Minimal immersions into space forms with two principal curvatures art lpmj minimal

Math. Z. 187, 325-333 (1984)

Mathematische

Zeitschrift

9 Springer-Verlag 1984

Minimal Immersions Into Space Forms

With Two Principal Curvatures

Luquesio P. J o r g e l and Francesco Mercuri 2 1 Dep. de Matem~ttica UFC, Fortaleza - Cearfi - Brazil 2 Dep. de Matemfitica, UNICAMP-Campinas - SP - Brazil

w 1. Introduction

In [2] Chern, do C a r m o and K o b a y a s h i consider minimal immersions of an n- dimensional manifold M" into the unit sphere S "+N of the ( n + N + l ) - dimensional euclidean space IR n+N+l with the property that the second funda- mental form has constant length n / ( 2 - 1 / N ) . (If M is compact, this is the smallest possible value for a non-totally geodesic minimal immersion as above.) They prove that locally M is either a piece of a Veronese surface (n = 2, N =2) or the product of two spaces of dimensions m and n - m and of constant curvatures n/m and n / ( n - m ) . They also prove the corresponding global result under a compactness assumption for M. The main feature of the situation is that the second fundamental form in any normal direction has two distinct eigenvalues and, in general, the respective eigenspaces will determine involutive distributions whose integral manifolds will give the local product structure. This leads naturally to the study of minimal immersions into spaces of con- stant curvature such that the second fundamental form has, in any normal direction, at most two distinct eigenvalues. This problem was studied by Otsuki in [9] for the case of hypersurfaces. H e proves the following result (see also R e m a r k (3.7)).

(l.1) Theorem. I f x: M " - ~ M ~ +l, n > 3 , is a minimal immersion into a simply connected space of constant curvature c, such that the second fundamental form has two distinct eigenvalues of multiplicity m and n - m, and 1 < m < n - 1. Then:

(i) c > 0 and M n is locally the riemannian product of two spheres of dimen- sions m and n - m with curvatures n/m and n / ( n - m ) respectively.

(ii) I f M ~ is closed and x an embedding, then M" is globally the product of two spheres as in (i).

constant curvature with two principal curvatures Z and #, with /z simple and /~ = #(2), is a " r o t a t i o n hypersurface".

The extension of the above mentioned result to higher codimension was considered by M a t s u y a m a in [7]. H e proves the following result:

n - - n + N

(1.3) Theorem. Let x: M --,M e be a minimal immersion into a simply con- nected space of constant curvature c >=O such that the second fundamental form in any normal direction has at most two distinct eigenvalues and, if exactly two, each with multiplicity bigger than 1. Then the second fundamental form is parallel and its norm S is either zero or n < S <__n2/4. Moreover if c = 0 , S = 0 and if S = n , M is either the product of two spheres, as in (1.1) part (i), or M is the complex projective plane ( i f it is complete).

(1.4) Remark. The p r o o f of T h e o r e m (1.3) does not depend on the fact that c > 0 but only on c being constant.

The aim of this paper is to give a rather complete classification of minimal immersions into space forms of the type considered above. Since (1.1) and (1.2) take care of the codimension one case, we state our result in the following form:

(1.5) Theorem. Let x: M'---,M'~ +N be a minimal immersion into a simply con- nected space of constant curvature c such that the second fundamental form, in any normal direction, has at most two distinct eigenvalues. I f x is substantial, N > I and n >= 3 we have :

(1) c > 0 ,

(2) M" is an open part of a projective plane over the complex, quaternions or Cayley numbers,

(3) x is a standard embedding (in the sense of [5-1).

Moreover the standard embeddings of the spaces in (2) verify the hypothesis of the theorem.

(1.6) Remark. We observe a certain analogy with the result of Chern, do Carmo, K o b a y a s h i quoted at the beginning: A minimal immersion with our condition on the second foundamental form is either a codimension one immersion (and, in general product of spheres by (1.1)) or a Veronese type surface (by (1.5)).

We thank M. Dajczer who brought the problem to our attention and for his helpful comments and the referee for several improvements.

w 2. Notations and Preliminary Results

Minimal Immersions 327 by Ar the Weingarten operator in the ~ direction, i.e. (Ar X, Y) = (c~(X, Y), 4)

VX, YsTpM. Finally the mean curvature vector will be denoted, as usual, by H

and x is minimal if H - 0 , or equivalently, trace A t = 0 V~EvM.

