)

n(x1,x2) = i(x 1 ,5x 2 ^V'

= -p

x/T+^

v w

-l(XudvY) ,„' 2

2

(1 + V 2 ) § ( 1 - ^ 2 )

(l + v 2 ) i ( l - v 2 )

-r + (l + v2)5 (l + ^ 2 )5

V'2 + ( V 2 - 1 ) 2 c# (i + v 2 ) ( i - ^ ) In

= I(Xl,Xl)

= I{Xl,SXl)

-f>& - V>'P

=

--[{XhduY)

v'P

VV

1 -

(l +

1

2

v

)

2ip'2tp/3 -VJ9

:

yJT+^

L 1- V

2^'V/? (i -
vV/?' l-<£ 2

V

+

/3'-

¥>V/M l-v2

V (1 + ^ ) 5

y"/3 1 1-V 2 J ( v 'V + v,'2)/3 i-V

2
2 V

)(1-V2); (

,2

y^g

y

+(y 2 -i) 2 g\

■/2+(v2-i)2 q \(HV)(i-v' 2 ) p

0

2^'V

P 2

2

(1+V )(1-^ ) ^,2

+

(v,2_1)2

Since <^ ^ ±1, we have d e t l = —' T ^ i t f / 0, thus the immersion X is non-degenerate. X is then a regular immersion in S 3 (l), whose Gaufi-Kronecker curvature K is: K = ^ | = — 1 ^ 0; from Proposition 3.18 it is of Tchebychev type. Let's compute the mean curvature of X. Denote by (I'-') the inverse matrix such that PJLj = 5{. One has: H =

I t r I H = i l I w I w = i[lI 1 1 I l l + 2n12I12 + I 22 I 22 ]

p ( (

l_v2)2

+

(/

,,2)(1_¥,2)

[
Consequently, if the C°°-function ip is not constant, then the regular immersion X is nonisoparametric, but of Tchebychev type in S 3 (l). Thus, the deformation of the given im­ mersion x by a non-constant function is regular, non-isoparametric and of Tchebychev type.

166

4

Conformal Tchebychev vector field

Let (Mn,g), n > 3, be an n-dimensional semi-Riemannian connected manifold. Denote by R, Rice the Riemann curvature tensor and the Ricci tensor of g, respectively. The Riemannian curvature R is the (0,4)-tensor field defined by R(u, v, w, z) = g(R(u,v)w,z). For a (0,fc)-tensor field B on Mn, define the (0, k + 2)-tensor R ■ B and Q(g, B) by (R-B)(Xu...,Xk;X,Y)

= -B(R(X,Y)XuXa,... ,Xk) B{X1,...,Xk_1,R{X,Y)Xk); = ~B((X A Y)XuXt,... ,Xk) B(Xl,...,Xk_1,(XAY)Xk),

Q(g,B)(Xu...,Xk;X,Y)

respectively, where {X A Y)X{ = g{Xi:Y)X

-

g{Xi,X)Y.

Definition 4.1 ([24], [1]). A semi-Riemannian manifold (Mn,g) is said to be semisymmetric (respectively Ricci-semisymmetric) if and only if R- R = 0 (respectively R ■ Rice = 0) holds on Mn. Lemma 4.2 ([7]). Let x: (Mn,g) —> S n+1 (l) be an isometric immersion with n > 3. Denote by S the shape operator. If S 2 = a S + /3id, n

a , / 3 e l , is satisfied at p £ M , then R-R=(ln

(4.26)

/3)Q(g, R) holds at p.

n+1

Theorem 4.3. Let x: M —> S (l) be a regular immersion, with n > 3. Then the following assertions are equivalent: (i) (M",I) is semisymmetric; (ii) the shape operator S and its inverse S

satisfy

S s 1=

~~

{^ir)id'

(4,27)

where A is a nonzero principal curvature function; (iii) x has at most two distinct principal curvature functions Ai and X2 satisfying Cartan's formula in Lemma 3.8, i.e. A1A2 + 1 = 0, at non-umbilics; (iv) (Mn, H) is semisymmetric. Proof. Using a I-orthonormal frame of principal vector fields, straight forward calculations of R ■ R give the following relation between two given principal curvature functions A; and A,: (\ ~ Xj)(l + XiXj) = 0.

(4.28)

Clearly the equation (4.28) is equivalent to (4.27). If (ii) is true, it is clear that S fulfills the form of the equation (4.26) with /? = 1. This proves that (i) and (ii) are equivalent. The equivalence (ii)-$=>(iii) is obvious. The equation (4.27) gives the equivalence (ii)-s=^(iv). □

167 R e m a r k 4.4. As it is known [21] that semisymmetry and Ricci-semisymmetry for hypersurfaces of Riemannian space forms with non-zero constant sectional curvature are equivalent, Theorem 4.3 proves that semisymmetry (also Ricci-semisymmetry) is polarization invariant. Definition 4.5 ([10]). Let (Mn,g) be an n-dimensional semi-Riemannian manifold, and denote by V the Levi-Civita connection of g. A vector field X e X(Mn) is said to be g-conformalif and only if there exists

= 2ipg(Y,Z),

Y,Z&X{Mn).

for any

Locally, ViXkgkj + VjXkgki = 1<4>gij- The trace with respect to g gives: 2 div X = 2rup. Furthermore, if X is g-gradient, that is X = grad / for some / G C°°(M re ), then ip = - div" grad 9 f = - A / . On the other hand, 5(Vygrad

9

/ , Z) + g(Y, V ^ g r a d ' / )

= y f l (gradV, Z) - s(grad*/, V y Z ) + Z S (Y, grad"/) - s ( V z y , grad 9 /) = Y(Zf)

-

9(grad7,

VyZ) + Z(Y/) -

ff(Vy^

- [y, Z], grad 3 /)

= Y ( Z / ) + Z(Y7) - 2 5 ( g r a d ' / , V y Z ) + [Y, Z\f

=

2Y(Zf)-2VYZ(f)

= 2Hess 9 (/), where div , grad , Hess are the divergence, the gradient and the Hessian, with respect to the semi-Riemannian metric g; A 9 is the Laplacian associated to g and defined as follows: A s / = 3 W V/,V(/, with gMgk, = 5jf, p/t; = g(ek, e,) and (e;) a frame. So X = grad / is g-conformal if and only if Hess (/) = - ( A f)g, if and only if Vgrad / = T h e o r e m 4.6. Let x: Mn — i S n + 1 ( l ) be a regular semisymmetric Then the following statements are equivalent:

immersion,

with n > 3.

(i) x is of Tchebychev type; (ii) the Gaufi-Kronecker

curvature K satisfies the following

PDE:

V 2 grad 2 \n\K\ = (- A , In \K\) id;

(4.29)

(iii) the Tchebychev vector field T is 1-conformal; Proof. Since the immersion is semisymmetric, (4.26) with /3 = 1 implies that S — S is proportional to the identity map. Therefore, the immersion x is of Tchebychev type if and only if the Tchebychev operator T is proportional to the identity map. Calculate the trace of T to observe that, if r is proportional to the identity map, then r = ( - A , In \K\) id; so we proved (i) <=> (ii). Prom Proposition (2.1), the Tchebychev vector field T is a I-gradient field. The PDE (4.29) is equivalent (see above) to the fact that T is I-conformal, in other words, (ii) <=> (iii), see also [10], p.238. D

168 Corollary 4.7. Let x: Mn —► S" +1 (l) be a regular semisymmetric immersion. Assume that the Gaufi-Kronecker curvature function is not constant on Mn. If moreover the im­ mersion is of Tchebychev type, then the following statements hold for the semi-Riemannian manifold (Mn, I) : (i) In a neighborhood of any point with \\T\\2 ^ 0, 1 is a warped product metric 1 = rjdt2 + (In \K\)'2(t).g, (where \\ \\^ is the norm with respect to the indefinite metric H, r) is the sign of ||T||jj ^ 0), the trajectories of T/\\T\\t are 1-geodesics along which K satisfies {\n\K\)"' = l\\n\K\. (ii) The zeros of the Tchebychev vector field are isolated. In a neighborhood of such a zero p, E is a warped product in polar coordinates I = "qdt2 + .'"'„,l,,a,1,gv, where gv is the induced metric on the pseudo sphere {u € TpMn: \\u\\2 = r}}. In particular, near a critical point of\n\K\, the second fundamental form I is conformally flat. (' and " denote the first and the second derivatives along the trajectories.) Proof. Under these hypothesis, the non-constant function In K is a solution of the PDE (4.29). The corollary follows from Lemma 2.3, p.239 in [10]. □ Definition 4.8 ([20], p.92). Two semi-Riemannian manifolds (Mn,g) and {Mn,'g) are said to be anti-isometric if and only if (Mn,g) and (Mn, —g) are isometric. Corollary 4.9. Under the assumptions of Corollary 4-7, if moreover the semi-Riemannian space (Mn, I) is complete and of constant scalar curvature, then the following holds: (i) IfT has a zero, then (M,I) is isometric to §£,R£,H£, or a covering of S ^ or El^_1; (ii) ifT has no zero, then (M, I) is isometric or anti-isometric to: (a) a warped product (R x Mt,dt2 + f(t)g); (b) a twisted warped product (S 1 x Mt,dt2 + f(t)g), for some positive function f, and where (M,,g) is a complete (n — 1)-dimensional manifold of constant curvature; (iii) if T is a complete vector field with a zero, then (M, I) is isometric to S" or to RjJ. Proof. This corollary is a consequence of Theorem 4.6 and Theorem 4.3, p.244 in [10].

□

References [1] K. Arslan, Y. Celik, R. Deszcz, R. Ezentas: On the equivalence of Ricci-semisymmetry and semisymmetry, Coll. Math., Vol 76 No.2 (1998), 279-294. [2] F. Brito, H.L. Liu, V. Oliker, U. Simon, C.P. Wang: Polar hypersurfaces in spheres, Geometry and Topology of Submanifolds, IX (1999), World Scientific, 33-47.

169 [3] M.P. do Carmo: Riemannian Geometry, Birkhauser Boston (1992). [4] E. Cartan: Families de surfaces isoparametriques dans les espaces a courbure constante, Annali di Mat. 17, 177-191 (1938). [5]

: Sur des families remarquables d'hypersurfaces isoparametriques dans les es­ paces spheriques, Math. Z. 45 (1939), 335-367.

[6] T.E. Cecil, P.J. Ryan: Tight and taut immersions of manifolds, Pitman Advanced Publishing Program, Research Notes in Mathematics 107 (1985). [7] R. Deszcz, L. Verstraelen, S. Yaprak: Pseudosymmetric hypersurfaces in ^-dimensional spaces of constant curvature, Bull. Inst. Math. Acad. Sinica 22, No.2 (1994), 167-179. [8] S. Kobayashi, K. Nomizu: Foundations of Differential Geometry. Vol I, Intersc. Publ. N. Y. (1963). [9]

: Foundations of Differential Geometry. Vol II, Intersc. Publ. N. Y. (1969).

[10] W. Kuhnel, H.-B. Rademacher: Conformal vector fields on pseudo-Riemannian spaces, Diff. Geom. and its Appl. 7 (1997), 237-250. [11] LB. Kim, T. Takahashi: Isoparametric hypersurfaces in a space form and metric con­ nections, Tsukuba J. Math., 21, No. 1 (1997), 15-28. [12] H.L. Liu, U. Simon, C.P. Wang: Higher order Codazzi tensors on conformallyflatspaces, Beitr. Algebra Geom. 39, No.2 (1998), 329-348. [13] H.L. Liu, U. Simon, C.P. Wang: Codazzi tensors and the topology of surfaces, Ann. Global Anal. Geom. 16, No.2 (1998), 189-202. [14] A.M. Li, H.L. Liu, A. Schwenk-Schellschmidt, U. Simon, C.P. Wang: Cubic form meth­ ods and relative Tchebychev hypersurfaces, Geom. Dedicata 66 No2 (1997), 203-221. [15] A.M. Li, U. Simon, G. Zhao: Global Affine Differential Geometry of Hypersurfaces, De Gruyter (1993). [16] H.L. Liu, U. Simon, L. Verstraelen, C.P. Wang: The third fundamental form metric for hypersurfaces in nonflat space forms, J. of Geometry, 65 (1999), 130-142. [17] K. Nomizu, T. Sasaki: Affine Differential Geometry, Cambridge University Press (1994). [18] K. Nomizu, U. Simon: Notes on conjugate connections, F. Dillen et al (eds.), Geometry and Topology of Submanifolds, IV (1992). Proceedings of the Conference on Differential Geometry and Vision, Leuven, Belgium, 27- 29 June 1991. Singapore: World Scientific, 153 - 173. [19] K. Nomizu, B. Smyth: A formula of Simon's type and hypersurfaces with constant mean curvature, J. Diff. Geom., 3 (1969), 367 - 377. [20] B. O'Neill: Semi-Riemannian Geometry, Academic Press (1983).

170 [21] P.J. Ryan: Hypersurfaces with parallel Ricci tensor, Osaka J. Math., 8 (1971), 251-259. [22] U. Simon, A. Schwenk-Schellschmidt, H. Viesel: Introduction to the Affine Differential Geometry of Hypersurfaces, Lecture Notes, Science University of Tokyo (1991), ISBN 3 7983 1529 9. [23] M. Spivak: A Comprehensive Introduction to Differential Geometry, Vol IV, Boston, Mass., Publish or Perish (1975). [24] Z.I. Szabo: Structure theorem on Riemannian spaces satisfying R(X, Y)-R = 0, 3. Diff. Geom., 17 (1982), 531-582. TSASA LUSALA: Technische Universitat Berlin, Fachbereich 3 / Mathematik Sekr. MA 8-3, Strafie des 17. Juni 136, D-10623 Berlin Germany. Email address: [email protected] Received January 31, 2000

Geometry and Topology of Submanifolds X eds. W. H. Chen et al. (pp. 171-177) © 2000 World Scientific Publishing Co.

171

N I R E N B E R G ' S P R O B L E M I N 90'S LI MA ABSTRACT. In this paper, we discuss the recent existence results and compactness of solu­ tions in the study of Nirenberg's problem in the 90's. We will discuss some new results and will outline some key ideas used in proofs of Theorems old and new. We also present some new questions. KEYWORDS: Gaussian curvature, Morse theory, bifurcation. 1991-MSC: 53C, 35J

We are at the new century and it is a good time to retrospect what we know in the past and give a prospect for what we will do in the future. In the research of 2-dimensional Riemannian geometry, an interesting topic is to seek solutions of the prescribed Gaussian curvature problem on a 2-dimensional manifold, and in particular, to study Nirenberg's problem, which is a typical question with the conformal group (non-compact) action in nonlinear analysis. This two-dimensional problem does have many special features. Let (S2, c) be the standard unit 2-sphere in Euclidean space R3. Then Nirenberg's problem is to find functions K(x) on S2 such that they are the Gaussian curvature functions of metrics g, which are pointwisc conformally equivalent to c on S2. This amounts to finding a smooth function u : S2 —> R satisfying the following equation on S2: Au = l-Ke2u,

(1).

Here A is the Laplacian with respect to c. Assume that u is a solution of (1). Then the metric g = e 2u c is conformal to c and has K as its Gaussian curvature. Integrating both sides of (1) on S 2 , we find

f Ke2udo = 4ir. So a necessary condition for (1) to be solvable is that K be positive somewhere and we write such function class by C\(S2). We will assume this condition throughout the paper. Our discussion will be divided into two parts- the existence and the compactness. It might not be complete due to the limited space. P a r t I: E x i s t e n c e R e s u l t s Define A = {K e C\{S2);

(1)

is

solvable}.

The work is in final form and no version of it will be submitted for publication elsewhere. This work is partially supported by a grant from NSFC, SRF for Returned Overseas Chinese Scholars, State Education Commission, and a scientific grant of Tsinghua University at Beijing. I am grateful to Prof.K.C.Chang and Prof.W.Y.Ding for helpful discussions.

