Real Hypersurfaces of Type B in Complex Two-Plane Grassmannians

DOI 10.1007/s00605-005-0329-9

Real Hypersurfaces of Type B in Complex Two-Plane Grassmannians

By

Young Jin Suh

Kyungpook National University, Taegu, Korea Communicated by D. V. Alekseevski

Received September 3, 2004; accepted in final form January 29, 2005 Published online January 29, 2006#Springer-Verlag 2006

Abstract.In this paper we give a characterization of real hypersurfaces of type B, that is, a tube over a totally real totally geodesicHPⁿin complex two-plane GrassmanniansG₂ðC^mþ2Þ,m¼2nwith the shape operatorAsatisfyingAþA¼k,kis non-zero constant, for the structure tensor.

2000 Mathematics Subject Classification: 53C40; 53C15

Key words: Complex two-plane Grassmannians, real hypersurfaces of type B, tubes, shape operator, K€aahler structure, quaternionic K€aahler structure

0. Introduction

In the geometry of real hypersurfaces in complex space forms MmðcÞ or in quaternionic space forms there have been many characterizations of model hyper- surfaces of typeA1;A2;B;C;DandEin complex projective spacePmðCÞ, of type A0;A1;A2 andBin complex hyperbolic spaceHmðCÞ orA1;A2;Bin quaternionic projective space HP^m, which are completely classified by Cecil and Ryan [6], Kimura [7], Berndt [2], Martinez and Peerez [8], respectively. Among them there were only a few characterizations of homogeneous real hypersurfaces of typeBin complex projective spaceP_mðCÞ. For example, the condition thatAþA¼k, kis non-zero constant, is a model characterization of this kind of typeB, which is a tube over a real projective spaceRPⁿ inP_mðCÞ,m¼2n(see Yano and Kon [13]).

LetMbe að4m1Þ-dimensional Riemannian manifold with an almost contact structureð; ; Þand an associated Riemannian metricg. We put

!ðX;YÞ ¼gðX;YÞ; ð0:1Þ where!defines a 2-form on M and rank!¼rank¼4m2.

If there is a non-zero valued functionsuch that

gðX;YÞ ¼!ðX;YÞ ¼dðX;YÞ; ð0:2Þ

This work was supported by grant Proj. No. R14-2002-003-01001-0 from the Korea Research Foundation.

the rank of the matrixð!Þ being 4m2, we have ^!zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{^{2m1 times}^ ^!

¼^^ð2m1Þdzfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{^{2m1 times}^ ^d 6¼0:

Let us denote by G₂ðC^mþ2Þ the set of all complex 2-dimensional linear sub- spaces ofC^mþ2. We call such a setG₂ðC^mþ2Þ complex two-plane Grassmannians.

This Riemannian symmetric space G₂ðC^mþ2Þ has a remarkable geometrical struc- ture. It is the unique compact irreducible Riemannian manifold being equipped with both a K€aahler structureJand a quaternionic K€aahler structureJ¼SpanfJ1;J₂;J₃g not containing J. In other words, G₂ðC^mþ2Þ is the unique compact, irreducible, K€aahler, quaternionic K€aahler manifold which is not a hyperk€aahler manifold (see Berndt and Suh [4], [5]).

Now we consider að4m1Þ-dimensional real hypersurfaceMin complex two- plane Grassmannians G₂ðC^mþ2Þ. Then from the K€aahler structure of G₂ðC^mþ2Þ there exists an almost contact structureonM. If the non-zero functionsatisfies (0.2), we callM acontacthypersurface of the K€aahler manifold. Moreover, it can be easily verified that a real hypersurfaceMinG₂ðC^mþ2Þiscontactif and only if there exists a non-zero constant functiondefined onM such that

AþA¼k; k¼2: ðÞ The formulaðÞmeans that

gððAþAÞX;YÞ ¼2dðX;YÞ;

where the exterior derivatived of the 1-formis defined by dðX;YÞ ¼ ðrXÞY ðrYÞX for any vector fieldsX;Y onM inG₂ðC^mþ2Þ.

On the other hand, in G₂ðC^mþ2Þwe are able to consider two kinds of natural geometric conditions for real hypersurfaces M that ½ ¼Spanfg or D^? ¼ Spanf1; ₂; ₃g, _i¼ JiN,i¼1;2;3, whereN denotes a unit normal toM, is invariant under the shape operator A of M in G₂ðC^mþ2Þ. The first result in this direction is the classification of real hypersurfaces inG₂ðC^mþ2Þsatisfying both two conditions. Namely, Berndt and the present author [4] have proved the following Theorem A. Let M be a connected real hypersurface in G2ðC^mþ2Þ, m53.

Then both½ andD^? are invariant under the shape operator of M if and only if (A) M is an open part of a tube around a totally geodesic G2ðC^mþ1Þ in G2ðC^mþ2Þ,or

(B) m is even,say m¼2n,and M is an open part of a tube around a totally geodesicHPⁿ in G₂ðC^mþ2Þ.

In Theorem A the vectorcontained in the one-dimensional distribution½is said to be aHopfvector when it becomes a principal vector for the shape operatorAofM in G₂ðC^mþ2Þ. Moreover in such a situation M is said to be a Hopf hypersurface.