We will be interested in minimal immersions x: M ~ r ~ + N where ~r~ '+N is a simply connected space of constant sectional curvature c with the following additional condition:

2PC: For any normal vector ~evM, Ae has at most two

distinct eigenvalues.

A minimal immersion which verify 2PC will be called a 2PC-minimal immer- sion.

(2.1) Remark. If x is a 2PC-minimal immersion and Ar then Ar has exactly two distinct non zero eigenvalues, since trace A~_ = 0.

The space

N1, v = Span {~(X, X)[X ~ TpM }

= {~EvvMIA r = 0} •

is called the first normal space of x at p. If x is a 2PC-minimal immersion and A~=t=0 we will denote by 2(~) and #(4) the two distinct eigenvalues of Ar by l(~) and m(~) their multiplicity and by Ta(~) and Tu(~) the eigenspaces relative to 2(4) and/~(~) respectively.

(2.2) Lemma. Let x: M " ~ M "+N be a 2PC-minimal immersion. I f dim N1,p> l,

~eN1, p then l(~)---m(~) and 2(~)+p(~)=0.

Proof If d i m N l , p > l , given ~eN1, p there exists a smooth path/3: [0, 1]---~N1, p such that fi(0)=~, / ~ ( 1 ) = - 4 and /?(t)+0 for all tel0,1]. It is then possible to find continuous functions 21 .... ,2n: [0, 1 ] ~ such that the set {21(t)} is the set of repeated eigenvalues of Ap(t) (see [6] Theorem (6.8) page 122). By (2.1), 2i(t ) 4=0 for all te[0,1J, i = 1 ... n, and therefore the multiplicity of the positive eigenvalue of A~( 0 is constant. But Ap(~)= -Aa(o) and so the multiplicity of the negative eigenvalue of Ap(o) is equal to the multiplicity of the positive one and this proves the lemma.

(2.3) Remark. If follows easily from (2.2) that if ~, tleNLv, dimNa,p> 1, then (a) the minimal polynomial of A~ is t 2 +2(~)p(~).

(b) If trace A~A~=O then A~A~+A,Ar In particular, A~(T~(r T,(r and An( T,(r = Tz(r

This allows a certain simplification of Matsuyama's proof of the parallelism of the second fundamental form (see appendix).

(2.4) Proposition. Let x: M'~M'~ +~ be a 2PC-minimal immersion, n>=3 such that x(M") is not contained in any (n+l)-dimensional totally geodesic sub- manifold of ~_2+u. Then the dimension of N~ is a constant bigger than 1.

Proof. The proof will be based on Theorem (1.3) and on the following two

facts:

So if M is connected and U c M is a non empty open set we have:

(a) If x(U) is contained in a totally geodesic submanifold of ~ , + N , so is all of x(M').

(b) If the second fundamental form is parallel in U, it is parallel on all of M.

(2.4.2) (See [11] Lemma 2). If x: M " + M ~ +N is an isometric immersion, and U c M is a non empty connected open set such that the first normal space of x is 1-dimensional in U and for all peU there exists a 2-plane crcTvM with sectional curvature K(cr)~= c, then x(U) is contained in a totally geodesic (n + 1)- dimensional submanifold of .~"+ N

Let us now prove the proposition. If at some point pEM dimN~.p> 1, there exists an open neighborhood U of p where the same happens. By (2.2) and n > 3, Ar does not have simple eigenvalues and so (1.3) implies that the second fundamental form is parallel in U and therefore, by (2.4.1), it is parallel on all of M. In particular d i m N 1 is a constant greater than 1. Let us suppose now dim Nl,p < 1. If there exists an open set U c M, U ~ 4), such that dim N 1 = 0 on U, then xLU is totally geodesic and by analiticity x is a totally geodesic immersion. Suppose now that there exists an open connected non empty set U ~_ M such that dim N1 = 1 in U. Let { be a unit normal vector that spans N,. Since Ar there exist two orthonormal eigenvectors of A~,X, Y,, with eigen- values 2, # and 2 ~ 0 + # . By the Gauss equation the sectional curvature of the plane ~ = s p a n { X , Y} is K ( ~ ) = 2 # + c . c and so, by (2.4.2), x(U) is contained in an ( n + 1)-dimensional totally geodesic submanifold of ~;,+N and by (2.4.t) the same is true for x(M).

(2,5) Corollary. Let x: M~--,M~ +N be a 2PC-minimal immersion, n> 3, such that x(M") is not contained in any (n+l)-dimensional totally geodesic sub- manifoldof ~d2 +N. Then c > 0 .