172 To solve (1), we note that it has a variational structure. Namely, the critical points of the following functional J(u) = — f (\S7u\2 + 2u)da-log An JS2

— f Ke2ud(T, 47r JS2

which is defined on the set H = {ue

H'tS2);

Ke2uda

— f

> 0},

2 47T Is Jgi

correspond uniquely to the solutions of (1) up to scalar factors. In fact, If,

,

2-j5- L,

Ke2uvda

We remark that one may also introduce another variational structure for (1) on a constraint subset of H (see Theorem 10 below). Anyway, one can try to use variational methods to solve (1). Actually the first result about solving (1) was obtained by J.Moser [15], who used the direct method when K is an even function positive somewhere. Using Moser-Trudinger's inequality [15], J.Moser showed that J is bounded from below on H and the minimizer exists among even functions in H. Hence, we have Assertion 1. C 2 even functions positive somewhere are in A. From this we conclude that the prescribed Gaussian curvature problem on the real pro­ jection plane P 2 was completely solved. Interestingly, we have Assertion 2. The infimum of J on H can never be attained unless K =

constant.

This was showed by Hong [13] and Onofri [20]. Now we come to another key fact-the Kazdan-Warner obstruction, which motivates many works in Nirenberg's problem. In 1974, Kazdan-Warner (see [14]) found the following nec­ essary conditions: Proposition 3. The solution u of (1) should satisfy

f

< \/K, SJXj > e2uda = 0; Is22 Js where the coordinate functions Xj are the first spheric harmonic - A Xj = 2XJ,

j = 1,2,3, on

functions,i.e.

2

S.

These famous obstructions originate in the conformal group action of (1). In fact, this proposition can be proved by using the conformal diffeomorphism group of S2. Let F be a conformal diffeomorphism on S2 and let F' be the Jacobian of F. Then F*(c) = \F'\c. Note that every conformal diffeomorphism F can be generated by the rotations, involutions, and the maps of the form P~ltvP and P~ldsP, where ta is the translation by a € R2: U(y) =

y-a,

and ds is the dilation by s > 0: ds{y) = sy.

173 Here P is some stereographic projection from S2 to R2. We will consider (a, s) as the parameters for the family {|F'|} on S2. Sometimes, we write Fa>J for F obtained in the way above. Assume u is a solution of (1). Define u(s) = u o F0iS

+-log^J

\'

Since ^J(u(s)) = 0 at $ = 1, we obtain from this the Kazdan-Warner conditions. The following result is well-known to experts: (see [6] [8] [13] [20]): Proposition 4. all solutions of the equation Au = 1 - e2u,

on

S2

are of the form UF = \log\F'\ and they are the minimizers of J on H. More explicitly, written in the stereographic projection, we have u

r(y) = \lo9{1

+

JlSy_al2r

V^R2.

Therefore we know all solutions for (1) when K = 1. If one considers this as a uniqueness conclusion for solutions of (1) up to the symmetry, then I prefer to believe that this may be the only case which has such a uniqueness result (see also the comment after Theorem 13). For earlier results of Nirenberg's problem and its obstructions, one may see the article by J.P.Bourguignon and J.P.Ezin [3], A.Chang and P.Yang [6], W.Chen and W.Ding [8], and the references therein. Because of the lack of the Palais-Smale condition for the functional J on H, people would like to try to develop the Morse theory or minimax theory for (1). However, a PS sequence analysis tells us that the blow up behavior is simple (see [5][8]): Proposition 5. Assume (uk) is a PS sequence, i.e., J(uk) -> C and J'(uk) -> 0. Then there is a subsequence, still denoted by (uk), such that either (uk) converges strongly to a critical point of J, or there is a point Q and a sequence sk —> +oo such that

^Jjv(uk-FQ,Sk)\2da->0; furthermore, Q should be a critical point of K with — A K(Q) > 0 provided — A K(Q) ^ 0. Chen and Ding [8] used a mass center analysis (see [1]) to prove this Proposition. It is easy to observe from this result that for an even function K, J satisfies the PS condition on the subspace of even functions of H. In general, let T be a finite group of isometries with order > 2 acting without fixed point on S2 and let HT be the subspace of T-invariant functions in H. Then J satisfies the PS condition on HT provided the prescribed curvature function K is F-invariant. By this result one can see Assertion 6. The full Morse theory or other minimax theory is true for J on Hr provided K is a T-invariant function on S2.

174 As a side-remark, if T has finite fixed points {pi, ...,p m } and K satisfies the "non-degenerate" condition AK(pi) ^ 0 whenever v K{Pi) = 0 and Kfa) > 0, for i = l,..., m. Then p^s are the only possible bubbles of J, and we have a Morse lemma at infinity of J at each Pi just like in [5] and [11]. Therefore, we conclude [16] that Proposition A. the Morse theory (at infinity in the sense of A.Bahri [2]) is true for J on Hr provided the prescribed curvature function K is T-invariant. Similar to the Morse theory, A. Chang and P.Yang [5] and Z.Han [11] established the following. Theorem 7. Assume K is a Morse function on S2 satisfying AK(x) ^ 0 whenever \/K(x) = 0 and K(x) > 0. Let p =number of local maxima and q =number of saddle points with AK < 0. If p ^ q + 1, then (1) admits at least one solution. Using this result we can derive the following (see [18] for a proof): Assertion 8. ^4.s a subset ofC2 functions positive somewhere, A are dense in Cl'a-topology, where a £ (0,1). The I? version of this result is in [3]. In view-point of Morse theory one can pose the following: Question 9. Find all possible relations between the critical points of a non-trivial function K on S2 and solutions of (1). We can give a partial answer to this question when K = exp(th) for t near 0 (see [17]). Theorem 10. // h is a smooth function on S2 such that for the function H : R2 x R+ —> R defined by f h{y)e2^do-y, Js2 its gradient dH has a simple zero (a0, so) (i.e. the Jacobian of this vector function is invertible at this zero point (ao,So) ) , then there is a small positive constant e such that Nirenberg's problem of prescribing Gaussian curvature exp(th) has a solution, which is near to uF, for each small t : \t\ < e. Here F = Fa0tSo. H{a,s):=

This result can be considered as a first step in relating the critical points of the function K = h with solutions of (1) because the geometries of K = exp(th) for t small and h are the same. In getting a global result, one may use a suitable continuation argument to solve (1) with prescribed curvature functions in a homotopy from K = expith) to K = h. We remark that for suitably given functions H(a, s), we can use the Fourior transformation law to find h. Note that the original proof in [17] is for K — 1 + t.h, but it also works for K = exp(th), which corresponds to the original formulation of Nirenberg's problem [1]. It is clear that our result actually gives a multiplicity result on solutions of (1). So it can be considered as a global result provided K = exp(th) is fixed. In this sense, this result is very strong. The early perturbation results (see [4] for references) are obtained by degree counting method. We also ask the reader to notice that there is a change in the type of solutions of (1) when t changes from zero to non-zero. Actually according to Assertion 4 our solutions can not be minimizers of J on H for fixed K = exp(th) (t ^ 0) and they are isolated in C2'Q-topology in H, where a 6 (0,1).

175 One key part in our proof is that we restrict the functional J to the "sphere" of H denned by E = {u G H, -i- / e2u = 1}. Note that a critical point of J on E is a solution of (1) up to scalar translation. This is like in the Nehari's trick. In fact, assume u is such a critical point. Then by Lagrange's multiplier method we find a constant A € R such that Ke2n - A u + 1 - l . „ 7 = Ae2u, on S2. Integrating the equation above on S2 we have that A = 0. Another key part is that the solution family {UF} to (1) with K = 1 is a (non-compact) non-degenerate critical manifold of J on E in some sense of R.Bott when t = 0. We remark that this fact can be easily derived from the spectrum property of the Laplacian of the standard metric c on S2 (see [17]). Then one just follows the Lyapunov-Schmidt process [19] with a simple calculation. It is clear that our formulation is of variational nature. A related perturbation argument was used in [7]. Another remark is that one may use the Thorn-Mather theory or Conley-Morse theory to study this problem in the degenerate cases like that the function H(a, s) has good degeneracy somewhere. There should be some deep results. However, we will not do it. A formulation similar to our theorem above for the scalar curvature problem was in our work [18]. We now return to a beautiful Morse theory result. In 1993, K.C.Chang and J.Liu [4] found a complete Morse theory for (1): Theorem 11. Assume K is a function on S2 satisfying AK(x) ^ 0 whenever \jK{x) = 0 and K(x) > 0. Assume a < b are two regular values of K such that K has finite, number of critical points with values in [a, b] and assume J has only isolated critical points, then the following Morse inequalities hold:

^ K + ^-A,)** = (l+t)Q(f) where Q is a formal series with non-negative coefficients, mq is the Morse type number of J on J~1[—logb, -logo] and /3, is the Betti number for the pair (J-iogb, J-i09a)> and nq ^number of critical point set Crq(a, b). This theorem caps many previous results (see [4] for full exposition) and it should be considered as one of the main development in Nirenberg's problem. In [4] the authors used their Morse theory with general boundary conditions. In comparison with Assertion 6 and Theorem 11, we believe that there are local invariants which can guarantee the existence of solutions of (1). Now let's consider another important case that A" is a radial function ; by this we mean that K is invariant under the actions of rotations which fixes the north pole and south pole. In this respect, let me mention the works of Chen and Li [9-10]. First let's see the following result of X.Xu and P.Yang [22]: Theorem 12. Assume K is a "non-degenerate" radial function, i.e. it has the property that K"(r) / 0 where K'(r) = 0. If the derivative K' has positive and negative values in the region where K > 0, then (1) has a solution.

176 Using the technique of moving spheres, W.Chen and C.Li [9] improved this result and proved the following beautiful result (see Corollary 1 in [9]): Theorem 13. Assume K is a "non-degenerate" radial function, i.e. it has the property that K"(r) / 0 where K'(r) = 0. Then the necessary and sufficient for (1) solvable is that the derivative K' has positive and negative values in the region where K > 0. Their argument is based on the observation that if (1) has a solution and K' has only one sign on where K > 0, then one can use moving sphere method to show that the solution is oo at the north pole and thus derives a contradiction. This result also shows us that the geometry (not just topology) of the function K plays an important role in Nirenberg's problem. We may ask: Do we have the uniqueness of solutions in this theorem? The answer is no (see Theorem 3 in [9]). This means that if we can use Theorem 13 to find some solution, then it may not have any radial symmetry. One can also find other interesting results in [9]. Part Two: Compactness Now we go to the compactness question about the solution set of (1). Chen and Li (see Theorem 2 in [10]) gave a complete proof of Theorem 14. Assume K e C2'a changes sign and | V K{x)\ > c0 where \K(x)\ < SoFurther assume K satisfy | A K{x)\ > da > 0 where \/K(x) = 0. Then there is a positive constant C depending only on Co, So, \K\ci,<* and do such that for any solution u of (1), we have |M|CV. < C.

Note that the assumption that K changes sign plays a important role in their argument. Using an energy estimate method , A.Chang, M.Gursky and P.Yang [7] had a similar apriori bound for K > 0. The most general result is Theorem A in the work of Han [11]. However an explicit apriori estimate is still missing. It is a tempting guess that with a good use of mass center analysis and blow-up argument, one can get such an apriori estimate without the condition that K changes signs. At least, in view-point of Assertion 1 we ask Question 15. When K are Y-invariant functions except positive constants, do we have a better apriori bound for the solutions of (1) ? Remarks: Certainly we may ask the questions about prescribing Gaussian curvature functions on other 2-dimensional compact Riemannian manifolds. For such problems, a important ingredient is the following Aubin-Trudinger-Moser inequality [1] on closed Rie­ mannian 2-manifold (M,g) with the Euler-Poincare characteristic being less than two: Proposition 16. There exist two constants A and k such that for every u £ HX(M) satis­ fying JM u = 0, we have

[ eu
177 other hand, we should say that the study of Nirenberg's problem has many interactions with other problems, in particular the scalar curvature problem (see [1][2][21]). Definitely, the research of Nirenberg's problem will go further and variational methods will still play a important role. By this we are sure that the variational principles are promising in the future of nonlinear analysis. To close our report, we mentioned some nice survey articles on earlier works. For the development of Nirenberg's problem in 80's, please see the article of A.Chang and P.Yang: The conformal deformation equation and isospectral set of conformal metrics; in "Recent Development in Geometry", Contemporary Math. 101, S.Y.Chang, et.ed.pp 165-178, 1989. For 70's, see Th.Aubin's paper: Meilleures constantes dans le theoreme d'inclusion de Sobolev et un theoreme de Fredholm nonlineaire pour la transformation conforme de courbour scalaire ; J.Funct. Anal. 32(1979) 148-174, and his book Nonlinear Analysis on Manifolds, MongeAmpere Equations. Springer-Verlag, Berlin, 1982; and there for complete references. REFERENCES [1] Th.Aubin, The Scalar Curvature, Differential Geometry and Relativity, edited by M.Cahen and M.Flato, Reidel Publ. Co. Dord., Holland, 1976, pp5-18. [2] A.Bahri, Critical Points at infinity in Some Variational Problems, Pitmann Research Note in Math. Vol.182, Longman-Pitman, London, 1989. [3] J.P.Bourguignon and J.P.Ezin, Scalar Curvature Functions in a Conformal Class of Metric and Con­ formal Transformations, Trans. AMS 301(1987), 723-736. [4] C.K.Chang and J.Q.Liu, On Nirenberg's Problem, Inter. J. Math. 4(1993), pp35-58. [5] A.Chang and P.Yang, Conformal metrics on S2, J. Diff. Geom. 23(1988), 259-296. [6] A.Chang and P.Yang, Prescribing Gaussian curvature on S2, Acta Math. 159(1987), 214-259. [7] A.Chang, M.Gursky, and P.Yang, The scalar curvature equations on S- and S-sphere, Calc. Var. and PDE, 1(1993), 205-229. [8] W.Chen and W.Ding, Scalar curvature on S2, Trans.A.M.S. 303(1987), 365-382. [9] W.Chen and C.Li, A necessary and sufficient condition for the Nirenberg problem, CPAM, 1997. [10] W.Chen and C.Li, A priori estimates for prescribing gaussian curvature equations, Ann. of Math., 145(1997), 547-564. [11] Z.Han, Prescribing Gaussian Curvature on S2, Duke Math. J. 61(1990), 679-703. [12] Z.Han and Y.Y.Li, A Note on the Kazdan-Warner type condition, nonlinearie analyse, 1996. [13] C.Hong, A best constant and the Gaussian curvature. Proc. A.M.S. 97(1986), 737-747. [14] J.Kazdan and F.Warner, Curvature functions on compact 2-manifolds, Ann. Math. 99(1974), 14-47. [15] J.Moser, On a nonlinear problem on differential geomtry, Dynamical systems, (peixoto, ed.) Academic Press, New York, 1973, 273-280. [16] L.Ma, Some inequalities on manifolds, to appear. [17] L.Ma, Bifurcation in Nirenberg's problem, C.R.Acad. Sci.Paris, t326, p583-588, 1998. [18] L.Ma, Bifurcation in the scalar curvature problem, To appear in Acta Math. Sinica. [19] L.Nirenberg, Topics on nonlinear functional analysis, Lecture notes in CIMS at NYU, 1974. [20] E.Onofori, On the positivity of the effective action in the theory of random surfaces, Comm. Math. Physics 86(1982), 321-326. [21] R.Schoen and S.T.Yau, Differential Geometry (in Chinese), Science Press, Beijing, 1988. [22] X.Xu and P.Yang, Remarks on prescribing gauss curvature, Trans. AMS, Vol.336(2)(1993), 831-840. [23] E.Hebey and M.Vaugon, The Best Constant Problem in the Sobolev Embedding Theorem for Complete Riemannian Manifolds, Duke J. Math, 79(1995), 235-279. DEPARTMENT O F MATHEMATICAL SCIENCES, TSINGHUA UNIVERSITY, BEIJING 100084, CHINA

E-mail address: lma9math.tsinghua.edu.cn RECEIVED JANUARY 6, 2000

178

Geometry and Topology of Submanifolds X eds. W. H. Chen et al. (pp. 178-199) © 2000 World Scientific Publishing Co. A N E W PROOF OF THE HOMOGENEITY OF ISOPARAMETRIC HYPERSURFACES WITH (g, m) = (6,1)

REIKO MIYAOKA ABSTRACT. A geometric proof of the homogeneity of isoparametric hypersurfaces with six single principal curvatures is given. This method is successfully applied t o the multiplicity two case [M3]. The present proof for single multiplicity case clarifies the idea of the proof as well as the geometry of the hypersurfaces. KEY W O R D S : Isopametric hypersurfaces, Homogeneity 1991 MATHEMATICS S U B J E C T CLASSIFICATION: Primary 53C40

§1. Introduction Isoparametric hypersurfaces with less than four principal curvatures are classically known to be homogeneous [Cl,2], while the discovery of infinitely many non-homogeneous examples with four principal curvatures by Ozeki-Takeuchi ([OT], see also [FKM]) has shown the richness, and difficulty of the classification of these objects [Y, Problem 34]. U. Abresch [A] got a remarkable result on the multiplicities of principal curvatures in the case of four and six principal curvatures. In the latter case, he showed that the multiplicities are limited to one or two. Then Dorfmeister-Neher [DM] proved the homogeneity in the single multiplicity case, and we proved it in the multiplicity two case [M3]. It is worthwhile to describe the proof for single case in our strategy, because it is much easier and clarifies the idea of the proof as well as the geometry of the hypersurfaces. The idea is a use of the characterization of the homogeneity given in [M2], i.e. an isoparametric hypersurface with six principal curvatures is homogeneous when the null direction of the shape operators of the focal submanifold is independent of the normal directions. Assuming that the null direction changes, we show a contradiction by using the isospectrality of the shape operators and the Lax equation which satisfy.