Besides this, a real hypersurface M in G₂ðC^mþ2Þ also admits the 3-dimensional distribution D^?, which is spanned by almost contact 3-structure vector fields f1; ₂; ₃g, such that T_xM¼DD^?. Also in the paper [5] due to Berndt and

the present author we have given a characterization of real hypersurfaces of typeA when the shape operatorAofM inG₂ðC^mþ2Þcommutes with the structure tensor , which is equivalent to the condition that the Reeb flow on M is isometric.

Moreover, in the paper due to the present author [12] we have also given a characterization of type A by vanishing Lie derivative of the shape operator A in the direction of the structure vector field.

Real hypersurfaces of type B in Theorem A is just the case that the one di- mensional distribution½is contained inD^?. It was shown in the paper [11] that the tube of typeBsatisfies the following formula on the orthogonal complement of the one-dimensional distribution½

AA¼0; ¼1;2;3:

From this view point, the present author [11] has given a characterization that the almost contact 3-structure tensorsf1; ₂; ₃g and the shape operatorAof a real hypersurfaceM inG₂ðC^mþ2Þcommute with each other as follows:

Theorem B.Let M be a Hopf real hypersurface in G₂ðC^mþ2ÞsatisfyingðÞon the orthogonal complement of the one-dimensional distribution ½. Then M is locally congruent to an open part of a tube around a totally geodesic HP^m in G2ðC^mþ2Þ,where m¼2n.

Now in this paper as another characterization of real hypersurfaces of typeBin complex two-plane GrassmanniansG2ðC^mþ2Þin terms of thecontacthypersurface we want to assert the following remarkable fact:

Theorem. Let M be a real hypersurface in G2ðC^mþ2Þ with constant mean curvature satisfying

AþA¼k;

where the function k is non-zero and constant. Then M is congruent to an open part of a tube around a totally geodesicHPⁿ in G2ðC^mþ2Þ,where m¼2n.

1. Riemannian Geometry ofG₂ðC^mþ2Þ

In this section we summarize basic material about G₂ðC^mþ2Þ, for details we refer to [3], [4] and [5]. The special unitary groupG¼SUðmþ2Þacts transitively on G₂ðC^mþ2Þ with stabilizer isomorphic to K ¼SðUð2Þ UðmÞÞ G. Then G₂ðC^mþ2Þ can be identified with the homogeneous space G=K, which we equip with the unique analytic structure for which the natural action ofGonG₂ðC^mþ2Þ becomes analytic. Denote bygandkthe Lie algebra ofGandK, respectively, and bymthe orthogonal complement ofkingwith respect to the Cartan-Killing form B of g. Then g¼km is an AdðKÞ-invariant reductive decomposition of g.

We put o¼eK and identify ToG2ðC^mþ2Þ with m in the usual manner. Since B is negative definite ong, its negative restricted tommyields a positive definite inner product onm. By AdðKÞ-invariance ofBthis inner product can be extended to a G-invariant Riemannian metric g on G2ðC^mþ2Þ. In this way G2ðC^mþ2Þ becomes a Riemannian homogeneous space, even a Riemannian symmetric space.

For computational reasons we normalize g such that the maximal sectional cur- vature ofðG2ðC^mþ2Þ;gÞis eight. SinceG₂ðC³Þis isometric to the two-dimensional complex projective spaceCP²with constant holomorphic sectional curvature eight we will assumem52 from now on. Note that the isomorphism Spinð6Þ ’SUð4Þ yields an isometry betweenG₂ðC⁴Þ and the real Grassmann manifoldG^þ₂ðR⁶Þ of oriented two-dimensional linear subspaces ofR⁶.

The Lie algebra k has the direct sum decomposition k¼suðmÞ suð2Þ R, whereR is the center of k. Viewingk as the holonomy algebra ofG2ðC^mþ2Þ, the centerR induces a K€aahler structureJ and the suð2Þ-part a quaternionic K€aahler structure J on G2ðC^mþ2Þ. If J1 is any almost Hermitian structure in J, then JJ1¼J1J, and JJ1 is a symmetric endomorphism with ðJJ1Þ² ¼I and TrðJJ1Þ ¼0. This fact will be used frequently throughout this paper.

A canonical local basisJ1;J2;J3 ofJconsists of three local almost Hermitian structures J in J such that JJ_þ1 ¼J_þ2¼ Jþ1J, where the index is taken modulo 3. Since J is parallel with respect to the Riemannian connection rr of ðG2ðC^mþ2Þ;gÞ, there exist for any canonical local basisJ₁;J₂;J₃ ofJthree local one-formsq₁;q₂;q₃ such that

r

rXJ ¼q_þ2ðXÞJþ1q_þ1ðXÞJþ2 ð1:1Þ for all vector fieldsXonG₂ðC^mþ2Þ.

Letp2G₂ðC^mþ2ÞandWa subspace ofT_pG₂ðC^mþ2Þ. We say thatW is a quater- nionic subspace of TpG2ðC^mþ2Þ ifJWW for allJ2J_p. And we say thatW is a totally complex subspace ofTpG2ðC^mþ2Þ if there exists a one-dimensional sub- spaceVofJ_p such thatJW W for all J2VandJW ?W for allJ2V^? J_p. Here, the orthogonal complement of V inJ_p is taken with respect to the bundle metric and orientation onJfor which any local oriented orthonormal frame field of Jis a canonical local basis ofJ. A quaternionic (resp. totally complex) submanifold ofG₂ðC^mþ2Þ is a submanifold all of whose tangent spaces are quaternionic (resp.

totally complex) subspaces of the corresponding tangent spaces ofG₂ðC^mþ2Þ.