Proof. By (2.4) and (2.2) x verifies the hypothesis of (1.3). In this case Mat- suyama calculated, using Simons' formula, the norm S of the second funda- mental form obtaining, for a suitable choice of an orthonormal frame ~a .... , ~N ~vM,

n i = 1

Now clearly (2.5.1) implies that if c < 0 then S = 0 and therefore x is totally geodesic.

(2.6) Remark. Again using (2.3), we can get a simplification of the argument used to prove (2.5.1).

w 3, 2PC-Minimal Immersions in S "+N

Minimal Immersions 329 denote by D the riemannian connection of IR "+N+x and by ~, A, ~ the second fundamental form, the Weingarten operator and the normal bundle of the immersion 2 = i o x : M n ~ N "+u+~. If X, Y are tangent vectors of M at p, since Dx Y - Vx Y = - ( X , Y ) x(p), we have:

(3.1) ~(X, Y)= co(X, Y ) - ( X , Y ) x(p). If ~ we have:

(3.2) d ~ = A ~ . - ( x ( p ) , ~ ) I , where ~r = ~ - (x(p), ~) x(p). The following is a well known fact:

(3.3) The second fundamental form of x is parallel if and only if the second fundamental form of ff is parallel.

Isometric immersions in IR n with parallel second fundamental form are rather well known (see [10]). We will comment briefly some of the properties of such immersions that we will use to prove Theorem (1.5). F o r a large class of compact symmetric spaces which includes the classical spaces and few of the exceptional ones, it is possible to construct rigid embeddings in N K such that the second fundamental form is parallel. Those embeddings, called the stan- dard embeddings have, among others, the following properties:

(3.4) After a possible normalization of the metric, the image of a standard embedding is contained in the unit sphere and the induced embedding in that sphere is minimal.

(3.5) The standard embedding iM: M ~ N 1r is tight, i.e., if ~_ is a regular value of the Gauss normal map then the height function h~: M--,N, given by

he(P) = ( iM(P), 4),

has the minimum number of critical points compatible, in the sense of Morse theory, with the topological structure of M.

In [5], Theorem 1, D. Ferus proves that if x: M " ~ S "+N is a minimal immersion, non-totally geodesic and with parallel second fundamental form, then x(M) is an open part of the image of a standard embedding. So the last part of Theorem (1.5) follows easily from the following result.

(3.6) Theorem. Let iM: M"--~S "+u be a standard embedding of a compact symmetric space such that i u is a 2PC-minimal immersion and iM(M ) is not contained in any (n+ 1)-dimensional totally geodesic submanifold of S n+N. Then i M is the standard embedding of a projective plane over the complex, quaternions or Cayley numbers. Moreover the standard embedding of those spaces verify the hypothesis of the theorem.

(3.6.1) Assertion. Let i~ be a 2PC-minimal immersion of a compact rieman- nian manifold M" into S "+N such that the first normal space at each point has dimension at least two. If ~ES "+N is a regular value of the Gauss normal map the function h~: Mn-+IR

h~(p) = (iM(p), ~7

has only non-degenerate critical points and the index of such a points is either 0, n or n/2.

Proof. It is well known (see [103) that h~ has only non-degenerate critical points if ~ is a regular value of the Gauss map. Moreover the hessian of h~ at a critical point is given by the matrix of A~. If {X~ . . . X~} is an orthonormal basis which diagonalizes A ~ we have, from (3.2),

(X,~ x,,

x j) = a,~(~,- (x(p), ~>).

F r o m (2.2) up to a reordering of the X~'s, we have

21 . . .

2nl2----

--'~n/2+ 1 " " --'~n

and the conclusion follows easily.

The above assertion and Morse theory applied to h~ imply that M has the h o m o t o p y type of a CW-complex with only cells in dimensions 0, n or n/2. If n > 2, we have the following immediate consequences:

(3.6.2) M is simply connected, (3.6.3) H i ( M , ~ ) = 0 if i+-O, n, n/2. (3.6.4) Assertion. M is irreducible.

In fact, if M ~ M ~ l x M'22, from (3.6.3) and the Kiinneth formula for the ho- mology of a product, M~ '~ is homeomorphic to S" and hence isometric, since M is symmetric (see [14]). It follows that M = S " / 2 x S "/2 and i M being the standard embedding has image contained in S" + 1 c IR ~ + s* 1

(3.6.5) If H , / a ( M , Z ) = 0 , then M is h o m o t o p y equivalent to S" and therefore i M is a totally geodesic immersion of S n into S ~+N (see [14]).