§2. Preliminaries Let M be an isometrically immersed orientable hypersurface in the unit sphere 5 n + 1 having a unit normal vector field £. With respect to the Riemannian connection V on Partially supported by Grants-in-Aid for Scientific Research, The Ministry of Education, Japan This paper is in final form and no version of it will be submitted for publication elsewhere.

179 S"+1, the second fundamental tensor A of M is given by

AX = -vxf,

X e TPM,

peM,

of which eigenvalues Ai > • • • > A„ are called the principal curvatures. For A 6 {Ai,..., An} define Dx(p) = {XeTpM\AX = XX}, and let mxip) = dimDx(p)- When mx is constant on M, A is a differentiable function on M, and Dx is a completely integrable differentiable distribution on M. Moreover, if A is constant in the direction DA, which is the case for mx > 2, then Lx is locally an m,\-sphere of Sn+1 [R]. Assume from now on that mx is constant on M for all A. Then we can choose a local orthonormal frame (ei,..., e„) so that each ea is a unit principal vector with respect to Aa, 1 < a < n. We express (2.1)

Veaep = h."af}ea + Xa6a/3£,

where 1 < a, f3, a < n, using the Einstein convention. Obviously we have A7

-

-A'3

The curvature tensor Rap1(, of M is given by (2.2) RaPjS = (1 + XaXp)(6p~,6as - SaySpt) = e*(A£7) - ep(A6ay) + A ^ A ^ - A ^ A ^ - A£„A*7 4- A ^ X 7 The covariant derivatives of the coefficients of the second fundamental tensor hap = (Aea,ep) = Xa5ap are given by (2.3)

ha0n

= ey(haP) - k"iaha0 - b.°phaa = e 7 ( A a ) ^ + A ^ a ( A a - \p).

From the equation of Codazzi (2.4)

^a/3,7 = ^ 0 7 , a = ^ya,P

we obtain (2.5)

ep{\a) = kia{\a-Xp),

for a / j9.

For distinct principal curvatures A a ,A^,A 7 , we get from (2.3) and (2.4), (2.6)

Klp{\p - A7) = A^Q(Aa - Xp) = A£7(A7 - A0).

Moreover from (2.3) and (2.4), we have (2.7)

A2 6 =0,

A2a = A^,

if Aa = A 6 # A 7

and a ± b.

180 If A„ is constant on M, from (2.5) we obtain (2.8)

Ala = 0

if

A7^Aa.

Definition. When all principal curvatures are constant on M, M is called isoparametric. Isoparametric hypersurfaces are level surfaces of an isoparametric function / : S"+1 —» R that is a restriction of the degree g homogeneous polynomial F on E"+2, satisfying |VF| 2 = A

^

flV*"2, „ 9 2

r{x) = \x\, mi -m.2 -m2 mi

n,

c = " 22 ~ V9, m = mXi using the Euclidean inner product and the Laplacian [Mii]. AF = cr<>- ,

2.2. Focal submanifolds. Let M be an isoparametric hypersurface with (g, m) = (6,1), i.e. a hypersurface with six (i — l)7r single constant principal curvatures. As is well known, A; = cot#j, 6t = Q\ + —, 1 < 6 7T

i < 6, 0 < 6\ < —. Denote Dj = D^. We can choose a local orthonormal frame field 6 e i , . . . e6, so that e* is a unit vector of £>,. By (2.7) and (2.8), a leaf L* = £* of Di(p) is a geodesic in the corresponding curvature sphere Si, since Tjj~Ll n TpSj = @jfrDj(p). Define a map / ' : M -> S 7 by / ' (p) = cos 6>;p + sin 6i£p, and denote p = /'(p). Then we have (2.9)

/iej^sine^Ai-A^-,

where the right hand side is considered as a vector in TpS7 by a parallel translation in R8. In the following, we always use such identification. The rank of /* is constant and we obtain the focal submanifold Mj of M corresponding to A* = cot#,: Mi = {cos#ip + sin^^p | p e M}. By (2.9), TpMi = (Bj^iDj(q), for any q £ / ' space of Mi at p is given by r)v and Cp, where

(p). An orthonormal basis of the normal

% = - sin 6ip + cos 6»j£p,

CP = e; (p).

We consider the connection V on M, induced from the Euclidean connection D, that is

181 where X is a tangent field on R 8 in a neighborhood of p, and X is a tangent field near p defined by X(q) = X(q) for q = f(q). In particular, we have 1 sm6i(Xi-

V e ,e fc

We denote by vj.ej, S7. Then we have

'i>=e'

+ S

Mx^p-P)}-

(V^.e*,, resp.) the tangential (normal, resp.) component of Ve.efc in

TEA5*e'' sin 9i(Xi - Xj) Ifr 1 -{A}fcei 4- sin0j(l + sindi(Xi - X}

Vl.e*

HA

(2.10)

X

Xj, {J2

XjXi)5jkT)p},

since {Xj£p — p,r)p) = sin#j(l + A.,-A;). Note that in the latter, i is fixed and we take no sum. For later use, we give the matrix representation of the second fundamental tensors Bnp and B$ of M; corresponding to the normal vectors nv and £ p , respectively: L e m m a 2 . 1 . Identifying TpMi with ($^=1Di+j(p) where the indices are modulo 6, the second fundamental tensors BVp and B^p atp are given respectively by symmetric matrices: I^Jl 0 B

iv

I

Be

0

0

0

73 °

0

°

° 0 0

0 0

0 0

0 0

0 - 4 173

V 0

0

0

o

=

\

-Vs/

0 bi+li+2 frj+li+3 &i+lt+4

bi+n+2 0 &i+2i+3 6j+2i+4

^i+l>+3 6j+2i+3 0 &t+3i+4

&i+l i+4 6i+2i+4 &i+3i+4 0

\6i+li+5

6i+2t+5

&t+3i+5

&i+4j+5

sin#;

where

frj+li+5 \ 6j+2i+5 ^1+3 i+5 6i+4 i+5 0

1 bjk =

—A Aj — Aj

i-

Proof. We have ^p(e,) =

-vjj =

1 D sin #; (A* - Aj r e>

1 + XjX, Aj

Aj

/

182 where Djn denotes the component of Dejn tangent to XpMj. Recall that

3

Aj

Aj

1— t

cot(6j — 6i). Then from 8j — 8i = —-—n, we get BVp. Next for j ^ i, from

we obtain B{ , which is symmetric via (2.6).

□

By this lemma, Mt is minimal. The following is important. Lemma 2.2 [Mii, M2]. For any unit normal vector n of Mi atp, Bn is isospectral, i.e. the eigenvalues of Bn are ±-\/3, ±-7=1 0, and each eigenspace is of dimension one. v3 Fix a point p G M, put i = 6 and parametrize L6 (p) by (2.11)

p(t) = cos 9ep + cos t(p — cos dep) + sin t sin 86es(p).

Then p(t) = sin8eee(jp(t)), where ■ = J^. Consider a one-parameter family of shape operators Bm of M6 at p, where nt = -sin# 6 p(i) + cos#6& e TpM6, & = £p(t). We denote Bv = BVo,B^ = B^0, where we put (t = ee(jp(t)). It is easy to see that p(t) - cos 9eP = — sin 86rjt

for each t.

Then from p = cos 96p + sin#6£p = cos#6P(<) +

sm

#6£t,

we get (2.12)

fh =

__L_( p (t)_co8fl 6 p) smpfi

(cos i(p — cos 8ep) + sin i sin 8e() sm»6 = cos tn — sin t£

_ P W = —% = sint// 4- cost£ IPWI %-IT/2-

Lemma 2.3. LetSp = {s/iope operators of Mi atp}. ThenSp is a two dimensional vector space spanned by Bn and B^. Moreover, with respect to the metric given by (X, Y) =

183 tr(XY) on the space of symmetric with the same norm.

matrices, B^ and B^ are mutually orthogonal

matrices

Proof. Because of codim MQ = 2, dimSp < 2. Each shape operator has the constant norm ^ because of the isospectrality. By the expressions of Bn and B^ in Lemma 2.1, we get

P u t L{i) = BVt, i.e. (2.13)

L(t) =

costBv-smtB0

a one parameter family of shape operators of TpMg. Since L(t) is isospectral and symmet­ ric, we have orthogonal matrices U(t) e 5 0 ( 5 ) such t h a t (2.14)

BVt = L{t) = U{t)L{Q)U(t)-1

IKfyB^Uit)-1.

=

Define H{t) G o(5) by (2.15)

(±U(t))U(trl

H(t) =

then L satisfies the Lax equation (2.16)

jtL{t)

=

[H{t),L(t)l

i.e. - s i n t B , , - c o s t B c = cost[H(t),Bv]

- sin<[fl"(t),B c ].

In particular, we obtain (2.17)

BC =

-[H{0),B„].

Let U(t) be the solution of (2.15) with U(0) = / . Now the eigenvector ei(t) of L(t) with respect to the eigenvalue Aj is given by ei(t)

=

U(t)ei(0),

hence we obtain (2.18)

ei(t)=H(t)ei(t).

§3. Global p r o p e r t i e s Any isoparametric hypersurface M can be uniquely extended to a closed one [Mii]. We treat now a global properties of M. Let p 6 M and let 7 be the normal geodesic at p. We know t h a t 7 n M consists of twelve points p\,...,pw which are vertices of a dodecagon with 6 symmetric axes. Moreover, using the fact that M is taut, we have shown in [Ml, Lemma 6] (see Fig.l):

184 Lemma 3.1. (see also [M2, p. 197]) (3.1)

A(Pi) = De-i(p2) = A+a(Ps) = D^iipi) D

iiPj) = Di(pj+e),

= Di+i(j>b) = D2_i(p6)

j = 1,..., 6

where the equality means "be parallel to with respect to the connection of S7", and the indices are considered modulo 6. Choosing e5(p2) parallel with ei(pi), let p(t) be the geodesic of Llpi in the direction ei(pi) and let q(t) be the geodesic of Lp2 in the direction e$(p2). Extend ei and e 5 as the unit tangent vectors of p(t) and q(t), respectively. Consider the normal geodesic yt at p(t), and put pi(t) = p(t) and p2(t) = q(t) (Ijf By (3.1), we can take e,i(pi(i)) parallel with e2(p2)- Then we obtain 1 d sin*?! dte4[P[

_ sin6>5 11 d , , ., e2(q(t)). >> ~ s i n ^ sin0 5 dt

Therefore the Dj component of {Veiei)(p\) is the De-j component of {Vebe-2){p2) multi­ plied by sin0 5 /sin^i. We denote such a relation by Ai4(pi)■

i 6-j

0»2

A similar argument at every pa implies the global correspondence among Alg's: Lemma 3.2. For suitable frames aroundpa, we have correspondences A'-fc(pa) ~ A'-,fc,(pt,) where i,j, k at pa correspond to i',j', k' at p& in the following table: Pi

Pi

1 2 3

5

4 4

s

4 2 5 6

Pz 3

1 6

5 6 1 2

Pi

3 2 1 6 5

P5

5 6 1 2 3

4 4

Pe 1 6 5

/r\>

4

Y

3 2

i

v

r

'-y/]\

Vk

i?y ;Vi yvr

§4. Some formulas and key lemmas. Lemma 4.1. For distinct three indices a,i,j, (4.1)

we have

(V e o V e i e 0 , e i ) + (An - A j )sinfl 0 (B ea (V eo e i ),e i ) = 0,

185 which implies for X = YU& (4.2)

^ei'

{Vea V e j e a , X) + (A« - A,) sin0a<.Bea(Ve,>ei), X) = 0.

Moreover, we have (4.3)

l + AaAi = 2(V e a e i ,V e j e a ).

Proof. From the Gauss equation (2.2) and the relation (2.6), we obtain (4.4)

0 = Raiai = ea(Ai) + AfttA^ - A^AL + A?0A£ k \k f*a ~ *k _

(VeoVe,e0,e,-)-^A^A*0(

Aj

A

i — Afc Ag — Afc

A a — Afc A a — Aj

= (v e .v ei e olCj >-E A Si A *-r-4 i fc

a

~

?

= (V e a V e j e 0 ,ej) + (Aa - A,)sin6la(Be 2, since span{e3(p),Ve6e3(p)} C Ep where e3(p) and Ve6e3(p) are mutually orthogonal. We denote E instead of Ep, when it causes no confusion. Hereafter, we denote M+ = M 6 and M_ = Mj. The following lemmas are basic and significant. Lemma 4.2. Let E1- be the orthogonal complement of E in TpM+, and let N = span{V e3e6(p),V esV e3es(p)} ■ ThenN c £ x . Proof. Let p G M and express Bv and B^ in the basis e;(p) as in Lemma 2.1. Then we have f y/Sc

B„

—

S613

—S&14

—S&15 \

^ C

-S&23

-S*24

-5625

-S&13

-S&23

0

-5^34

-5*35

-S&14

-5^24

-S&34

~~73C

~*&45

V —S&15

-5^25

-S*35

-«&45

-\/3c/

-Sbu

—Sfcl2

C c = cosi, s = sint,

186 When e3(t) = *(ui3,..., U53) = U(t)e3(p) is its (zero-eigenvector, the third component of BVt(e3(t)) is given by

-« £&«««, = - - ^ £ ^ 3 = __i__l_AJ 6 (p) U a (0 = 0, hence we have (4.5)

=0

for all t, or equivalently <e3(p),Ve,e6(t)> = 0. Differentiating this by t, we obtain (e3(p),V e6 Ve 3 e 6 W) = 0, equivalently <e3(t),Ve6Ve3e6(p)) = 0, for all t.

a

By definition, E 2> span{e3(p), Ve6e3{p), V e 6 V e 6 e 3 (p),...}, thus from this lemma, we immediately obtain at each point p G M (4.6)

0 = (V e ,c 6 (p),V£,e 3 (p)),

* = 0,1,2,...,

0=(V e6 V e3 e 6 (p),V* 6 e3(p)),

* = 0,1,2,....