The Riemannian curvature tensorRR ofG₂ðC^mþ2Þis locally given by R

RðX;YÞZ¼gðY;ZÞXgðX;ZÞYþgðJY;ZÞJXgðJX;ZÞJY2gðJX;YÞJZ þX³

¼1

fgðJY;ZÞJXgðJX;ZÞJY2gðJX;YÞJZg

þX³

¼1

fgðJJY;ZÞJJXgðJJX;ZÞJJYg; ð1:2Þ whereJ₁;J₂;J₃ is any canonical local basis ofJ.

2. Some Fundamental Formulas for Real Hypersurfaces inG2ðC^mþ2Þ

In this section we derive some fundamental formulas which will be used in the proof of our main theorem. LetM be a real hypersurface in G₂ðC^mþ2Þ, that is, a hypersurface in G₂ðC^mþ2Þ with real codimension one. The induced Riemannian

metric onM will also be denoted byg, andrdenotes the Riemannian connection ofðM;gÞ. LetN be a local unit normal field ofMandAthe shape operator ofM with respect toN.

The K€aahler structureJ of G₂ðC^mþ2Þ induces on M an almost contact metric structure ð; ; ;gÞ. Furthermore, let J₁;J₂;J₃ be a canonical local basis of J.

Then eachJ induces an almost contact metric structureð; ; ;gÞonM. Using the above expression (1.2) for the curvature tensorRR, the Gauss and the Codazzi equations are respectively given by

RðX;YÞZ¼gðY;ZÞXgðX;ZÞYþgðY;ZÞXgðX;ZÞY2gðX;YÞZ þX³

¼1

fgðY;ZÞXgðX;ZÞY2gðX;YÞZg

þX³

¼1

fgðY;ZÞXgðX;ZÞYg

X³

¼1

fðYÞðZÞXðXÞðZÞYg

X³

¼1

fðXÞgðY;ZÞ ðYÞgðX;ZÞg

þgðAY;ZÞAXgðAX;ZÞAY and

ðrXAÞY ðrYAÞX¼ðXÞYðYÞX2gðX;YÞ þX³

¼1

fðXÞYðYÞX2gðX;YÞg

þX³

¼1

fðXÞYðYÞXg;

þX³

¼1

fðXÞðYÞ ðYÞðXÞg;

whereRdenotes the curvature tensor of a real hypersurface M inG₂ðC^mþ2Þ.

The following identities can be proved in a straightforward method and will be used frequently in subsequent calculations:

_þ1 ¼ _þ2; _þ1 ¼_þ2; ¼; ðXÞ ¼ðXÞ;

_þ1X¼_þ2Xþ_þ1ðXÞ; _þ1X¼ þ2XþðXÞþ1:

ð2:1Þ

Then in this section let us give some basic formulas which will be used later.

Now let us put

JX¼XþðXÞN; JX¼XþðXÞN

for any tangent vectorXof a real hypersurfaceMinG2ðC^mþ2Þ, whereN denotes a normal vector ofMinG2ðC^mþ2Þ. Then from this and the formulas in Section 1 we have that

ðrXÞY¼ðYÞAXgðAX;YÞ; rX¼AX; ð2:2Þ rX ¼qþ2ðXÞ_þ1qþ1ðXÞ_þ2þAX; ð2:3Þ ðrXÞY¼ qþ1ðXÞþ2Yþq_þ2ðXÞþ1YþðYÞAXgðAX;YÞ: ð2:4Þ

Summing up these formulas, we know the following rXðÞ ¼ ðrXÞþðrXÞ

¼ qþ1ðXÞþ2þq_þ2ðXÞþ1þðÞAXgðAX; ÞþAX:

ð2:5Þ Moreover, fromJJ ¼JJ,¼1;2;3; it follows that

X¼XþðXÞðXÞ: ð2:6Þ

3. Some Key Propositions

Before going to give the proof of our main Theorem in the introduction let us check that ‘‘What kind of model hypersurfaces given in Theorem A satisfy the formulaðÞ.’’ In other words, it will be an interesting problem to know whether there exist any real hypersurfaces inG₂ðC^mþ2Þ satisfying the conditionðÞ.

In this section we will show that only real hypersurfaces of type B in G₂ðC^mþ2Þ, that is, a tube over a quaternionic projective spaceHPⁿ inG₂ðC^mþ2Þ satisfies the formulaAþA¼k,m¼2n, where the functionkis non-zero and constant.

Now in order to solve such a problem let us recall some Propositions given by Berndt and the present author [4] as follows:

For a tube of typeAin Theorem A we have the following

Proposition A.Let M be a connected real hypersurface of G2ðC^mþ2Þ.Suppose that ADD,A¼,andis tangent toD^?.Let J12Jbe the almost Hermitian structure such that JN¼J1N. Then M has three(if r¼=2 ffiffiffi

p8

) or four(other- wise)distinct constant principal curvatures

¼ ffiffiffi p8

cotð ffiffiffi p8

rÞ; ¼ ffiffiffi p2

cotð ffiffiffi p2

rÞ; ¼ ffiffiffi p2

tanð ffiffiffi p2

rÞ; ¼0 with some r2 ð0; =4Þ.The corresponding multiplicities are

mðÞ ¼1; mðÞ ¼2; mð Þ ¼2m2¼mðÞ;

and the corresponding eigenspaces we have T ¼R¼RJN ¼R₁;

T ¼C^?¼C^?N ¼R₂R₃; T ¼ fXjX?H;JX¼J1Xg;

T¼ fXjX?H;JX¼ J1Xg;

whereR,CandQrepectively denotes real,complex and quaternionic span of the structure vectorandC^? denotes the orthogonal complement ofCin H.