(3.6.6) If M" is Lie group, n < 6 , then M = S 3 x S 3 as follows from (3.6.2), (3.6.3) and the classification of compact Lie groups. Moreover HS(M, IR)4=O for n- dimensional Lie groups with n > 3 and this, if n >6, contradict (3.6.3) (see [13]). After all this information we take a look at the list of the Poincar~ poly- nomials of compact irreducible symmetric spaces which are not Lie groups, computed in [12], and we easily recognize that the only such spaces with a homology structure compatible with (3.6.2), (3.6.3), etc., are the projective planes over the complex, quaternions and Cayley numbers.

Let now i~: M'~S~+N~_IR "+N§ ~ be the standard embedding of one of the mentioned planes.

Minimal Immersions 331 In fact, in this case the Gauss normal map of the immersion iM: M " ~ N "+N+ ~ maps an open neighbourhood of (p, t/) in the unit normal bundle of T~t onto an open set in S "+N. It follows that, arbitrary near (p,t/), there exists (q,~) such that index (A,)=index(Ar and that ~ is a regular value of the normal map; therefore the function he is a Morse function. By the tightness of i-M and the cohomological structure of the projective planes we get index(.4~)

= 0, n o r

n/2.

We prove now, that i M is a 2PC-minimal immersion. F o r that let

~VpM

be a unit normal vector with Ar and 21 __<22__< ... __<2~ the eigenvalues of Ar Consider ~o =(cos O) ( - ( s i n O). iM(p)~gp. Then, according to 3.2, /1~o has the eigenvalues 2i(0)=(cos 0). 2 i + s i n 0. Since trace Ae=0, we have 21 < 0 < 2 , . If on the other hand

2j<2k,

then 2j(O)<2k(O ) for 10l<zc/2. As 1 i ( ~ / 2 ) = - 2 1 ( - ~ z / 2 ) = 1 for all i, there exists a 0 E l - g / 2 , rc/2[ such that_Ar is non-singular and 2j(O)<O<2k(O ). By our Claim it follows that index (A~o)=n/2, i.e. j < n / 2 < k . In this way we obtain 21 =-.- = 2,/2 < 0 < 21 +,/2 = . . . 2,.

(3.7) Remark. Theorem (1.1) may be proved using the parallelism of the second fundamental form guaranteed by (1.3), by a direct inspection of stan- dard embeddings in S "+ 1. Once we know that the sectional curvatures are non- negative, Theorem (1.1) can be deduced from a result of Erbacher, who cla- ssifies complete immersions in space forms with non-negative sectional curva- ture, parallel mean curvature and fiat normal bundle (see [1]).

(3.8) Remark. If M is complete and x: Mn'---~S n+p is a 2PC-minimal immersion such that x(M") is not contained in any (n+l)-dimensional totally geodesic submanifold of S "+~, then x factors as composition of an isometric covering map and a standard embedding (see [4]). By (3.6) the isometric covering is actually an isometry.

A p p e n d i x

In this appendix we will give a proof of the parallelism of the second foundam- ental form of a 2PC-minimal immersion since the proof given in [7] was not completely clear to us.

Theorem. Let x: 1vl ~1vl c be a 2PC-minimal immersion and suppose that for any ~.evqM", A has no simple eigenvalues. Then ~ is parallel.

Proof If dim N 1 < 1 it is easily seen that we can reduce the codimension to 1 and the theorem follow from Otzuki's theorem ((1.1) in our introduction).

we have, by 2.3:

and therefore (A.1)

(~, c~(X, Y))=(A~X,

Y ) = 0 if

i#=i o

~(X, Y)= 2(X, Y) ~

for X, Ye Tx.

Now, if

ZeT~M,

since

A~=)3, Ar162162

- 2 ( ( A r and therefore:

(A.2) (A~_ - 2) Z e T_;~

VZeTqM.

Claim.

2 is constant.

In fact let us consider the Codazzi equation projected in the ~ direction

(1.3)

(VxA~) Y - A % r Y= (VrAr

-A%~X.

If X, Ye T;. we have

(A.4)

(A~- )O[X, Y] + Av~,r Y - Av}~X= X(2) Y - Y(2) X.

Let {X~ ... X,} be a basis which diagonalizes A~ and

ZeTM.

F r o m (A.1) we get:

(Av~Xi,A~Xi)--<Vzl{,o:(Xi,A~Xi))=)o 2

(Vz~,~> = 0 which implies

(A.5) trace Av}~A~ = 0.