Note that (4.7)

V e3 e 6 = A 36ei = A ^ ^ — ^ = (2 - v ^ A ^ e , , A6 — Aj

where /ii = -fi5 = V3, fi2 = ~m = -j-. Lemma 4.3. Both Bn and B^ map span{e3(j>), V e6 e 3 (p), V e6 Ve 6 e 3 (p)} into Ex. In par­ ticular, Bm maps span{e3(p), V e6 e 3 (p), V e6 V e6 e 3 (p)} into Ex for any t. Proof. It is easy to see B„(e3(p)) = 0, Bn(Vsee3(p))

= kl3(p)^ei

= ciV e3 e 6 (p) £ JV c B.p1 , '

ci1 = _

2--y3'

and hence = 0,

187 where we use (4.6). On the other hand, we have 1 sin 06 A3 - A6 1 1 = C3Ve6Ve3e6(p) £ N c E^, c3 = p ' sin 0$ A3 - A6

Bi{e3(p)) = c 2 V e3 e 6 (p) &NC E}; B((Veee3{p))

C2

where we use (4.2), hence using (4.6), we obtain (Bc(Ve6Ve6e3(p)),e3(p)) = = c 2 (V e8 V e8 e 3 (p), V e3 e 6 (p)) = 0, (B c (V e6 V e8 e 3 (p)),V e8 e 3 (p)) = c 3

L(t - |)(V e .e 3 (t)) G E\

(i(*-J)(Ve6Ve,e3(0),C3(0> = 0, (4-9)

(L(t - ^)(V e 6 V e 6 e 3 )(t), V e ,e 3 (t)) = 0,

Now because we have — L(t) = -sintBr, - costB( = L(t + ^ ) , d_ = sin# V , 6 e6

It

differentiating (4.8) at t = 0, we obtain 0 = e 6 {I(<)(V e6 V e6 e 3 (i)), Ve6e3(t))]t=o sin6>6

( i ( ^ ) ( V e 8 V e6 e 3 ), V e6 e 3 ) + (B,(V e6 V e6 V e6 e 3 ), Veee3> + (B„(V es V e8 e 3 ), V C6 V e8 e 3 ) 2 63), V ee e 3 ) + (B^(V es V e6 e 3 ),

= (V e8 V e6 V e6 e 3 ,B„(V e6 e 3 )) + (B„(Ve6Ve6e3),Ve6Ve6e3> where the last equality follows from (4.9). Since the first term of the last line vanishes because of (4.6), we get (B t) (V e6 V e8 e 3 (p)), V ee V e6 e 3 (p)) = 0.

188 Next, differentiating (4.9) at t = 0, we obtain 0 = ee(L(t - ^)(V e6 Ve 6 e 3 ), Ve6e3)|*=o = ^ V ( ^ ( V e 6 V e 6 e 3 ) , Ve6e3> + (B c (V e6 V e6 V e6 e 3 ), V e6 e 3 ) + (5 c (V e 6 V e 6 e 3 ), Ve6Ve6e3> = e3)> + (B c (V e6 V e6 e 3 ), V e6 V e6 e 3 >. Using (4.6) and B^(Ve6e3) = C2VesVe3e6, the first term of the last line vanishes, and we obtain (S c (V e6 V e6 e 3 (p)), Ve6Ve6e3(p)> = 0. Now, using Bm = cos tBn - sin tB^, we have (B„(V e( ,V e ,es(p)), V e6 V e6 e 3 (p)) = 0. Finally, Bm maps span{e3(p), V ee e 3 (p), V e6 V ee e 3 (p)} into E^.

D

Lemma 4.4. dimi? < 3. Proof. When dimiV = 0, V e3 e 6 = 0 hence dimE = 1. When dimiV = 1, we see that dimB c (span{e 3 (p), Veee3(p)}) = 1 = dimB,(span{e 3 (p), V e6 e 3 (p)}). Therefore we get dimS, t (span{e 3 (p), V ee e 3 (p)}) = 1 and KerS,,, = Me3(t) € span{e3(p), V ea e 3 (p)} for any t. This implies dimE = 2. When dimiV > 2, from Lemma 4.2 follows dimE < 3. D Let d = maxpgM dim £■(/)). We know that 1 < d < 3 and M is homogeneous when d = 1. In the next sections, we show that d = 2 and d = 3 cannot happen. §5. dim E Here we consider the both focal submanifolds M±, and denote E+(p) = Ep and E_(q) = span£i(g){e4(<7(£))}, where Lx{q) = {
d_=maxE_(q).

Proposition 5.1. When d± ^ 1, either d+ = 3 or d_ = 3 occurs. Proof. Suppose d + = cL = 2. Note that dimi5 + (p) and dimE_(q) are lower semicontinuous, and U = {p € M | dim£ + (p)} = 2 and V = {q € M | dimE-(q)} = 2 are non-empty open subset of M. At a generic point p € U and g € V, we have (5.1)

E+(p) = span{ e3 (p), V e6 e 3 (p)}, E_(g) = span{e 4 ( g ), Veie4(q)},

V e6 V e6 e 3 (p) e £+(p), V e i V e i e 4 (p) € £ - ( p ) .

189 On the other hand, since KerB^ e E+(p) implies dimB((E+(p)) = 1, we have N+ = span{V e3 e 6 (p)}, N_ = span{V e4ei (g)},

V e6 V e3 e 6 (p) e N+, V ei V e4 ei(g) £ AL.

By (5.1), es{t) 6 span{e3(p), V es e 3 (p)} for all t. Note that f{L3(i)) is a geodesic of S 7 in the direction /*(e3(£)). Put X = e3(p) and Y = V e6 e 3 (p). Claim. If dim E+(p) = 2, the union \Jp^ei,e^f(L3(t)) S7 lying in M+.

is a totally geodesic 2-sphere of

In fact, from Bv{aX + bY) = bc\Ve3e6(p) and B^(aX + bY) = (c2a + czb)Ve!ie§(ip) for some constant c\, 02,03, we get Bm(xX + yY) = costciyV es e 6 (p) - sint(c 2 a; + c4f/)Ve3e6(p) = {-C2xsint + y{c\ cosi — C4sin£)}Ve3e6(p), which is zero when x 1 — = —(ci cott — C4), y c2v hence Kerf?,,, exhausts all the directions of span{e3(p), Ve6e3(p)} as t changes. This means that the union U p (t) ei 6( p )/(L 3 (i)) is a totally geodesic 2-sphere of S7 lying in M + . The claim implies that when d+ — 2, Ve6e3(p) = 0 at some p e M cannot happen. In fact, suppose there exists {pn} C U with \impn £ U. Since dimi? + (p) < 2, e3(p(i)) is constant on Le(p) = {p(t)}, and the union Uj,(()££6(g)/(L3(i)) is a single geodesic of S7 in the direction ftez(j>), hence cannot be a limit of totally geodesic 2-spheres. Thus we can define a distribution on M + by D+(p) = f,{E+(p)) = span{/»(e 3 (p)),/ t (V e6 e3(p))},

p = /(?)•

Similarly because dimE_(q) = 2 for all q ۥ Af, V ei V e4 ei(p) e iV_ holds along L 1 ^ ) , and by the global correspondence, we see Ve3Ve6e3(p) G span{V e6 e 3 (p)} along L3(p). Therefore both /*(e 3 ) and /*(Ve6e3) are parallel along f(L3(p)), which implies that the leaf of D+(p) is nothing but the totally geodesic 2-sphere given by U p ( t ) 6i ,6(p)/(L 3 (t)), i.e. D+ is a totally geodesic integrable distribution. Then f~1(D+) = {e$, Ve6e3(p),e6} is also integrable, which contradicts [e3,e6] = V e6 e 3 (p) - Ve3e6(p) e / - 1 ( D + ) since V e3 e 6 (p)_L/- 1 (D+).

D

190 Now, by Proposition 5.1, we may consider the case where d = d+ = 3. We emphasize that the space E is independent of ( by the definition in §4. Because of d = 3, at a generic point p, Ep is given by e

Ep = span{e3(p), V ea e 3 (p),

3(p)}-

Recall E^ = span{V e3 e 6 (p), V e6 V e3 e 6 (p)}, which is orthogonal to Ep. Since Ep and E^ span TM+, from Lemma 4.3 and its proof follows: Lemma 5.2. Bn and B^ map Ep onto Ep . Thus BVt maps Ep onto Ep for any t. Lemma 5.3. With respect to a fixed frame (e\(p),..., (5.4) COS (D\t) J

V e6 e 3 (i) = a{t)(

J2

ei(p)

e 6 (p)), we can express

+ smtp(t)e2(p) + smi/>(t)e4(p) +

COS t/>(£} ;

J}

e 5 (p)) + b{t)e3(p),

where

(5.5)

m=Mt)

+ c(p), e = ±l.

Proof. We write e, = e,(p) for short. By Lemma 4.3, we have (B^xex + ye.2 + ze4 + we$),xe\ + ye2 + ze4 + we5) = 0, for xei + ye2 + ze4 + wes e E, hence 3a;2 + y2 - (3w2 + z2) = 0. Then we can put cos tp x = a—r=^, v3

. y = asmip,

z = asmip,

w=

cos ib a—^, V3

and get (5.4). Moreover, we have (5.6) sin0 6 Ve 6 V e6 e 3 W = ^ V e 6 e 3 W . , . ,cosip(t) = a(t)(—-^-a

. ,. . ,,. cosiA(i) . ... + smip(t)e2 + smtb(t)e4 + ^-es)+b(t)e3

. . . . . sin¥>(t)e /,.,•, ;, , ,v + a{t){ip( " 7 p i + cos>f(t)e2) + ^{cosib{t)e4

sint/)(i) ., "Tf^)}'

191 so from
2(t). and il>(t) = e
D

Remark 5.4- When V e „e 3 (p) = 0 at a point p e M , the Gauss equation 0 = #636, = e 6 (A 3 6 ) + Af6A*„ - A- 3 At 6 + A ^ A ^ . implies V e „e 3 = 0 on Le(p), since this is an exponential type differential equation for ( A 3 6 , . . . , A 36 ) (see (4.7)). Thus a(t) never vanishes unless V e 6 e 3 = 0, and a(t) is uniquely determined by , a(t)sm
= (Ve6e3(0,e2(p)),

tp(t) is uniquely determined modulo 2n. Similarly, ip(t) and c(p) are uniquely determined modulo 2ir. Now, extend e;(p), i = 1 , . . . , 5 to a frame a(t) along p(t) by ei(t)

= U(t)ei(p),

U(t) 6 5 0 ( 5 ) ,

where L(t) = [/(t)L(0){7(t) _ 1 (see (2.14)). Here by £><(< + T ) = A s - i W , if we choose e 4 (p) = e 2 (g) and e 5 (p) = ei(g) where q = p(ir), we may assume (5.7)

ei(t + 7r) = e 5 (t), e 3 (i + 7r) = £ 3 e 3 ( t ) ,

e 2 (t + TT) = e 4 (i), e 4 (i + w) = e 4 e 2 (i),

e 5 (t + 7r) = £ 5 ei(i),

where e* = ± 1 , and e 3 e 4 es = 1. Now define a function / : Le(p) —> R by /(*) = (Aj 3 ) 2 (p(t)) - (A^ 3 ) 2 (p(t)) with respect to the above frame. Then f(t) we have

does not depend on £;. On the other hand,

192 because of (5.7). Thus there exists to at which /(to) = 0. Now at this p = p(t0), we fix the frame (e;(p)) and apply the argument above. Then it implies cos2 ip(p) = cos2(eip(p) + c{p)) sin2 ip(p) = sm2(sip(p) + c(p)) and hence 2
h E Z.

[Case 1] e = 1 implies c(p) = 0 mod IT and

In this case, we have (5.8)

V e .e 3 (0 = a ( t ) ( £ 2 ^ ( e i ( p ) ± e5(p)) + sin
for all £, and moreover from (5.6), we obtain sinfl 6 V e ,V e ,c 3 (0 = ^yV e 6 e 3 (<) + 0 ( t ) v { ( - ^ ^ ( c i ( p ) ± e5(p)) + cos V (t)(e 2 (p) ± e 4 (p))} + o(t)ea(p). Thus we get E = span{e3(p), ej(p) ± e5(p), e2(p) ± e 4 (p)}, 11

E - = span{ ei (p) =F e5(p), e2(p) T e 4 (p)} and in particular,

BJI:E
B^-.E^E1-

[Case 2] e = — 1 implies 2tp(p) = c(p) mod 7r and VW = -(*) + 2p(p) + fcTT. In this case, using <^(0) = V'(O) mod ir, we obtain V e6 e 3 (p) = 0 ( 0 ) { ^ 2 i ( C l ( p ) ±

CB(p))

+ sinv>(0)(e2(p) ± e 4 (p))}.

193 Moreover putting t = 0 in (5.6), we get Ve.Ve,e3(0)=^|ve„e3(t) + a ( 0 ) v i ( 0 ) { ^ ^ ( - e 1 ( p ) ± e 5 ( p ) ) + c o s ¥ > ( 0 ) ( e 2 ( p ) T e 4 ( p ) } + i(0)e3(p), and so putting a = cosi£>(0),

j3 = sin ip(0),

we obtain a 8 E = span{e3(p), -^=(ei(p)±e 5 (p))+/3(e2(p)±e 4 (p)), - ? =(-ei(p)±e 5 (p))+a(e 2 (p)=Fe4(p))}-

§6. Am\E = 3 does not occur. Recall (2.18)

e j (t) = F(Oe J (t)(=sine 6 V e 8 e j (t)), tf(t) 6 o(5),

and put fl" = (sin^gAg^)), where ffe* = s i n ^ y , , ^ . Lemma 6.1. = 0. Proof. From (2.17) B c = [Bv, H], we obtain (B((Z),Z)

=

({BV,H](Z),Z)

= ((B„H - tf B„)(Z), Z) = (HZ, BnZ) + {BnZ, HZ) = 2(HZ,Br,Z).

a Lemma 6.2. If we take an orthonormal basis ofTpM+ Zx = e3(p), Z2, Z2 e E,

W2, W3 e E1-

satisfying B(Zi) = CiWi, i = 2,3, we /ioue / 0 -a; ff = -y . 0 \ 0

a; y 0 0 r 0 —r 0 — u 0 u 0 -w 0 - t

0\ u 0 t . 0/

194 Proof. The first line follows from He3 e E. Since B( : E -» E-1, we obtain ( 5 ^ , ^ =0 and this implies {HZi,Wi)=0,

i = 2,3

by Lemma 6.1. Finally from {Bc(Z2 + Z3),(Z2 +

Z3))=0,

we get hence putting u = {HZ2, W3), we obtain the lemma.

□

[Case 1] We can choose an orthonormal basis Zi=e3,

Z2 = - ^ ( e i +e 5 ),

W2 = - L ( e i - e 5 ) ,

Z 3 = - = ( e 2 + e4) e £ ,

W3 = -L( C2 - e4) e £ x

which satisfies the assumption of Lemma 6.2 (The case where Z2 = -4?(ei — es) be similarly treated). It is easy to see that /0 ' 0 £„ = 0 0

0 0 0 /

0 0 0\ 0 I 0 ' 0 0 n 0 0 0

l = V5, n = - L ,

Vo 0 n 0 0 / and from (2.17) and Lemma 6.2, we obtain ' (6.3)

B( =

0

0 -zZ \-yn

0 0 (/ - n)« 0 It — nr

0 (I- n)u 0 Ir — tu 0

—xl —yn \ 0 It-nr Ir-tu 0 0 (/ — n)u (Z — n)u 0 /

thus we get u = 0 because Br : i? —» i?-1. So we can write 0 0 B < = 0 0 a 0 \b d

f°

0 0 0 e 0

a 0 e 0 0

<\ 0 0 0/

195

and. we have

(6.4)

0 0 0 0 L(t) = cos tBv — sintB^ = 0 0 -sa cl V — sb —sd

0 -so 0 cl 0 — se — se 0 en 0

-sb\ -sd en 0 0 /

where c = cosi and s = sint. In order that L(t) is isospectral, i.e. has eigenvalues 0, ±\/3, ±-i=, we must have det(A7 - L(t)) = A(A2 - 3)(A2 - | ) for all t. Since we have det(A7 - L(t)) = A[A4 - {c2{l2 + n2) + s2{a2 + b2 + d2 + e 2 )}A 2 + c2l2n2 + cs(n2a2 + l2b2) + s2{a2d2 + 6 V + d2e2} + 2cs3ab(ld + ne) -

2c2s2lnde],

putting s = 1, we obtain

a2d2 + b2e2 + d2e2 = 1. Together with I = \ / 3 ,

73' 2

n a + Pfc2 + 2s ab(ld + ne) - 2cs£n
|V e „e 3 r °

W2

|V,,e 6 |'

3

u r 7

„——Tjj

l(V e6 V e6 e 3 )

e

ii

'

|(Ve f l V e a e 6 )^|

±

where by ( ) we means Z 3 is orthogonal to Z j , Z 2 , and W 3 is orthogonal to W%. At p, we have a Z2 = 9{-7=(ei + e 5 ) + /3(e2 + e 4 )}, j3

W 2 = /i{a(ei - e 5 ) + -^=(e 2 - e 4 )},

B Z 3 = / i { - ^ ( - e i + e 5 ) + a ( e 2 - e 4 )} a

W 3 = fl{/3(-ei - e 5 ) + -y=(e 2 + e 4 )},

196 where

s=\/ 2 (y+/5 2 )

2(«2 + ^ )

(The case where Z2 = 9{-jzie\ ~ e s) +/3(e2 - 64)},... are similarly treated.) In particular, we have (6.4)

Bn(Z2) = | l ^ 2 ,

BV(Z3) =

-W3.