For such kind of real hypersurfaces of type Amentioned above let us check whether this type satisfies the formulaðÞor not.

Now let us assume that real hypersurfaces of typeAsatisfies the formulaðÞ.

In Proposition A let us put X¼₂2T, ¼₂¼₃, and ¼₁. Then by the formula (2.1) we have

A₂þA₂¼A₂₁þA₂

¼ A3þ₂₂

¼ 3323

¼ 2 ffiffiffi p2

cot ffiffiffi p2

r3: From this, together with the formulaðÞwe know

k3¼k₂ ¼ 2 ffiffiffi p2

cot ffiffiffi p2

r ₃ which meansk¼2 ffiffiffi

p2

cot ffiffiffi p2

r.

On the other hand, by the paper [5] of Berndt and the present author we know that the distributionsT andTin Proposition A are-invariant, that isT T and TT respectively. By virtue of this fact we know that for any X2T ,

¼ ffiffiffi p2

tan ffiffiffi p2

r

AXþAX¼ 2 ffiffiffi p2

tan ffiffiffi p2

rX:

Then in this timek¼ 2 ffiffiffi p2

tan ffiffiffi p2

r. From this, together with the above formula we get cot² ffiffiffi

p2

r¼ 1, which makes a contradiction. So real hypersurfaces of typeAcan not satisfy the formulaðÞ.

Moreover, for a tube of typeBin Theorem A we introduce the following Proposition B.Let M be a connected real hypersurface of G2ðC^mþ2Þ.Suppose that ADD,A¼,andis tangent toD.Then the quaternionic dimension m of G₂ðC^mþ2Þ is even, say m¼2n, and M has five distinct constant principal curvatures

¼ 2 tanð2rÞ; ¼2 cotð2rÞ; ¼0; ¼ cotðrÞ; ¼ tanðrÞ with some r2 ð0; =4Þ.The corresponding multiplicities are

mðÞ ¼1; mðÞ ¼3¼mðÞ; mð Þ ¼4n4¼mðÞ and the corresponding eigenspaces are

T¼R; T¼JJ; T¼J;T ;T;

where

T T¼ ðHCÞ^?; JT ¼T ; JT¼T; JT ¼T: Of course we have proved that all of the principal curvatures and its eigen- spaces of the tube of typeA(resp. the tube of typeB) in Theorem A satisfies all of the properties in Proposition A (resp. Proposition B).

Now by using this Proposition B we show that a tube of typeBin Theorem A, that is, a tube over a totally geodesic HPⁿ in G₂ðC^mþ2Þ, m¼2n satisfies the formulaðÞfor a constantk¼2 cot 2ras follows:

For any 2T,¼ 2 tan 2r, we have

AþA¼0¼k:

For any 2T, ¼2 cot 2r, the eigen space T¼J gives 2T. This implies A ¼0 for any ¼1;2;3. From this we have the following for k¼2 cot 2r

AþA ¼2 cot 2r: For any X2T, ¼ cot r we know thatJT ¼T gives

AXþAX ¼ tanrXþ cotrX¼2 cot 2rX:

This means that the formulaðÞholds fork¼2 cot 2r.

Finally, for the case 2T, ¼1;2;3, the formula ðÞ also holds for k¼2 cot 2r.

4. Some Key Lemmas and Theorems

In order to give a characterization of typeBamong the classes of real hyper- surfacesMin complex two-plane GrassmanniansG₂ðC^mþ2Þwe will prepare some lemmas and a proposition as follows:

Lemma 1. Let M be a real hypersurface in G₂ðC^mþ2Þ satisfying AþA¼k;

where the function k is non-zero and constant. Then Tr A¼þnk, where n¼2m1

Proof. Now suppose thatAþA¼k. By applyingto the left, we have ²AþA¼k²:

Then it follows that

AXAXkXþ ðkÞðXÞ¼0:

Now let us take an orthonormal basis feiji¼1;. . .;4m1g for M in above formula. Then we have

TrATrA ð4m1Þkþ ðkÞ ¼0: ð4:1Þ On the other hand, we know

TrA¼TrA²¼ TrAþ:

Because we have

A²X¼ AXþðXÞA¼ AXþðXÞ: From this, together with (4.1), we have

TrA¼ ð2m1Þkþ;

which completes the proof of Lemma 1. &

Now let us assume that the structure vector is principal and denote by H the orthogonal complement of the real span ½ of the structure vector in TM.

Then taking an inner product of the Codazzi equation in section 2 withand using A¼imply

2gðX;YÞ þ2X³

¼1

fðXÞðYÞ ðYÞðXÞ gðX;YÞðÞg

¼gððrXAÞY ðrYAÞX; Þ

¼gððrXAÞ;YÞ gððrYAÞ;XÞ

¼ ðXÞðYÞ ðYÞðXÞ þgððAþAÞX;YÞ 2gðAAX;YÞ: ð4:2Þ PuttingX¼, we have

Y¼ ðÞðYÞ 4X³

¼1

ðÞðYÞ ð4:3Þ

for any tangent vector fieldY onM. Substituting this formula into (4.2), then we have

2gðX;YÞ þ2X³

¼1

fðXÞðYÞ ðYÞðXÞ gðX;YÞðÞg

¼4X³

¼1

fðXÞðYÞ ðYÞðXÞgðÞ þgððAþAÞX;YÞ 2gðAAX;YÞ:

From this formula we are able to assert

Lemma 2. If A¼and X2H with AX¼ X,then 0¼ ð2 ÞAX ð2þ ÞXþ2X³

¼1

f2ðÞðXÞðXÞ ðXÞðÞXg:

Lemma 3. Let M be a real hypersurface in G2ðC^mþ2Þ satisfying AþA¼k, k is non-zero and constant. Then is principal. Moreover, the principal curvature functionis constant provided that 2D^? or2D.