In particular, from 2.3,

Av}~(Tz)c_T_ a

and this, together with (A.2), implies that the left hand side of (A.4) belongs to T_ x. But the right hand side of (A.4) belong to Tx and is therefore zero. Since d i m T ) > 2 , for any

XeTx

we can choose

YeTx

independent from X and therefore X(2)=0.

In the same way we see that - 2 is constant in any T_ x direction and this completes the proof of our Claim.

Claim.

For all

X, Y,,ZeTM, (#zC~)(X, Y)eN(.

It is sufficient to prove that for i = 1, ..., k

<(Vz ~)(x, y), ~) =o.

As in (A.3), this is equivalent to

(A.6)

((VzAr162

Y, X )

= 0 .

Let ~=~i and + 2 be the eigenvalues of Ar If

XeT+x

and

ZeTM,

since 2 is constant we get:

(VzAr X = Vz()~X ) - Ar

)

= - (Ar ~ 2)

V z X

and therefore by (A.2),

(VzAr

"

and using (A.5), we have

Minimal Immersions 333

and therefore (A.6) holds for X, Y in the same eigenspace. Let now

XeTx,

YeT_ x. From (A.7) and (A.3) we get

(1.8)

(VxAr162

Y=O=(VrAr162

F r o m the above, using the Codazzi equation we get:

O=((VxAr162

Y , Z ) = ( ( V z A r

Y,,X),

hence (A.6) is proved for

X e T x

and YeT_ x, and conversely. Therefore our claim is proved.

Claim. N 1

is parallel (in U).

Let ~_ = ~io be one of our frame field for N 1 choosen at the beginning of the proof and X, Y orthonormal eigenvectors of Ar relative to the same eigenvalue. F r o m (A.1), we get c~(X, Y)=0 and therefore for all

Z e T M , (~rzO:)(X, Y)eN 1

and by our second claim,

(~7zcO(X, Y)=0.

In particular

0 =(l~y e)(X, Y) = (l~x e)(Y,, Y)=

+_2Vx~-2c~(Vx Y, Y)

and therefore

V ~ e N 1

for all

X e T x w T_ x

and therefore for all

X e T M .

A standard argument concludes the proof of the Theorem: From the second claim

(VzcQ(X, Y)

is orthogonal t o N 1. If t/ is a unit section of Ni L, by our last claim it follows that

V z tl~N~, V Z e T M

and therefore

((P~ ~)(x,

Y), ~ ) = (~/(c~(x, Y)) - ~(F~ x , Y) - ~ ( x , F~ Y), ~ ) = - ( ~ ( x , Y), ~ ' ~ ) = 0 .

References

1. Chen, B.Y.: Geometry of Submanifolds. New York: Marcel Dekker, Inc. 1973

2. Chern, S.S., Carmo, Do, Kobayashi, S.: Minimal submanifolds of a sphere with second fundamental form of constant lenth. Functional Analysis and Related fields, pp. 59-75. Berlin- Heidelberg-New York: Springer 1970

3. Carmo, M., Do, Dajczer, M.: Rotational hypersurfaces in spaces of constant curvature. To appear in Trans. A.M.S.

4. Ferus, D.: Immersions with parallel second fundamental forml Math. Z. 140, 87-93 (1970) 5. Ferus, D.: Symmetric submanifolds of euclidean space. Math. Ann. 274, 81-93 (1980)

6. Kato, T.: Perturbation Theory of Linear Operators. Berlin-Heidelberg-New York: Springer 1966

7. Matsuyama, Y.: Minimal submanifolds of S N and F, u. Math. Z. 175, 275-282 (1980) 8. Mitnor, J.: Morse Theory. Princeton, N.J.: Princeton Academic Press 1963

9. Otsuki, T.: Minimal hypersurfaces in a Riemannian manifold of constant curvature. Amer. J. Math. XCII, (1) 1970

10. Rodriguez, L : Geometria das Subvariedades. Monografias, IMPA, 1976

11. Rodriguez, L., Tribuzy, R.: Reduction of Codimension of regular immersion. Preprint IMPA, 1982

12. Takeuchi, M.: On Pontryagin classes of compact symmetric spaces. J. Fac_ Sci_ Univ. Tokyo Sect. 19, 313-328 (1962)

13. Wolf, J.A.: Spaces of Constant Curvature. New York: McGraw-Hill 1967

14. Wolf, J.A.: Symmetric spaces which are cohomology spheres. Journal Differential Geometry 3, 59-68 (1969)