It is easy to see that

(° 0 0

Br,

Vo

0 0 0 I 0

0 0 0 0 n

0 I 0 0 v

0 n

5

V

and by (2.17), W(3 have (6.5)

-Be

=

[•^ii H\

0 —xl

z

V 0 Note that

0 0 0 —vu It — nr

0 0 0

-tn + Ir uv

—xl —vu -tn + Ir -2tv 0

It — nr uv 0 2tv )

/°0 0 0

Vo

0 0 0 b d

0 0 0 e

/

0

6 e P 0

° \.

d

f 0

-J

o = {B c (e3),V e 3 e 6 )^0, d=(B((Ve,c3),Ve,Ve,e£)?iO.

Moreover, we have

f L(t) = cos tBv — s'mtB^

0 0 0 -sa 0

V

0 0 0 cl - sb —sd

0 0 0 —se en — sf

0 —sa cl — sb —se en-sd - sf —sp cv cv sp 1

A

where c = cost and s — sin*. As before (here we use Mathematica), det(A/ - L(t)) = A[A4 - {c2(l2 + n2 + v2) + s2(a2 + b2 + d2 + e2 + f2 + p2) - 2cs(lb + nf)}\2 4- {c2s(p(l2 - n2) - 2v(dl + en)) + 2cs2(p(-bl + fn) + v(bd + ef)) + s3p(a2 d2 + e2-f2)}\ + c4l2n2 - c?s(fl2n + bin2) + c2s2(f2l2 - 2lnde + Alnbf + a2n2 + b2n2) + 2cs3(defl - bf2l + bden - a2fn - b2fn) + s\a2d2 + d2e2 + a 2 / 2 + b2f2 - 2bdef)} = A(A2-3)(A2-i)

hence we have (6.6) (6.7)

;2 + n 2 + u2 = a2 +

62 +

H

d2+e2 +

/

2

+

p

2

=

^

(6.8)

Ib + nf = 0

(6.9) (6.10)

p(l2 - n2) - 2v{dl + en) = 0

(6.11)

p(a2 + b2 - d2 + e2 - f2) = 0

(6.12)

l2n2 = 1

(6.13)

ln(fl + bn) = 0

(6.14)

f2l2 - 2lnde + Alnbf + a2n2 + b2n2 = 2

(6.15)

(de - bf)(fl + bn) - a2fn = 0

(6.16)

(o2 + e2)d2 + (a2 + 6 2 )/ 2 - 2bdef = 1.

p(-bl + fn) + v{bd + ef) = 0

From (6.13) and (6.15), we obtain / = 0. Thus from (6.8) we have b p / 0 implies v =£ 0 by (6.5). If p / 0, from (6.11) and (6.16), we get a2 + e2 = d2 = l,

(6.17)

V2 = \,

the latter follows from (6.7). Thus (6.9) and (6.14) are written as (6.18)

p2(J2 - n2)2 = 4v2(l2 + e2n2 + 2dlen)

(6.19)

n 2 - e 2 n 2 - 2dlen = 2,

and these with (6.17) imply (I2 -

2 2 n

)

= (10 - 3l2 - 3n2)(l2 + n2 - 2).

Noting that In = 1 by (6.4), we obtain I = n, and then by (6.9) e2 = d2 = 1, contradicts o ^ 0. Thus we obtain p = 0. When v =£ 0, (6.9) implies (6.20)

dl = -en

hence by (6.7) and (6.16), we get ,

s (6 21)

'

f o 2 + e2 = 3 |d2=I '

, 6

1 =3^'

198

f a2 _i_ „2 { rf 3.

(6.22)

_

I

3

e2 = —4 n

by using (6.20). On the other hand, from (6.14) and (6.20) follows a2n2 + 2e 2 n 2 = 2. Thus (6.21) implies 2 = 3n 2 + e V = 3ra2 + —r2 -H- n2 = - , I2 = 3 3n 3 and (6.22) implies 2 = — + —2 o n 2 = 3, I2 = 3 n 3 where we use (6.12). In both cases, we obtain v = 0 by (6.6), a contradiction. As a conclusion, we obtain b = f = p = v = 0, which is a special case of Case 1. Thus this can not happen. Finally we obtain Theorem [DN], Isoparametric hypersurfaces in Sn with (g, m) = (6,1) are homogeneous. Remark. 1. Homogeneous hypersurfaces with (g,m) = (6,1) are known to be principal orbits of the linear isotropy action of the symmetric space G2/SO(4). For more details, see [M2]. 2. We discuss the multiplicity two case in [M3]. This is by no means a simple extension of this argument, because, for instance, the elements Bij of Bv and B^ now become 2 x 2 matrices and the calculation is no more commutative. The strategy of the proof is almost the same.

REFERENCES

[A]

U. Abresch, Isoparametric hypersurfaces with four and six principal curvatures, Math. Ann. 264 (1983), 283-302.

[Cl]

E. Cartan, Families de surfaces isoparametriques dan les espaces a courbure co nstante, Ann. di Mat. 17 (1938), 177-191.

[C2]

, Sur des families remarkables d'hypersurfaces isoparametriques Math. Zeit. 45 (1939), 335-367.

dan les espaces spherique,

199 [DN]

J. Dorfmeister and E. Neher, Isoparametric 13 (1985), 2299-2368.

hypersurfaces,

case 5 — 6,

m = 1, Comm. in Alg.

[FKM] D. Ferus, H. Karcher and H. F. Munzner, Cliffordalgebren und neue isoparametrische Math. Z. 177 (1981), 479-502.

Hyperflachen,

[Ml]

R. Miyaoka, Dupin hypersurfaces with six principal curvatures, Kodai Math. J. 12 (1989), 308315.

[M2]

, The linear isotropy group of G2ISO(A), faces, Osaka J. of Math. 30 (1993), 179-202.

[M3]

Homogeneity , Homogeneity

the Hopf fibering and isoparametric

hypersur­

of isoparametric isoparametric hypersurfaces hypersurfaces with with six principal principal curvatures, curvatures, Preprint of P r e p r i n t (2000).

[Mii]

M . F. F. M i i n z n e r , Isoparametrische M. Munzner, Isoparametrische

Hyperflachen Hyperflachen

in M a t h . Ann. Ann. 2 57-71. in Sphdren, Sphdren, I, I, Math. 2 5 1 (1980), 57-71.

[OT]

H. H . Ozeki O z e k i and a n d M. M . Takeuchi, T a k e u c h i , On On some some types types of of isoparametric isoparametric hypersurfaces hypersurfaces in in spheres spheres II, 77, Tohoku Tohoku Math. M a t h . JJ.. 2 28 8 ((1976), 1 9 7 6 ) , 7-55. 7-55.

[R]

H. Reckziegel, Reckziegel, Kriimmungsflaschen Krummungsfldschen

von isometrischen isometrischen Immersionen Immersionen in in Rdumen Raumen konstan konstan ter ter Krummung, Krummung,

M a t h . Ann. Ann. 2 2 3 (1976), 169-181. 169-181. Math. 223 [Th] [Y]

T h o r b e r g s s o n , Dupin Dupin hypersurfaces, hypersurfaces, L o n d o n Math. M a t h . Soc. Soc. 1 5 ((1983), 1 9 8 3 ) , 493-498. G. Thorbergsson, Bull. London 15 S. T a u , Open T.. Y Yau, Open problems problems

in in geometry, geometry,

C hern - A reat g e o m e t e r of tthe h e ttwentieth w e n t i e t h century, century, Chern A ggreat geometer

IInternational n t e r n a t i o n a l Press, P r e s s , 1992, 1992, pp. p p . 275-319. 275-319. DEPARTMENT OF MATHEMATICS, SOPHIA UNIVERSITY, K I O I - C H O , CHIYODA-KU, T O K Y O 1028554, JAPAN

E-mail address: r-miyaok8hoffman.cc.sophia.ac.jp RECEIVED FEBRUARY 1, 2000

200

Geometry and Topology of Submanifolds X eds. W. H. Chen et al. (pp. 200-215) © 2000 World Scientific Publishing Co. HARMONIC MAPS AND NEGATIVELY CURVED HOMOGENEOUS

SPACES

SEIKI NISHIKAWA

ABSTRACT. AS a general class of negatively curved homogeneous Riemannian manifolds, we define a category offc-stepCarnot spaces, which includes the rank one symmetric spaces of noncompact type. Fundamental properties of fc-step Carnot spaces and the Dirichlet problem at infinity for harmonic maps between these spaces are discussed. KEYWORDS. Harmonic map, homogenous manifold of negative curvature, Dirichlet problem at infinity, Carnot group. 1991 MS-CLASSIFICATION. Primary 58E20; Secondary 53C43, 53C21, 53C30, 58J32. 1.

INTRODUCTION

Let M = (M,g) and M' = (M',g') be a Riemannian m-manifold and a Riemannian m'manifold, respectively, and u : M -> M' a C2 map from M to M'. In local coordinate systems (x') and (x'a) of M and M', the Riemannian metrics g and g' are given locally by S=^2

gijdx'dx3,

g1 = "52

gapdx'adx'l3,

and u is written as u(x) = ( « V , . . . ,xm),...

,um'(x1,...

,zm)) = ( u ' V ) ) -

Then the energy density e(u) of u is defined by e(n)(x) = ^ E E

5iJW5^(«W)^-W^(x) =

\\du\\x),

\du\ being the norm of the differential du of u, and for a relatively compact domain fi <s= M the energy En(u) of u over fl is defined to be En(u) = / Jn

e(u)dng,

where /ig denotes the canonical measure on M given by the Riemannian metric g. Let C2(M,M') be the space of C 2 maps from M to M'. Then the energy En{u) of u defines a functional En : C2(M, M') —> R, and u is called a harmonic map if it is a critical point of the functional En for all variations with compact support for each fi <= M. Namely, Partly supported by the Grant-in-Aid for Scientific Research, Japan, (A) No. 10304004, and the program "Analysis on Manifolds" of the Chinese University of Hong Kong. This paper is in final form and no version of it will be submitted for publication elsewhere.

201 u is a harmonic map if it satisfies on M the Euler-Lagrange equation r(u) = 0 for the energy functional, which is expressed in local coordinate systems as "i

(1.1)

~ p

m'

a

J

x

x

/} 7 x

Au (x) + J2J29' ^nM ))^ )^l( )=0'

« = l,--->m',

i j = l ,9,7=1

where A denotes the Laplace-Beltrami operator on M and Tg 7 are the Christoffel symbols of M'. Geometrically, the quantity r(u) defines a section of the pullback bundle u~lTM' of the tangent bundle of M' by u, that is, a C 1 vector field on M' along u, and is called the tension field of u. Indeed, if we regard the differential du of u as a section of the bundle T'M ® u~xTM'', T'M being the cotangent bundle of M, then in terms of the canonical connection V on the bundle T'M ® u~lTM' induced from the Levi-Civita connections of M and M', T(U) is defined as the divergence of du, that is, T(U) = Divdu = Trace Vdu. Since the harmonic map equation (1.1) is a system of semilinear elliptic partial differential equations of second order, any C 2 harmonic map u is in fact a smooth map. Now suppose that M is a Hadamard m-manifold, that is, a complete, simply connected Riemannian m-manifold of nonpositive curvature. It is well-known by the Cartan-Hadamard theorem that M is then diffeomorphic to the Euclidean m-space R m . Moreover, we can define a point at infinity of M to be an equivalence class of asymptotic geodesic rays in M. Here two unit speed geodesic rays 7 and a in M are said to be asymptotic if there exists a positive constant C such that the distance d(-y(t),a(t)) of ^(t) and a(t) in M satisfies d(-y(t), cr(t)) < C for all t > 0. Let M(oo) denote the set of points at infinity of M, and set M = MUM(oo). Then it is known that M(oo) is homeomorphic to the (m— l)-sphere S™ - 1 and M to the closed m-ball Bm, relative to a natural topology, called the cone topology, of M which induces on M the original topology of M. We call M the geometric compactification or the Eberlein-O'Neill compactification of M and M(oo) the ideal boundary or the sphere at infinity of M. See [EO] for details. Let M be a Hadamard m-manifold with the ideal boundary M(00) and M' a Hadamard m'-manifold with the ideal boundary M'(oo), respectively. Given a continuous map / : M(oo) —> M'(oo), the Dirichlet problem at infinity for harmonic maps then consists of finding a harmonic map u : M —> M' that assumes the boundary value / continuously. This is a fundamental important problem in geometric analysis, which seems to be quite hard in general. To see the nature of the problem, let us first look at the most typical case. Suppose m = m' = 2, and let M and M' both be the real hyperbolic 2-space Rff 2 , defined to be the open unit disk equipped with the Poincare metric gp of constant negative curvature —1, that is,

RH2=({zeC\\z\
9P

=.

4|dz|2

U-kl2

In this model, two geodesic rays are asymptotic if and only if they meet the boundary circle Sl in the same point, and hence M(oo) and M'(oo) can be identified with S 1 . Recall that the real hyperbolic 2-space RH2 is also represented as the upper half plane model H ^ with the Poincare metric gn, that is,

202 and these two models are isometric under the Cayley transform $ : H^ —> RH2 given by

(i.2)

* w = v crip^ >

zen\.

z + V-l Noting that the Cayley transform $ is well-defined even on the real axis {z G C | y = 0}, we may regard $ as defining a coordinate neighborhood of RH2 around the ideal boundary S1. Consequently, taking a family of these Cayley transforms as a system of coordinate neighborhoods at the ideal boundary, the geometric compactification RH2 = RH2 U S1 of the real hyperbolic 2-space RH2 admits a structure of smooth manifold with boundary, relative to which it is diffeomorphic to {z S C | \z\ < 1}. Then the Poincare metric gH of the upper half plane model H^ yields a local coordinate representation of the Poincare metric gp of RH2 with respect to a coordinate neighborhood given by the Cayley transform $ around S 1 , that is,

It is easily observed from (1.3) that the Poincare metric gp blows up at the ideal boundary S1 as \z\ —> 0. As a result, the corresponding Laplace-Beltrami operator is an operator that is degenerate everywhere on the ideal boundary S 1 . Therefore the Dirichlet problem at infinity in this case is a degenerate boundary value problem of the harmonic map equation (1.1) on the geometric compactification KH2, which is a smooth manifold with boundary, of RH2. Regarding the Dirichlet problem at infinity for harmonic maps between real hyperbolic spaces, Li-Tam [LT1, LT2, LT3], Gromov [G] and Akutagawa [Akl] (in the two-dimensional case) proved around 1990 that any C1 map / between ideal boundaries with positive energy density e(f) > 0 admits a harmonic extension u. More precisely, identifying the ideal boundaries of RH™ and RH™' with Sm~l and S"*'-1, respectively, the following theorem was proved. Theorem 1.1 ([LT3], [G]). Let f : S™_1 —> 5 m ' _ 1 be a C 1 map with positive energy density e(f) > 0 relative to the canonical metrics of 5 m _ 1 and 5 m _ 1 . Then there exists a unique proper harmonic map u : RHm —> RHm which has f as the boundary map and is C1 up to the ideal boundary. It should be remarked that for proper harmonic maps u : RHm —> RHm which are not C up to the ideal boundary, we do not in general have the uniqueness of solutions for the Dirichlet problem at infinity. Indeed, Li-Tam [LT2] constructed a two-parameter family of harmonic diffeomorphisms of RH2 that are only 1/2 Holder continuous at a boundary point and smooth everywhere else, and whose boundary value is the identity map of S1. This example was further generalized by Economakis [E] to higher dimensions. For the background of these results and related topics, see Akutagawa [Ak2]. Besides real hyperbolic spaces, familiar examples of Hadamard manifolds are the other rank one symmetric spaces of noncompact type, that is, complex or quaternion hyperbolic spaces and the Cayley hyperbolic plane. In 1993 Donnelly [Dol] studied the Dirichlet prob­ lem at infinity for harmonic maps in the context of the rank one symmetric spaces of noncompact type (see Section 2). Subsequently, it has been proved by Ueno [U] that these can be further extended to the case of Damek-Ricci spaces, which are a generalization of rank one symmetric spaces of noncompact type ([BTV]). 1

203 With these understood, our primary object of this paper is to study the Dirichlet problem at infinity for harmonic maps from a broader point of view.