Proof. Then by applying the structure vector to the above assumption in the right side, we knowA¼0. This meansA¼, that is, the structure vector

is principal. Then we are able to use (4.2) and (4.3). The formula (4.3) means that

grad¼ ðÞþ4X³

¼1

ðÞ: ð4:4Þ For the case where2D^?. We may put¼₁. Then (4.4) implies

grad¼ ðÞ: ð4:5Þ

For the case where 2D. Then naturally the formula (4.4) gives (4.5). Now differentiating (4.5), we have

rXðgradÞ ¼XðÞþ ðÞAX:

Then this implies

0¼gðrXðgradÞ;YÞ gðrYðgradÞ;XÞ

¼XðÞðYÞ YðÞðXÞ þ ðÞgððAþAÞX;YÞ:

This gives

kðÞgðX;YÞ ¼YðÞðXÞ XðÞðYÞ:

From this, puttingX¼, we haveYðÞ ¼ðÞðYÞ. Then it follows that kðÞgðX;YÞ ¼0:

By virtue ofk6¼0, we have¼0. From this, together with (4.5) we have grad¼0;

which means that the principal curvatureis constant. &

Then by using Lemmas 1 and 3 we have the following Proposition.

Proposition 4. Let M be a real hypersurface in G₂ðC^mþ2Þ satisfying AþA¼k,k is non-zero and constant.Then we have

2A²X2kAXþ ðkþ2ÞX

ðXÞð2²kþ2Þ þ4X

ðÞðXÞ 4X

²ðÞðXÞ

2X

ðXÞðXÞþðXÞðÞþðÞX

¼0;

whereP

denotes the sum from¼1 to¼3.

Proof. Now substituting (4.3) into (4.2), we have 2gðX;YÞ þ2X

fðXÞðYÞ ðYÞðXÞ gðX;YÞðÞg

¼ 4X

ðÞðXÞðYÞ þ4X

ðÞðYÞðXÞ

þgððAþAÞX;YÞ 2gðAAX;YÞ: ð4:6Þ

On the other hand, from the assumption we have

gðAAX;YÞ ¼kgðAX;YÞ gðA²X;YÞ: Then from this together with formula (4.6) we have

2A²X2kAXþ ðkþ2ÞX4X

ðÞðXÞ4X

ðÞðXÞ

¼2X

ðXÞðXÞðÞXg:

ReplacingXbyX, we have

2A²X¼2ðXÞA²þ2kAX2kðXÞA ðkþ2ÞX þðXÞðkþ2Þþ4X

ðÞðXÞ4X

²ðÞðXÞ þ2X

fðXÞðXÞþðXÞðÞþðÞXg

¼

ðXÞð2²kþ2Þ þ4X

ðÞðXÞ 4X

²ðÞðXÞ

ðkþ2ÞXþ2kAXþ2X

fðXÞ

ðXÞþðXÞðÞ þðÞXg:

&

Now we are going to prove a key Lemma which will be useful in the proof of our Main Theorem.

Lemma 5. Under the same assumption as in Proposition 4 we have XðkTr AÞ ¼ðXÞðkTr AÞ þ4X

ðXÞðÞ:

Proof. Differentiating ðAþAÞX¼kXcovariantly, we have

ðrYÞAXþðrYAÞXþ ðrYAÞXþAðrYÞX¼ ðYkÞXþkðrYÞX:

Then substituting the formula (2.2) into the above equation, we have ðXÞfA²YþAYkAYg gðA²XþAXkAX;YÞþðrYAÞX

þ ðrYAÞX¼ ðYkÞX:

From this, using Proposition 4, we have ðXÞ

AYkþ2

2 Yþ

ðYÞ

²

2kþ1 þ2X

ðÞðYÞ 2X

²ðÞðYÞ

þX

fðYÞðYÞþðYÞðÞþðYÞYg

g

AXkþ2

2 X;Y

þg

ðXÞ

²

2kþ1 þ2X

ðÞðXÞ 2X

²ðÞðXÞ

;Y

þX

gðfðXÞðXÞþðXÞðÞþðÞXg;YÞ þðrYAÞXþ ðrYAÞX

¼ ðYkÞX:

From this, contracting, we have X

i

ðEikÞEi ¼Akþ2

2 X

i

gðAEi;E_iÞ þkþ2

2 X

i

gðEi;E_iÞX

i;

ðEiÞgð;E_iÞ þX

i;

ðEiÞðEiÞX

i;

ðEiÞðÞðEiÞ

X

i;

ðÞgðE_i;E_iÞþðrEiAÞEiþ ðrEiAÞEi; whereP

i(resp.P

) denotes the sum fromi¼1 toi¼4m1 (resp. from¼1 to¼3). Then by virtue of formulas defined in (2.1) we have

X

i

ðEikÞE_i¼Akþ2

2 ðTrAÞþkþ2

2 ð4m1Þþ62X

²ðÞ

X

ðÞðTrÞþX

i

ðrEiAÞEiþX

i

ðrEiAÞEi: ð4:7Þ On the other hand, the first term in the fourth line of (4.7) becomes

X

i

gððrEiAÞEi;XÞ ¼ X

i

gððrEiAÞEi; XÞ ¼ X

i

gðEi;ðrEiAÞXÞ:

Also by virtue of the Codazzi equation in section 2 the last term of the above equation can be changed into

X

i

gððrEiAÞX ðrXAÞEi;EiÞ

¼X

ðXÞ²ðÞ X

ðXÞðÞ þX

ðXÞTr ðXÞX

ðÞTr X

ðXÞðÞ þðXÞX

²ðÞ: ð4:8Þ

Also let us use the Codazzi equation in the final term of the fourth line of (4.7).

Then it follows that X

i

gðEi;ðrEiAÞXÞ ¼X

i

gðEi;ðrXAÞEiÞ ðXÞX

i

gðEi; E_iÞ þX

;i

ðEiÞgðX; E_iÞ þX

;i

fðEiÞgðX; E_iÞ ðXÞgðE_i; E_iÞ 2gðE_i;XÞðEiÞg

þX

;i

fðEiÞðXÞ ðXÞðEiÞgðEiÞ: ð4:9Þ Now from the third term in the right side of (4.9) let us calculate term by term as follows:

X

ðEiÞgðX; E_iÞ ¼ X

ð²XÞ ¼X

ðXÞðÞ X

ðXÞ²ðÞ;

X

i;

ðEiÞgðX; EiÞ ¼ X

ðXÞ ¼3ðXÞ X

ðXÞðÞ;

X

i;

ðXÞgðE_i; E_iÞ ¼X

ðXÞTr;

2X

i;

gðE_i;XÞðEiÞ ¼ 6ðXÞ þ2X

ðÞðXÞ;

and

X

ðXÞðEiÞðEiÞ ¼ 3ðXÞ þX

ðXÞðÞ²: Substituting all of these formulas into (4.9), we have the following X

i

gððrEiAÞEi;XÞ ¼X

i

gðEi;ðrXAÞEiÞ ð4m1ÞðXÞ

þ3ðXÞ X

ðXÞðÞ þX

ðXÞTr6ðXÞ þ2X

ðÞðXÞ 3ðXÞ þX

ðXÞðÞ² þX

ðXÞðÞ X

ðXÞðÞ²: ð4:10Þ Now substituting (4.8) and (4.10) into (4.7), then by Lemma 1 and Lemma 3 we have X

i

ðEikÞgðEi;XÞ ¼ ð4mþ4ÞðXÞ 2X

²ðÞðXÞ X

ðÞTrðÞðXÞ

X

i

gðEi;ðrXAÞEiÞ X

ðXÞ²ðÞ þX

ðXÞðÞ

X

ðXÞTrþðXÞX

ðÞTrþX

ðXÞðÞ ðXÞX

ðÞ²þX

i

gðEi;ðrXAÞEiÞ ð4m1ÞðXÞ þ3ðXÞ X

ðXÞðÞ þX

ðXÞTr6ðXÞ þ2X

ðÞðXÞ 3ðXÞ þX

ðXÞðÞ²þX

ðXÞðÞ X

ðXÞðÞ²

¼ ðXÞ 4X

²ðÞðXÞ þ4X

ðXÞðÞ TrðrXAÞ;

where we have used that the structure vector is principal and TrðrXAÞ¼0.

Then it can be written as follows:

XðkÞ ¼XðTrAÞ þðXÞ þ4X

²ðÞðXÞ 4X

ðXÞðÞ:

From this, replacingXbyX, we have ²XðkTrAÞ ¼ 4X

ðXÞðÞ: Finally, we have arrived at the following formula

XðkTrAÞ ¼ðXÞðkTrAÞ þ4X

ðXÞðÞ:

From this we complete the proof of Lemma 5. &

By Lemma 1 we know that the mean curvature is constant if and only if the function is constant. By the result in Lemma 5 we know that if the function kTrAis constant, then

X

ðXÞðÞ ¼0 ð4:11Þ

for anyX2TxM. Then the formula (4.11) is equal to X

ðÞ ¼0: ð4:12Þ

On the other hand, the formula P

ðÞ² ¼0 is equivalent to X

ðÞ ¼0;

becauseP

ðÞis orthogonal to the structure vector field. From this, (4.12) is equivalent to

ðYÞX

²ðÞ ¼X

ðÞðYÞ ¼0

for anyY2D. By virtue of this formula (4.11) is also equivalent to

2D or 2D^?: ð4:13Þ

Accordingly, by Lemma 5 we know that the constancy of the functionimplies the formula (4.13). Moreover, conversely, by Lemma 3 we are able to see that (4.13) implies that the functionis constant. Now we summarize this content as follows:

Theorem 4.1.Let M be a real hypersurface in G₂ðC^mþ2Þsatisfying the formula AþA¼k;

where the function k is non-zero and constant.Then the following are equivalent to each other

(1) the mean curvature is constant, (2) the function is constant, (3) 2Dor2D^?.

By virtue of this theorem we also assert the following

Theorem 4.2. Let M be a real hypersurface in G2ðC^mþ2Þwith constant mean curvature satisfying the formula

AþA¼k;

where the function k is non-zero and constant.Then we have the following (1) The structure vector field is principal,

(2) The function is constant, (3) 2Dor2D^?.