2.

H O M O G E N E O U S MANIFOLDS O F NEGATIVE CURVATURE

In order to study the Dirichlet problem at infinity in more general context, we begin with a brief review of the geometry of homogeneous Hadamard manifolds. Let M = (M,g) be a Hadamard m-manifold with the ideal boundary M(oo). Suppose now that M is homogeneous, that is, the isometry group I\M, g) of M acts transitively on M. Then, by a result of Wolf [Wo] and Heintze [H], it is known that there is a solvable subgroup G of I0(M,g), the identity component of I(M,g), that acts simply transitively on M (see also [Al, AW]). Therefore we can identify M with a solvable Lie group G and the Riemannian metric g of M with a left invariant metric ( , ) on G. Consequently, any homogeneous Hadamard manifold M can be represented as a simply connected solvable Lie group G endowed with a left invariant metric ( , ) of nonpositive curvature. For instance, if M is a rank one symmetric space of noncompact type, then I0(M,g) has the Iwasawa decomposition N ■ A ■ K, where A?" is a nilpotent subgroup, A is a abelian subgroup and K is a maximal compact subgroup, respectively. Since K coincides with the isotropy subgroup of a point p e M by a theorem of Cartan ([KN]), by setting G = TV ■ A, the semidirect product of TV with A, we obtain a solvable Lie group G which acts simply transitively on M. If we further assume that M is of strictly negative curvature, then it follows that G is a one-dimensional solvable extension of a nilpotent Lie group TV. In fact, let g denote the Lie algebra of G and n = [g, g] be its derived algebra. Then we know, since g is solvable, that n is a nilpotent subalgebra of g. Moreover, it follows from the curvature condition that the orthogonal complement n x of n must be one-dimensional, that is, tv 1 = R { # } with a choice of a generator H (see Heintze [H] for details). Hence g decomposes as g = n + R{H}. Since G is a simply connected solvable Lie group, this implies that G is a semidirect product J V x R of a nilpotent Lie group N with the abelian group R, where AT = [G, G] is the commutator subgroup of G, which is a simply connected subgroup with Lie algebra n. As a result, G is diffeomorphic to the product manifold TV x R of TV with the real line R, and hence, identifying R with the positive half line R + = { j / 6 R | j / > 0 } b y R 9 s i - > j / = e s € R + , we obtain a diffeomorphism (2.1)

$ : N x R + 3 (n,y) i-> n - l o g y e G = TV x R,

identifying G with the half space N x R + , the product manifold of TV with R + . Consequently, a homogeneous Hadamard manifold M of strictly negative curvature can be represented as the half space TV x R + of a simply connected nilpotent Lie group TV with the positive half line R + . Moreover we may regard the diffeomorphism $ as a generalized Cayley transform. Indeed, if M is the real hyperbolic 2-space HH2, then TV is isomorphic to the abelian group R so that the product manifold Ar x R + is identified with the upper half plane model H ^ of RH2, and the diffeomorphism $ in (2.1), composed with the diffeomorphism given by the simply transitive action of G on R i ? 2 , is identical with the Cayley transform * : H% -> M 2 in (1.2). Summing up these observations, we obtain

204 Proposition 2.1. (1) Each homogeneous Hadamard manifold M can be represented as a simply connected solvable Lie group G endowed with a left invariant metric (, ) of nonpositive curvature. (2) // M is, in particular, of strictly negative curvature, then the solvable Lie group G is a semidirect product N x R of a nilpotent subgroup N with the abelian group R. Moreover, M is diffeomorphic to the half space N x R + , the product manifold of N with the positive half line R + , under a generalized Cayley transform $ : N x R + -> G given by (2.1). If M is a rank one symmetric space of noncompact type, then we see that the nilpotent group N in Proposition 2.1 is very restricted. In fact, we have the following result. Example 2.1. Let M be a rank one symmetric space of noncompact type, or equivalently a connected symmetric Riemannian manifold of strictly negative curvature, and identify M with a solvable Lie group G with left invariant metric. Recall that the curvature tensor R of M is parallel with respect to the Levi-Civita connection V of M, that is, VR = 0. Then it follows from this curvature condition that the nilpotent subgroup N described in Proposition 2.1 must be at most a 2-step nilpotent group. Namely, the Lie algebra n of TV satisfies [n,[n,n]] = {0}. Consequently, if we set n2 = [n, n] and n! to be the orthogonal complement of n2 in n, then the Lie algebra fl of G is decomposed as 0 = nt + n2 + R{H}, and n = rt! + ri2 becomes a graded Lie algebra. Indeed, letting n< = {0} for i > 3, we have [tij.iv,-] C tii+j,

l
Moreover, with a suitable choice of the generator ff of n 1 = R{i7}, we see that ni and n2 are the eigenspaces of the adjoint representation ad H on n with eigenvalues 1 and 2, respectively: nt = {X e n | adH(X) = iX},

t = 1,2.

For details, we refer the reader to Heintze [H]. Remark 2.1. (1) In Example 2.1, each rank one symmetric space of noncompact type is distinguished by the dimension of n2, which is equal to 0,1,3 or 7 corresponding to real hyperbolic spaces RiJ™, complex hyperbolic spaces CHm, quaternion hyperbolic spaces HHm, or the Cayley hyperbolic plane C&H2, respectively. (2) In this context, real hyperbolic spaces H.Hm are exceptional in the sense that n is abelian, that is, 1-step nilpotent so that n = tii and tv2 = {0}. (3) It is proved by Kobayashi [K] that any connected homogeneous Riemannian manifold of strictly negative curvature is simply connected. 3. fc-STEP CARNOT SPACES

Motivated by the observations in Section 2, we now consider a more general class of homogeneous Riemannian manifolds of negative curvature that arise as a one-dimensional solvable extension of certain fe-step nilpotent Lie groups, called Carnot groups ([P]). More precisely, let G be a simply connected solvable Lie group satisfying the following conditions:

205 (1) G is a semidirect product of a nilpotent Lie group N and the abelian group R. (2) If n and g = n + R{H} denote the Lie algebras of TV and G, respectively, then n has a decomposition k

intofc-subspacesgiven by (3.1)

rH = { X e n | adH(X) = iX},

i=

l,...,k.

It is easy to see that, since the adjoint representation ad H is a derivation of n, the above decomposition of n defines a graded Lie algebra structure of n, that is, [tij, a,] C n<+.,-,

1 < i, j < k

with the convention n^ = {0} for i > k. In particular, rt is a fc-step nilpotent Lie algebra, that is, the fc-th derived algebra of n vanishes: [n,---[n,[n,n]]-■•] = {<)}. Moreover, since the eigenvalues of ad H are all positive, we see immediately from a theorem of Heintze [H] that G admits a left invariant metric g of strictly negative curvature. As a consequence, we obtain a homogeneous Riemannian manifold M = (G, g) of strictly negative curvature. With these understood, we give the following definition. Definition 3.1. A homogeneous Riemannian manifold M = (G,g) of negative curvature obtained as above is called a k-step Carnot space. For example, real hyperbolic spaces are 1-step Carnot spaces, and complex or quaternion hyperbolic spaces and the Cayley hyperbolic plane are 2-step Carnot spaces. Also, it should be noted from Example 2.1 that fc-step Carnot spaces are not symmetric if fc > 2. Example 3.1 (fc-step Carnot space with fc > 2). Let fc > 2 and SL(k + 1,R) be the special linear group of degree fc + 1 with Lie algebra sl(fc + 1,R). Denote by Eij the matrix unit having 1 for the (i,j) element and 0 elsewhere. Define n = SpanR{i?y | i < j} = {upper triangular matrices of degreefc+ 1}, (k '

\ fc-2

H=l

esl(fc + l,R), -(fc - 2)

-kj

V

and set g = n + R{ff}, which is a solvable Lie subalgebra of sl(fc + 1,R). It is easy to see that n = [g, g], and [H,Eij] = {j-i)Eij,

l
Therefore we have the graded Lie algebra decomposition of n as k

g = 2^r4 + R{ff}, «=i

nt = {X en\

adH(X) = iX}.

206 Let G be a simply connected Lie subgroup of SL(k +1, R) with Lie algebra 0. Then G gives rise to a fc-step Carnot space M = (G,g). Now, let M = (G, g) be a fe-step Carnot space, where G is a simply connected solvable Lie group which is a semidirect product TV x R of a nilpotent Lie group TV with R, and g is a left invariant metric on G of strictly negative curvature. Then, as seen in Proposition 2.1, M is represented as the product manifold TV x R + of TV with the positive half line R + via a generalized Cayley transform $ : TV x R + —> G given by (3.2)

$ : TV x R+ 9 (n, y) i-+ n ■ exp sH G G = TV x R,

where s = log y and H is the generator of the Lie algebra of R. This half space model TV x R + of M is convenient in studying the geometry of M at infinity. In fact, since M is a Hadamard manifold, we have the ideal boundary M(oo) and the geometric compactification M = M U M(oo) of M, respectively. Then, regarding the Cayley transform $ as defining a coordinate neighborhood near the ideal boundary M(oo), we have the following proposition which describes the behavior of g near M(po). Proposition 3.1. Under a generalized Cayley transform $ : TV x R + —> G, the metric g is expressed on the half space model TV x R + of M as a k-ply warped product metric 1 3 3

(-)

$

S = ^n1

1 1 dy2 + ^Sn 2 + "-- + ^ f f „ t + -^-,

where y denotes the coordinate on R + , and gni + g„2 + ■ —I- g„k is a left invariant metric on TV. In fact, let <j>: TV x R —> G be a diffeomorphism defined by (j>: TV x R 9 (n, s) i-> n ■ exp sH € G = TV x R. Then, by a direct calculation, it is easy to see that the pullback metric *g on TV x R is given by 4>'g = <(expad(-stf ))(•), (expad(-sff))(-))« + ds2, where ( , )„ is a left invariant metric on TV. On the other hand, it follows from (3.1) that on each subspace tij of n we have (expad(-sff))(X) = e- i s -X,

X £ n^.

Hence we obtain n, + e~4s( , ) „ + - • + e-™°( , ) n t + ds\ where ( , ) ni denotes the metric on each tij induced from ( , ) n . Consequently, by setting s = log?/, we see that the pullback metric $*g on TV x R + under a generalized Cayley transform $ : TV x R + —> G is written as 1

1 I dy2 r jr jr* y2 When M is, for instance, the complex hyperbolic m-space CHm, these observations can be carried out explicitly as follows. Since the argument is parallel in any dimension, let us look at the case m = 2 for simplicity.

207 Example 3.2. Let CH2 be the complex hyperbolic 2-space, defined to be the open unit ball B = {z = (zl,z2) e C 2 | \z\ < 1} in the complex Euclidean 2-space C 2 , equipped with the Bergman metric gs of constant holomorphic sectional curvature —4 defined by

9B=f

_*w±zm*&

.4-;

dz*dzi

',3=1 1

__i__

fl^f

" .rl|2

1^,2 _ ^ ^ 1 _ U^ j2 „| 22| 2)\

, |J,2|2 _ | , 2 j „ l _

+

Then it is well-known ([CG]) that 5(7(1,2), the special unitary group of signature (1,2), acts transitively on B as isometries, which are linear fractional transformations on B preserving gB. The isotropy subgroup K at o = (0,0) G B is isomorphic to (7(2) and is a maximal compact subgroup in the Iwasawa decomposition of 5(7(1,2). Indeed, this decomposition is given as 5(7(1,2) = N■ A-K, where TV is a 2-step nilpotent subgroup, called the Heisenberg group, and A is a one-dimensional abelian subgroup. If we define G = TV x A, which is a solvable subgroup of 5(7(1,2), then G acts simply transitively on B as isometries. Consequently, CH2 is identified with a solvable Lie group G with a left invariant metric ( ,

>■

Let g = n + o be the Lie algebras of G = TV x A. Then, as in Example 2.1, g is decomposed into the orthogonal direct sum g = n 1 + n 2 + R{ff}, where H stands for a generator of o with (H, H) = 1, and ni and n2 are the eigenspaces of the adjoint representation ad-ff with eigenvalues 1 and 2, respectively. Explicitly, they are determined as follows: m = { ull + vV

n2 = ItT Note that the only nontrivial bracket relation of the Lie algebra n of TV, the Heisenberg Lie algebra, is [U, V] = —2T. Moreover, a direct calculation shows that, setting z = u + y/—Tv, the Heisenberg group TV can be identified with the vector space C x R with the group multiplication given by (zi, ft) • (z2, t2) = (zi + z2, h + t2 + Im(zizi)). Now, let $ : TV x R + —> G = TV x R be the generalized Cayley transform given by (3.2), and t : G = T V x R 9 a = exp(uU + vV + tT) ■ exp(sH) H-> a-oe B be the diffeomorphism defined by the action of G on B, where a acts as a linear fractional transformation. Then, by identifying the half space model TV x R + of CH2 with the upper half space H^_ = C x R x R + and $ with t o $, it is verified by a straightforward calculation that the Cayley transform * : H^. ->• CH2

208 from H+ to the complex hyperbolic 2-space CH2 = (B,gB) is given by

where (z,t,y) £ H^_ and y = es. Consequently, the pullback metric $*
1

$* 5iJ = -Adz\2 + - j IA + >/=T(*dz - zdz)/2\ yl

yi\

o

A,2

+ AT, yl

where dz and dt + */^l(zdz — ~zdz)/2 are left invariant 1-forms on TV, which corresponds to the expression (3.3) with k = 2. In a half space model TV x R + of a ft-step Carnot space M, the ideal boundary M(oo) of M is easily seen to be identified with the one-point compactification of the nilpotent Lie group TV. Namely, we have the following Proposition 3.2. In a half space model TV x R + of a k-step Carnot space M = (G,g), R + directions (n = const, y) define asymptotic geodesic rays of M and hence give rise to a point at infinity oo € M(oo). Moreover, M(oo) \ {oo} is naturally identified with N x {0}. Indeed, recall the basic formula for the Levi-Civita connection V of M: 2g(VxY, Z) = Xg(Y, Z) + Yg(X, Z) - Zg(X, Y) - g(X, [Y, Z\) - g(Y, [X, Z]) + g(Z, [X, Y]). Then it is immediate from (3.3) that the unit vector field X = y(d/dy) along R + directions satisfies VxX = 0 a r , d hence defines asymptotic geodesic rays. On the other hand, the second claim is a consequence of the fact that, since M is of strictly negative curvature, any two distinct points in Af(oo) can be joined by a unique geodesic of M (see [EO] for details). Since asymptotic classes of geodesic rays are preserved under isometries, the isometry group I(G,g) of a k-step Carnot space M = (G,g) acts also on its ideal boundary M(co). Indeed, each isometry of M is well-defined on M(oo) and becomes a homeomorphism of M = M U M(oo) relative to the cone topology. Concerning this extended action of I(G,g), we have the following proposition obtained by a combination of results due to Chen [C] and Druetta [Dr]. Proposition 3.3. Let M = (G, g) be a k-step Carnot space. Then the following hold. (1) // M is symmetric, that is, a rank one symmetric space of noncompact type, then I{G,g) has no common fixed point in M(oo). (2) If M is not symmetric, then I(G,g) has a unique common fixed point 7(00) G M(oo), and for any p / 7(00) in M(oo), under the left translations by the nilpotent subgroup N of G, the orbit N(p) of p coincides with M(oo) \ {7(00)}. As a consequence, we see that if M is a rank one symmetric space of noncompact type, then half space models TV x R + of M yield, via generalized Cayley transforms $ : TV x R + —> G, local coordinate neighborhoods at the ideal boundary M(oo), and by taking these as a system of coordinate neighborhoods at the boundary, the geometric compactification M = M U M(00) admits a structure of a smooth manifold with boundary. However, if M is not symmetric, then we have a single half space model N x R + of M and a unique generalized Cayley transform $ : TV x R + —> G, under which the point at infinity 00 e M(oo) determined