5. Proof of the Main Theorem

LetMbe a real hypersurface in a two-plane complex GrassmanniansG₂ðC^mþ2Þ with constant mean curvature. Now let us denote by H the orthogonal compo- nent of the structure vector in the tangent space of M in G₂ðC^mþ2Þ. Then by Theorem 4.2 let us consider the following two cases:

Now we consider the first case2D^?. In this case we may put¼1. Then by Proposition 4 we have for anyX2H¼ ½^?

2A²X2kAXþ ðkþ2ÞX2X

fðXÞðXÞþðÞXg ¼0:

ð5:1Þ From this formula we are able to assert the following

Proposition 5.1. Let M be a real hypersurface in G2ðC^mþ2Þ satisfying the formula ðÞ with constant mean curvature. Then the principal curvature is constant and for all X2H with AX ¼ X one of the following two statements holds:

(1) 2 ²2k þk¼0andDX¼ 1DX, (2) 2 ²2k þ ðkþ4Þ ¼0and₁DX¼DX.

Proof. In order to prove this Proposition we use the formulas in (2.1) to the formula (5.1). Then for any principal vectorX2Hsuch thatAX¼ Xthe equation (5.1) can be given by

f2 ²2k þ ðkþ2ÞgXþ4f2ðXÞ2þ₃ðXÞ3g 2₁X¼0: ð5:2Þ Now we decompose the vectorX2H as follows:

X¼DXþ₂ðXÞ₂þ₃ðXÞ₃;

where DX denotes the D component of the vector X2H. Then by the formula (2.1) again we have

₁X¼₁DXþ₂ðXÞ2þ₃ðXÞ3: From this, together with (5.2) it follows that

f2 ²2k þkþ2gDXþ f2 ²2k þkþ4g₂ðXÞ₂ þ f2 ²2k þkþ4g3ðXÞ22₁DX¼0:

From this, together with the fact that₁DDwe have the following f2 ²2k þ ðkþ2ÞgDX2₁DX¼0;

f2 ²2k þ ðkþ4Þg2ðXÞ2¼0;

f2 ²2k þ ðkþ4Þg3ðXÞ3¼0:

If 2 ²2k þ ðkþ4Þ ¼0, from the first equation we know DX¼ 1DX, that is,₁DX¼DX. Thus we assert the formula (2) in our Proposition.

Now when we consider 2 ²2k þ ðkþ4Þ 6¼0, then ₂ðXÞ ¼₃ðXÞ ¼0.

This means X2D. Then by the first equation we know that ₁DX andDXare proportional. From this we have

₁DX¼ DX:

If 1DX¼ DX, then ð2 ²2k þkþ4ÞDX¼0, which makes a contra- diction. So ₁DX¼DX, that is DX¼ ₁DX. Then we have our assertion (1). From this we complete the proof of our Proposition. &

In this section we have assumed that the mean curvature ofMinG₂ðC^mþ2Þis constant. Then by Theorem 4.2 we know that the functionis constant. Accord- ingly, all principal curvatures satisfying the formulas (1) and (2) in Proposition 5.1 are constant. Also by virtue of these two formulas the number of principal curva- tures in the subspaceHis at most four. Since the functionkis given by 2as in the introduction, the formulas in Proposition 5.1 can be written by

2k þ¼0 ð5:3Þ

and

2k þþ2¼0: ð5:4Þ

In (5.3) the function k¼2 is given by the sum of two roots of the quadratic equation. Then it follows that two roots are equal to each other, that is¼. By

virtue of this fact we also know that there cannot exist any roots satisfying the formula (5.4). So we are able to assert that H¼T, where T¼ fX2HjAX¼Xg.

Since we know that the structure vectoris principal with principal curvature , we assert that M is locally congruent to a totally umbilic hypersurface in G₂ðC^mþ2Þ. But in a paper [10] due to the present author it is proved that there does not exist such a real hypersurface in G2ðC^mþ2Þ. So we conclude here that the first case2D^? cannot appear.

Now let us consider the second case2D. Then by Lemma 2 for anyX2H andA¼, we have

0¼ ð2 ÞAX ð2þ ÞX2X

fðXÞþðXÞg; ð5:5Þ where H¼ ½1; ₂; ₃; ₁; ₂; ₃ G and G is the orthogonal complement of the subspace½1;. . .;. . .; ₃ inH. Then any vectorX2H can be expressed by

X¼GXþX

ðXÞX

ðXÞ;

whereGXdenotes theG-component of the vectorX2H. IfAX ¼ X, then by the assumption ðAþAÞX¼kX we know that AX¼ ðk ÞX. From this, together with (5.5) we have

0¼ fð2 Þðk Þ ð2þ ÞgX2X

fðXÞþðXÞg: ð5:6Þ From this, multiplying and using¼, ¼1;2;3, we have

f2 ²2k þ ðkþ2ÞgGXþ f2 ²2k þ ðkþ2Þ þ2gX

ðXÞ

f2 ²2k þ ðkþ2Þ þ2gX

ðXÞ ¼0;

where we have used the above decomposition for the expression ofX2H. Accord- ingly, we are able to assert the following:

f2 ²2k þ ðkþ2ÞgGX¼0;

f2 ²2k þ ðkþ4ÞgðXÞ ¼0; ¼1;2;3 f2 ²2k þ ðkþ4ÞgðXÞ ¼0; ¼1;2;3:

ð5:7Þ

From these equations we know that if 2 ²2k þ ðkþ4Þ 6¼0, then the vectorX is orthogonal to and for any¼1;2;3. Then naturallyX¼GX. From this fact we know that all of principal curvatures corresponding to eigenspaces in the spaceH satisfy one of the following equations:

2 ²2k þ ðkþ2Þ ¼0 or 2 ²2k þ ðkþ4Þ ¼0:

On the other hand, by Theorem 4.2 the functions and k are known to be constant. From this together with the above equation all of the principal curvatures are constant.

Now without loss of generality we may put¼ 2 tan 2rand ¼ cotrfor a real numberrwith 0<r<₄. Then by (5.6) (also by Lemma 2), we know for any X2GwithAX¼ Xthat

AX¼X; ¼ þ2

2 : Then the function¼ tanr. So it follows that

k¼ þ¼ cotr tanr¼2 cot 2r;

which impliesk¼ 4. Then its principal curvatures in H satisfy

2k 1¼0 or ²k ¼0:

Then, including principal curvature the real hypersurface has at most five dis- tinct constant principal curvatures. Then by the above formulas and the quadratic equations the other possible principal curvatures are

¼2 cot 2r; ¼0; ¼ cotr; ¼ tanr:

Note that the principal curvature andare two different roots of the equation 2x²2kxþ ðkþ2Þ ¼0;

wherek¼2 cot 2r.

A basic role in the geometry of Riemannian symmetric space is played by the so-called maximal flats. In the case of G2ðC^mþ2Þ, a maximal flat is a two- dimensional totally geodesic submanifold isometric to some flat two-dimensional torus. A non-zero tangent vector X of G₂ðC^mþ2Þ is said to be singular if X is tangent to more than one maximal flat of G₂ðC^mþ2Þ. InG₂ðC^mþ2Þthere are two types of singular tangent vectors X which are characterized by the properties JX?JX and JX2JX. We will have to compute explicitly Jacobi vector fields along geodesics whose tangent vectors are all singular. For this we need the eigen- values and eigenspaces of the Jacobi operatorRR_X:¼RRð:; XÞX, whereRRdenotes the curvature tensor ofG₂ðC^mþ2Þmentioned in Section 1. IfJN?JN then the eigen- values and eigenspaces ofRR_N are given by (see Berndt and Suh [4])

0 RNJJN ¼N ½1; ₂; ₃;

1 ðHCNÞ^? ¼ ½N; ; 1; 2; 3; 1; 2; 3^?; 4 RJNJN ¼R ½1; ₂; ₃;

ð5:8Þ

whereHCN ¼RNRJNJNJJN and½. . .;. . .;. . .denotes the linear real span of the given vectors.

For p2M denote by cp the geodesic in G2ðC^mþ2Þ with cpð0Þ ¼p and _

c

c_pð0Þ ¼N_p, and byF the smooth map

F:M!G₂ðC^mþ2Þ; p7!c_pðrÞ:

Geometrically,Fis the displacement ofMat distancerin direction of the normal fieldN. For eachp2Mthe differentiald_pFofFatpcan be computed by means of Jacobi vector fields by

dpFðXÞ ¼ZXðrÞ:

Here, Z is the Jacobi vector field along c_p with initial values Z_Xð0Þ ¼X and Z_X⁰ð0Þ ¼ AX. Using the explicit descriptions (5.8) of the Jacobi operator RR_N for the caseJN ?JN mentioned above and of the shape operatorAofM we get

ZXðrÞ ¼

fcosð2rÞ ₂ sinð2rÞgEXðrÞ if 2 f; g fcosðrÞ sinðrÞgEXðrÞ if 2 f ; g f1gEXðrÞ if 2 fg 8<

:

where E_X denotes the parallel vector field alongc_p withE_Xð0Þ ¼X. This shows that the kernel of dF is TT ¼JNT and that F is of constant rank dimðTTTÞ ¼4n. So, locally, F is a submersion onto a 4n-dimensional submanifoldBofG₂ðC^mþ2Þ. Moreover, the tangent space ofBatFðpÞis obtained by parallel translation of ðTTTÞðpÞ ¼ ðHTÞðpÞ, which is a quater- nionic and real subspace ofTpG2ðC^mþ2Þ.

Since bothJandJare parallel alongcp, alsoT_FðpÞBis a quaternionic and real subspace of T_FðpÞG2ðC^mþ2Þ. Thus B is a quaternionic and real submanifold of G₂ðC^mþ2Þ. Since B is quaternionic, it is totally geodesic in G₂ðC^mþ2Þ (see Alekseevski [1]). The only quaternionic totally geodesic submanifolds of G₂ðC^mþ2Þ,m¼2n54, of half dimension areG₂ðC^nþ2ÞandHPⁿ (see Berndt [3]).

But only HPⁿ is embedded inG₂ðC^mþ2Þ as a real submanifold. So we conclude thatBis an open part of a totally geodesic HPⁿ inG₂ðC^mþ2Þ. Rigidity of totally geodesic submanifolds finally implies that M is an open part of the tube with radius r around a totally geodesicHPⁿ in G₂ðC^mþ2Þ. Thus we have proved our

main Theorem. &

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Birkh€aauser

Author’s address: Y. J. Suh, Department of Mathematics, Kyungpook National University, Taegu 702-701, Korea, e-mail: yjsuh@mail.knu.ac.kr