209 by the R+ directions in TVxR+ corresponds to the unique common fixed point 7(00) e M(00) of the action of I(G, g). In a half space model TV x R+ of a fc-step Carnot space M, each subspace r^ of the graded Lie algebra decomposition n = X^,=i n t °f t n e Lie algebra n of TV defines, under the left translations of TV, a distribution on TV, that is, a subbundle of the tangent bundle of TV. Indeed, each t^, being identified with a subspace of the tangent space of TV at the identity e 6 TV, defines by left translations of TV a subspace (n^p of the tangent space of TV at each p e TV. We denote this distribution also by tij, which may be regarded, under the identification of TV x {0} with M(oo) \ {00} as in Proposition 3.2, as defining a geometric structure on the ideal boundary M(oo) of M. Example 3.3. Let M be the complex hyperbolic m-space CHm, defined to be the open unit ball B2m = {z e C™ I \z\ < 1} in the complex Euclidean m-space C m equipped with the Bergman metric gs- Let TV x R + be a half space model of CHm obtained as in Example 3.2, where TV is the Heisenberg nilpotent subgroup of SU(l,m). Then the graded Lie algebra decomposition n = nj + n2 of the Lie algebra n of TV defines in a natural way the canonical contact structure on the boundary sphere 5 2 m _ 1 , which is identified with the ideal boundary M(oo) ofCHm. In fact, if we identify 5 2 m _ 1 with the one-point compactification of the Heisenberg group TV as in Proposition 3.2, and consider the Hopf fibration S2m~l -> C P m _ 1 of S*2m_1 over the complex projective (m — l)-space C P m _ 1 , then, under the left translations by TV, the distribution n2 corresponds to the vertical subspaces along the fibre and nx to the distribution of horizontal subspaces defined by the canonical contact structure of S 2m_1 . Remark 3.1. It should be remarked that in the definition of a generalized Cayley transform $ : TV x R + -> G given in (2.1) and (3.2), we identify the real line R with the positive half line R + by the diffeomorphism R 3 s i-t j = e1 e R + . However, for a given fe-step Carnot space M = (G, g), the choice of a diffeomorphism between R and R + is not unique, and in general it is not clear a priori which choice should be canonical. In fact, if we choose the diffeomorphism R 3 s i-> y = e2s € R + in (3.2), then the pullback metric $*p in Proposition 3.1 takes the form 1

1 ! dy2 y y y V which implies that the metric g blows up at the ideal boundary M(oo) of M with different orders in y from those in (3.3). 4. HARMONIC MAPS BETWEEN /C-STEP CARNOT SPACES

Let M = {G,g) and M' = (G',g') be both A;-step Carnot spaces with k > 2. By definition, G and G' are simply connected solvable Lie groups that are semidirect products G = TV » R and 67' = TV' x R of fc-step nilpotent Lie groups TV and TV' with R, and g and g' are left invariant metrics of strictly negative curvature on G and 67', respectively. Recall that the Lie algebras n and n' of TV and TV' have graded Lie algebra decompositions n = 5Zi=1 r^ and n ' = Z)*=i n';> respectively. Now, let u : M —> M' be a proper C°° map from M to M'. Here the properness of u means that if {pj} is a sequence of points in M diverging to the ideal boundary M(oo) of M, then {u(pj)} is also diverging in M' to the ideal boundary M'(oo) of M'. We are interested in the case when u extends to a map u : M -> M' from the geometric compactification

210

M = MU M(oo) of M to the geometric compactification M' = M' U JW'(oo) of M'. Our primary concern is to investigate the asymptotic behavior of the boundary value / = u|M(oo) : M(oo) ->■ JW'(oo) of w at the boundary M(oo), when u is a harmonic map in the interior M. To see the nature of the problem, let us first look at the simplest case with k = 2. Thus, for example, let M and M' be the complex hyperbolic m-space CHm and the complex hyperbolic m'-space CHm', respectively. Let u : CHm —)• CHm be a proper harmonic map from CHm to CHm'. We suppose that u extends to a C 1 map up to the boundary, and plan to investigate necessary conditions satisfied by the differential of the boundary value / = ulS2™-1 : S ,2m - 1 -+ S2m'~l of u, where S2™"1 and S2™'-1 are identified with the ideal boundaries of CHm and CHm , respectively. To this end, let JV x R + and JV' x R + be half space models of CHm and CHm' as in Example 3.3, and consider the corresponding generalized Cayley transforms as defining coordinate neighborhoods at the boundaries S 2 " 1-1 and S2m'~l, respectively. Since JV and JV' are Heisenberg 2-step nilpotent groups, their Lie algebras decompose as n = ni + it2 and n' = n'i +n' 2 , respectively. Here n2 and n'2 are one-dimensional, and we have [rti, rii] = n2 and [n'i,n'i] = n'2. Let y be the coordinate on the second factor of JV x R + , and set e0 = d/dy. Choose an orthonormal basis {ei,... , e2 m - 2 } for tii and {e 2m -i} for n2, relative to the left invariant metric on JV (see Proposition 3.1), and extend them to left invariant vector fields on JV. We choose a frame field {eo,..., e 2m -i} on JV x R+ to consist of these vector fields, which are orthogonal but not orthonormal for the metric g realized as a doubly warped product metric as (3.3) with k = 2. Similarly, on JV' x R + , we set e'0 — d/dy' and choose a frame field {e'0,... ,e?2mi_x}, where we denote the corresponding quantities on JV' x R + with a prime, for instance, {e[,..., e2m,_2} is an orthonormal basis of n[ relative to the left invariant metric on JV'. With respect to these frame fields, the differential du of u and df of / are written respectively as 2m-12m'-l

2m-12m'-l

e

du = J2 E < **® «' * = E E f?e>'®<> i=0

a=0

e

i=l

oc=l

where {e^} denotes the dual coframe field of {ei}. With these conventions understood, the following theorems were proved by Donnelly. Theorem 4.1 ([Dol]). Suppose that u : CHm —► CHm is a proper harmonic map which extends to a C 1 map up to the ideal boundary. Let f : S2™-1 —»• S2m ~1 be the boundary value ofu. Then f2m'~l = 0 for 1 < j < 1m - 2. Theorem 4.2 ([Dol]). Let u and v be proper harmonic maps from CHm to CHm' which extend to C2 maps up to the ideal boundary. Suppose that u and v have the same boundary value f : S2"*"1 -> S2™'-1 and f ^ 1 ^ 0 everywh ere. Then u and v are identical. It should be noted that when u is a proper harmonic self-map of the complex hyperbolic m-space CHm having a C 1 extension up to the boundary, then the condition in Theorem 4.1 means that the boundary value / must preserve the contact structure on the boundary 5 2 m _ 1 (see Example 3.3). This forms a striking contrast to Theorem 1.1, which shows that for real hyperbolic spaces, any / with nonvanishing energy density can occur as the boundary value of a proper harmonic map having a C 1 extension up to the boundary. Also, compare the

211 condition in Theorem 4.2, which assures the uniqueness of solutions of the Dirichlet problem at infinity for harmonic maps, with that in Theorem 1.1. Now, let us turn to the case of general fc-step Carnot spaces. For given &-step Carnot spaces M = (G, g) and M' = ( C , 1, and the boundary value / of u is a map from V n (N x {0}) to N' x {0}. Namely, we suppose that

u G C°°{V n M, M') n c'(v n M, W),

f e c'(V n (N x {o», N' X {O}).

Then our first observation is that the following result corresponding to Theorem 4.1 above holds for a harmonic map between A:-step Carnot spaces. Theorem 4.3. Let M and M' be k-step Carnot spaces, k > 2, and V a neighborhood of p G N x {0}. Suppose that u G C°°(V n M, M') n C ' t V f l ~M, M7) is a harmonic map with boundary value f G Cl(V n (N x {0}), N' x {0}). Then it satisfies (4-1)

4f,((ni),)c(ni)/h)

ot onyg G Vn(iV x {0}). It should be noted that when M and M' are complex hyperbolic spaces CHm and CHm respectively, the condition (4.1) is equivalent to assuming /? m ' _1 (g) = 0 for 1 < j < 2m - 2 at q G V n (iV x {0}), which is the conclusion in Theorem 4.1. We now give the following definition related to the nondegeneracy of boundary values, which plays an essential role in our discussion below. Definition 4.1. Let M and M' be k-step Carnot spaces, fcj^ 1, and V a neighborhood of p € N x {0}. Suppose that u £ C00(V n M, M') n Cl(V n M,W) and the boundary value / e C ' f V n j J V x {0}), W x {0}). We say that / is nondegenerate if it satisfies t-i

(4-2)

4f,((n»),)^X)(ni)/(«)

at any g G V n (TV x {0}). If fc = 1, then with the convention n0 = n(, = {0}, the condition (4.2) simply means that dfq ^ 0 at q G V n (TV x {0}). Note that when M and M' are real hyperbolic spaces, this is equivalent to the condition e(f)(q) > 0 assumed in Theorem 1.1. Moreover, if k = 2, then the condition (4.2) means that dfq{(n2)g) £ K ) / ( , ) at g G V fl (TV x {0}), which is equivalent to the condition fl^-iil) / 0 assumed in Theorem 4.2, when M and M' are complex hyperbolic spaces CHm and Cffm , respectively. The nondegeneracy condition of Definition 4.1 together with a sufficient regularity up to the boundary then implies the following necessary conditions for the boundary value of a harmonic map between k-step Carnot spaces.

212 T h e o r e m 4.4. Let M and M' be k-step Carnot spaces, k > 2, arid V a neighborhood of p G N x {0}. Suppose that u G C°°(V n M, M') n Ck{V n ~M,W) is a harmonic map with nondegenerate boundary value f G Ck(V n (TV x {0}), TV' x {0}). Then f satisfies for each 1 < i
(4-3)

dfq ( j > ) , ) C J2W)M

at any q e V n (TV x {0}). Theorem 4.4 claims that nondegenerate boundary values of proper harmonic maps between k-step Carnot spaces, having Ck regularity up to the boundary, must preserve the filtrations on the boundaries ni C ni + n 2 C • • ■ ,

ni c n[ + n!, C • • • ,

where n* and n't are the distributions denned by the graded Lie algebra decompositions n = X)i=i •*> a n d tv' = J2i=i n ii respectively. The proof of the condition (4.3) is done by straightforward but long computations with an induction argument. We refer the reader for the details to [NU2]. In the course of the proof of Theorem 4.4, we also see the following. Recall first that on the half space models TV x R + and TV' x R + of M and M', the metrics g and g' are expressed as fc-ply warped product metrics 1

dy2

1

9 = ^ + - - - + ^9nk

+ -JJJ-,

,

1 ,

1

dy'2

,

g = ^<7„< + • • • + ^gK

+ -^r.

respectively (see Proposition 3.1). As in the case of complex hyperbolic spaces, for local expressions and computations we define frame fields { e j on N x R + and {e'a} on N' x R + as follows: Let e0 = d/dy and e'0 = d/dy'. Set UA = dimn^ and n'p = dimn^, where 1 < A, P < k. For each n^, choose an orthonormal basis { e ^ } , 1 < i < UA, relative to the left invariant metric gnA, and extend them to left invariant vector fields on Ar. Similarly, for each n' p , choose an orthonormal basis {e'Pa}, 1 < a < n'P, relative to the left invariant metric g'a, , and extend them to left invariant vector fields on TV'. With respect to these frame fields, the differential du of u and the tension field r{u) of u are written respectively as m—lm'—l

du

rn'—l

Ue = »=0 YU2 i '* ® e«> T(") = aY^ ^("K. a=0 =0

where m = dim M = 1 + J2A=I nA and m! = dim M' = 1 + Y%=\ n'p> dual coframe field of {ej}. With these understood, we have the following

an

d {e«*} denotes the

L e m m a 4.1. Let u and f be as in Theorem 4.4. Then it holds that k

\

k

tip

n'p

( E4.nJw) -E^EEW:) Kr a

A=\

on Vn(N

x {0}).

/

2

P=l

«=1 a = l

p)

=o

213 This result is derived from the e'0 component r°(u) of the tension field of u, which vanishes on V by assumption that u is harmonic. Indeed, if we multiply r°(«) by (y' o u)2k~l ■ y'2k and let y —> 0, then the Ck regularity of u up to the boundary and the nondegeneracy of the boundary value / deduce the condition (4.4). Note here that at the boundary the quantities Up' for 1 < i < nP and 1 < a < n'p are determined by the boundary value / of u. Hence, denoting Up" by fp", we obtain from (4.4) the following relation

(E ^ k

A=l

\

/

k—\np

n'p

nk

n'k

Kr-E^EE(^)aWp)-^EE(/^)2 = 0' P=\

t=l a=l

i=l o=l

as a necessary condition satisfied by / on V n (N x {0}). Since nt

n

'k

2 EE(a >o by the nondegeneracy condition (4.2), the equation (4.5) in «§ has a unique positive solution •=1 a = l

UQ > 0. Namely, for acondition given nondegenerate Ck boundary we can positive reconstruct from by the nondegeneracy (4.2), the equation (4.5) in value «§ has/ a, unique solution the equation (4.5) for the aeo*®e[, component u§ of differential of u/ , atwethecan boundary. In fact, UQ > 0. Namely, given nondegenerate Ckthe boundary value reconstruct from moreequation elaborate work us to the following the (4.5) theleads eo*®e[, component u§ ofobservation. the differential of u at the boundary. In fact, more elaborate work leads us to the following observation. Proposition 4.1. Let M and M' be k-step Carnot spaces, k > 2, and V a neighborhood ofp^Nx {0}. Suppose that u G C°°{V (~l M,M') n Ck{V n M, W) is a harmonic map with nondegenerate boundary value f e Ck(V n [N x {0}), N' x {0}). Then the asymptotic behavior ofu and its higher derivatives in R+ directions at the boundary M(oo) is determined by the boundary value f. As an application of these necessary conditions, we can show, for instance, the following uniqueness result for proper harmonic maps between fc-step Carnot spaces. Theorem 4.5. Let M and M' be k-step Carnot spaces, k > 2, and oo e M(oo) and oo' € M'(oo). Letu,v e C°°(M, M')nCk(M\{oo}, M'\{oo'}) be proper harmonic maps having a nondegenerate boundary value f G Ck(N x {0}, N' x {0}). Assume that dM'(u(p),v(p)) —► 0 as p —> co, where dM'(u(p),v(j>)) is the distance between u(p) and v(p) in M'. Then u and v are identical. As in the case of real hyperbolic spaces remarked in Section 1, for proper harmonic maps between fc-step Carnot spaces which do not have sufficient regularity up to the boundary, the uniqueness of solutions for the Dirichlet problem at infinity does not hold in general. In fact, for arbitrary fc-step Carnot space M, Watabe [Wa] has recently constructed a one-parameter family of harmonic diffeomorphisms of M which induce the identity map on the boundary Af(oo). Examining the growth order of the harmonic diffeomorphisms constructed in [Wa], it is conjectured that Theorem 4.5 holds under a weaker assumption that * dM,(u(p),v(p)) = O (exp(<5 • r(p))), S = ^A-nA, A=l

where r(p) denotes the distance of p in M' from some reference point. As another application of Proposition 4.1, we can also see the following. Let M = Cffm and M1 = CHm be the complex hyperbolic m-space and the complex hyperbolic m'-space

214 with m,m' > 2, respectively. Then it has been known ([F]) that if u : CHm —> CHm' is a proper holomorphic map, then u induces a smooth CR map / : g 2 m _ 1 —> S2m - 1 between the ideal boundaries M(oo) = S2m~l and M'(oo) = S 2 " 1 ' - 1 , that is, u extends smoothly up to the boundary and the boundary value f of u satisfies the tangential Cauchy-Riemann equation. Conversely, we can see the following T h e o r e m 4.6 ([NUl]). Let M = CHm and M' = CHm' be thecomplex hyperbolic spaces of dimension m, m! > 2, respectively. Let u € C°°{M, M') n C4(M, M') be a proper harmonic map. If the boundary value f : S2m~l —> S2m'~x of u is a nondegenerate CR map, then u is a holomorphic map. Remark 4.1. Recently, Donnelly [Do3] has constructed a number of proper maps of rank one symmetric spaces of noncompact type, whose boundary nondegenerate. Indeed, these harmonic maps extend continuously up to the assume the boundary values / which are smooth but violate the condition these / do not preserve the filtrations defined on the boundary.

harmonic selfvalues are not boundary and (4.1), that is,

REFERENCES

[Akl] K. Akutagawa, Harmonic diffeomorphisms of the hyperbolic plane, Trans. Amer. Math. Soc. 342 (1994), 325-342. [Ak2] K. Akutagawa, Harmonic maps between hyperbolic spaces, Sugaku Expositions 12 (1999), 151-165. [Al] D. V. AlekseevskiT, Homogeneous Riemannian spaces of negative curvature, Math. USSR Sbornik 25 (1975), 87-109. [AW] R. Azencott and E. N. Wilson, Homogeneous manifolds with negative curvature I, Trans. Amer. Math. Soc. 215 (1976), 323-362. [BTV] J. Berndt, F. Tricerri and L. Vanhecke, Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces, Lecture Notes in Math. Vol. 1598, Springer, 1995. [C] S.-S. Chen, Complete homogeneous Riemannian manifolds of negative curvature, Comm. Math. Helv. 50 (1975), 115-122. [CG] S. S. Chen and L. Greenberg, Hyperbolic spaces, Contribution to Analysis, edited by L. Ahlfors et al., 44-87, Academic Press, 1974. [Dol] H. Donnelly, Dirichlet problem at infinity for harmonic maps: Rank one symmetric spaces, Trans. Amer. Math. Soc. 344 (1994), 713-735. [Do2] H. Donnelly, Harmonic maps between rank one symmetric spaces - regularity at the ideal boundary, Houston J. Math. 22 (1996), 73-87. [Do3] H. Donnelly, Harmonic maps with noncontact boundary values, Proc. Amer. Math. Soc. 127 (1999), 1231-1241. [Dr] M. J. Druetta, Homogeneous Riemannian manifolds and the visibility axiom, Geom. Dedicata 17 (1985), 239-251. [EO] R Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45-109. [E] M. Economakis, A counterexample to uniqueness and regularity for harmonic maps between hyper­ bolic spaces, J. Geom. Anal. 3 (1993), 27-36. [F] C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1-65. [G] M. Gromov, Foliated Plateau problem, Part II: Harmonic maps of foliations, Geom. Funct. Anal. 1 (1991), 253-320. [H] E. Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974), 23-34. [K] S. Kobayashi, Homogeneous Riemannian manifolds of negative curvature, Tohoku Math. J. 14 (1962), 413-415. [KN] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. II, Interscience Publishers, New York, 1969. [LT1] P. Li and L.-F. Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math. 105 (1991), 1-46.

215 [LT2]

P. Li and L.-F. Tarn, Uniqueness and regularity of proper harmonic maps, Ann. of Math. 137 (1993), 167-201. [LT3] P. Li and L.-F. Tarn, Uniqueness and regularity of proper harmonic maps II, Indiana Univ. Math. J. 42 (1993), 591-635. [NUl] S. Nishikawa and K.Ueno, Complex analyticity of proper harmonic maps between complex hyperbolic spaces, preprint. [NU2] S. Nishikawa and K.Ueno, Harmonic maps between fc-step Carnot spaces, in preparation. [P] P. Pansu, Metriques de Carnot-Caratheodory et quasiisometries des espaces symetriques de rang un, Ann. of Math. 129 (1989), 1-60. [U] K. Ueno, Dirichlet problem for harmonic maps between Damek-Ricci spaces, Tohoku Math. J. 49 (1997), 565-575. [Wa] D. Watabe, Dirichlet problem at infinity for harmonic maps, Thesis, Tohoku University, 2000. [Wo] J. A. Wolf, Homogeneity and bounded isometries in manifolds of negative curvature, Illinois J. Math. 8 (1964), 14-18. MATHEMATICAL INSTITUTE, TOHOKU UNIVERSITY, SENDAI, 980-8578, JAPAN

E-mail

address:

nisikawafflmath.tohoku.ac.jp

Received: April 3, 2000

216

Geometry and Topology of Submanifolds X eds. W. H. Chen et al. (pp. 216-230) © 2000 World Scientific Publishing Co.

ON T H E REALIZATION OF CONNECTIONS AFFINE MINIMAL SURFACES AND AFFINE

ON

SPHERES

BARBARA OPOZDA ABSTRACT. In the paper the following problem is studied: Let V be a torsion-free connection on a manifold M 2 . Under which conditions satisfied by V it can be realized on an affine sphere or an affine minimal surface. Keywords: affine sphere, affine minimal surface, affine connection 1991 Mathematics Subject Classification: 53 A 15

0. I n t r o d u c t i o n . In the theory of submanifolds a vector field or a bundle transversal to a submanifold induces on it some geometric objects. The realization problem is, roughly speaking, a question whether an object prescribed on an abstract, that is, not necessarily immersed, manifold can be realized as an induced object on a submanifold of some standard homogeneous space. The question can be formulated in a general setting, but usually, some specific conditions are imposed. For instance, one can ask whether the submanifold can be of special type, e.g. minimal, or a prescribed object is of special type, e.g. a prescribed connection is locally symmetric. One of the realization problems in affine differential geometry is the following: Let V be a torsion-free connection on a n-dimensional manifold Mn. Does there exist an immersion f : Mn —* R n + 1 and a transversal vector bundle which induces V ? We shall study this question for connections on 2-dimensional manifolds and we shall ask about local realizations on affine spheres and affine minimal surfaces. The realization of metrics on affine spheres in R 3 was studied in [SW]. Some other results dealing with realizations of connections on hypersurfaces are proved in [ 0 ] i _ 4 . The author is grateful to the organizers of the meetings in Beijing and Berlin for their hospitality and support. 1. P r e l i m i n a r i e s . This section contains some basic information about affine structures on hypersurfaces in the standard affine space R " + 1 . Let / : Mn —> R " + 1 be a hypersurface in R n + \ that is, / is an immersion of a connected n-dimensional manifold Mn. For a while we assume that M is orientable. In the paper we shall only deal with local problems and the orientability is not needed. Denote by D the standard connection on R " + 1 . For The research was partially supported by the KBN grant 2 P03A 020 16 This paper is in final form and no version of it will be submitted for publication elsewhere.

217 a vector field £ transversal to / and any vector fields X, Y tangent to M the following formulas of Gauss and Weingarten hold (1.1)

Dxf.Y

(1.2)

= UVxY)

+ h(X,

Dxt=-f.(SX)

+

Y)£,

T(X)£.

In this way we define the objects induced by / and £: a torsion-free connection V, a symmetric bilinear form h, a (l,l)-tensor field S and a 1-form r on Mn. A transversal vector field £ is called equiaffine if r = 0. The objects h and S are called the corresponding second fundamental form and shape operator respectively. It is known that the rank of h is independent of a choice of f. In particular, if h is nondegenerate at each point of Mn, then / is called nondegenerate. We can also define a volume element 9 by (1.3)

6 ( X 1 ; . . . , Xn) = det(f,Xu

..., / , X „ , 0 -

It is known that if / is nondegenerate, then there is a unique (up to sign) choice of an equiaffine transversal vector field f such that the apolarity condition is satisfied, i.e. the volume element 0^, determined by the nondegenerate metric h is equal to G. The transversal vector field is called the affine normal. The connection induced by the affine normal is called the Blaschke connection, the induced S - the affine shape operator and h - the Blaschke metric. Of course, the affine shape operator and the Blaschke metric are determined up to sign. The objects V, h, S constitute the Blaschke structure on Mn. A hypersurface equipped with the Blaschke structure is called a Blaschke hyperface. The apolarity condition can be also expressed in the following way: (1.4)

trhVh(X,-,■)

= (}

or (1.5)

V0h=O

provided h and V are the Blaschke metric and connection. Let us also write the fundamental equations for a hypersurface endowed with an equiaffine transversal vector field: (1.6)

(1.7)

(1.8)

(1.9)

R{X, Y)Z = h(Y, Z)SX

- h{X, Z)SY,

h(SX, Y) = h(X, SY),

-

-

Gauss

Ricci

V/i(X, Y, Z) = V/i(y, X, Z), - Codazzi

VS(X,

Y) = VS{Y, X)

-

Codazzi

I

II

218 for any X, Y, Z G TxMn, x G M", where R is the curvature tensor of V. By virtue of the Gauss equation we obtain (1.10)

Ric{X, Y) = tr Sh{X, Y) - h(SX, Y),

where Ric denotes the Ricci tensor of V. The fundamental theorem in afnne differential geometry says that if V is a torsion-free connection, h a symmetric bilinear form and S a (1, l)-tensor field on a manifold M", then the objects can be locally (that is, in a neighbourhood of each point of M") induced by some transversal vector field on a hyper­ surface in R™+1 providing that V, h, S satisfy the fundamental equations. If moreover h is nondegenerate and V©/, = 0, then the objects can be realized on a Blaschke hypersurface. A Blaschke hypersurface is called an afnne sphere if S = Xid. On a connected M™ A is automatically constant. A Blaschke hypersurface is called afnne minimal if and only if tr S = 0. For more detailed exposition of the theory of afnne hypersurfaces we refer to [NS] or [LSZ]. In this paper we shall deal with 2-dimensional manifolds. For such manifolds there is a special version of the fundamental theorem. Namely, for a torsion-free connection V on a 2-dimensional manifold M2 we define n1(X,Y)

=

(S/xR)(X,Y)X,

n2{X,Y)

= (VXR)(X,Y)Y

K3(X,Y)

=

+

{VYR)(X,Y)X,

(VYR)(X,Y)Y,

where X, Y is a basis of TXM2 or X, Y is a local frame. Since R(X, Y)Z = Ric(Y, Z)X Ric(Y, Z)Y, we have fti (X, Y) = VRic{X, Y, X)X - VRic(X, (1 11)

X,X)Y,

n2{X Y)

= + Vffic y Y FRi<X'Y'F) ( ' ' X^X> - [VRic(X, X, Y) + VRic(Y, X, X)]Y, 1l3{X, Y) = VRic(Y, Y, Y)X - VRic(Y, X, Y)Y.

'

The fundamental theorem for 2-dimensional manifolds can be formulated in the follow­ ing way, see for instance [SI], [0]s. Theorem 1.1. Let V be a torsion-free connection on a 2-dimensional manifold M2 and h - a nondegenerate symmetric bilinear form on M2. The pair V, h can be locally realized on a Blaschke surface in R 3 as the Blaschke connection and the Blaschke metric if and only if the following conditions are satisfied (1.12) (1.13)

hiY^ll^XiY)

- h(X,Y)K2(X,Y)

ve h = o,

+ h(X,X)1Z3(X,Y)

= 0,

219 (1.14)

V/i is

symmetric.

Throughout this paper we shall use the following conventions. The word "smooth" will mean "of class C°°". If (u, v) a coordinate system on a manifold M2, then U, V will stand for du, dv. If V is a torsion-free connection on a manifold M 2 and (it, v) is a coordinate system, then the Christoffel symbols of V relative to (u, v) will be denoted by A,B,C,D,E,F. More precisely,

(1.15)

VVU = AU + BV,

\7VV = CU + DV,

VVV = EU + FV

For later convenience let us also write the formulas for the Ricci tensor on a 2-dimensional manifold. If V is given by (1.15), then we have

(1.16)

Ric{U, U) = BV-DU

+ D(A -D) + B(F - C),

Ric{U, V) = DV-FU

+ CD-

BE,

Ric(V, U) = CU- AV + CD-

BE,

Ric{V, V) = EU-CV

+ E{A -D) + C(F - C).

Recall also that the Ricci tensor of V on a manifold M is symmetric if and only if around each point of M there is a volume element parallel relative to V. 2. Realizations on affine spheres. We shall begin with the following observation. Proposition 2.1. A torsion-free connection V on Mn with symmetric nondegenerate Ricci tensor can be locally realized on an affine sphere if and only if V is projectively flat and (2.1)

tr RicVffic(-, •, W) = 0 for every W.

Proof. It is obvious that on an affine sphere the Ricci tensor of the induced connection is proportional by a constant to the Blaschke metric, and therefore, the Weyl tensor of V vanishes, Vffic is symmetric and the apolarity gives (2.1). Conversely, if V is a projectively flat connection with symmetric Ricci tensor satisfying (2.1), then defining h and S by Ric = -jjh and S = Hid, where H is any non-zero constant, one gets the objects satisfying all fundamental equations and the apolarity condition. This proves Proposition 2.1. Because of the above proposition we shall state the following definition. Definition. A torsion-free connection is called o^ne spherical if SJRic is projectively flat and (2.1) is satisfied. In case of 2-dimensional manifolds we have

220 Theorem 2.2. Let V be a torsion-free Ricci symmetric connection on M2. Assume that the Ricci tensor of V is nondegenerate and \ VRic \mc^ 0. If Ric is definite, then the connection V is locally realizable on an affine sphere in R 3 if and only if around each point of M2 there is a coordinate system (u, v) such that

(2.2)

VUV =

VUV-^U,

where V is the Levi-Civita connection of the metric tensor Ric and $ is a function satis­ fying the equation (2.3)

ARicln | $ |= -(2 + A ) .

In case of indefinite Ric we have: The connection V is realizable on an affine sphere if and only if around each point there is a coordinate system (it, v) such that

(2.4)

VuV = VuV-±-y v

^ = VvV + r i T ,

where V is the Levi-Civita connection of Ric and <& satisfies the equation (2.5)

ARicln | $ |= ± - 2.

Proof. We shall first prove some lemmas. Lemma 1. Let C be a symmetric cubic form on M2 and tr h,C(-, ■, W) = 0 for some metric tensor field h on M2 and all W. If \ C \hj= 0 at x0, then there is a 1-dimensional smooth distribution V around xa such that C(X, X, X) = 0 for X G V. Moreover, in case of indefinite h we have h(X, X) ^ 0. Proof of Lemma 1. Assume first that h is definite. Let X, Y be an orthonormal frame around xa such that and C(X, X, X) ^ 0. Set X = aX + f3Y, where a = cosip, /3 = simp. We have C(X, X, X) = a(l - 4/32)C(X, X, X) - /3(4a2 - 1)C(Y, F, Y).

221 Hence C(X, X,X)=0 (2.6)

if and only if C(Y, Y, Y) C{X,X,X)

a{\ - 4/32 ,S(4a2-l)

Denote the function on the left hand side of the above equality by g(x). Denote the right hand side of (2.6) treated as a function of