Real Hypersurfaces in Complex Two-Plane Grassmannians

Mh. Math. 127, 1±14 (1999)

Real Hypersurfaces in Complex Two-Plane Grassmannians

By

JuÈrgen Berndt^1; andYoung Jin Suh^2;

1University of Hull, United Kingdom

2Kyungpook National University, Taegu, South Korea (Received 13 November 1996; in revised form 3 March 1997)

Abstract.The complex two-plane GrassmannianG2…C^m‡2†in equipped with both a KaÈhler and a quaternionic KaÈhler structure. By applying these two structures to the normal bundle of a real hypersurfaceMinG2…C^m‡2†one gets a one- and a three-dimensional distribution onM. We classify all real hypersurfacesMinG2…C^m‡2†,m53, for which these two distributions are invariant under the shape operator ofM.

1. Introduction

In Riemannian geometry, the general problem in submanifold theory is to determine or describe the submanifolds of a given Riemannian manifold which satisfy certain geometrical data. Submanifold theory in Euclidean spaces is classical, in spaces of constant curvature well-established. In more general ambient spaces one encounters two basic problems. On the one hand, the fundamental equations for submanifolds become rather complicated and are dif®cult to handle.

On the other hand, it is not easy to decide what data are natural to consider, they should somehow be related to the geometrical structure of the ambient space.

To give an example, consider a real hypersurface M in a complex projective spaceCP^m. The Codazzi equation is rather complicated in the generic case, but simpli®es considerably when requiring that the one-dimensional distribution J…?M†onMobtained by applying the KaÈhler structure JofCP^m to the normal bundle ?M of M is invariant under the shape operator A of M. The condition AJ…?M† J…?M† appears to be rather natural, and in fact there is a well- established theory for such hypersurfaces. Any tube around a complex sub- manifold inCP^m satis®es this geometrical condition. T. E. CECIL and P. J. RYAN

proved in [4] that these tubes are essentially characterized by this feature. Here the word essentially refers to some additional condition on the focal map. So, roughly speaking, the theory of real hypersurfaces inCP^mwithAJ…?M† J…?M†is the theory of tubes around complex submanifolds inCP^m.

1991 Mathematics Subject Classi®cation: 53B25; 53C15, 53C35

Key words: Complex Grassmannians, real hypersurfaces, tubes, shape operator, KaÈhler structure, quaternionic KaÈhler structurePartially supported by DFG Grant Be 1811/1-1Partially supported by BSRI-97-1404 and TGRC-KOSEF

The analogous question in the quaternionic projective space HP^m leads to a surprise. The corresponding geometrical feature is that the three-dimensional distribution J…?M†on M, which is obtained by applying the almost Hermitian structures in the quaternionic KaÈhler structureJofHP^mto?M, is invariant under A. In fact, every tube around a quaternionic submanifold of HP^m has this geometrical feature. (Note that by a result of D. V. ALEKSEEVSKII [1] such a quaternionic submanifold is necessarily totally geodesic.) But the converse is not true. The ®rst author proved in [2] that also every tube around a totally geodesic CP^m inHP^m satis®es AJ…?M† J…?M†, and that there are no other ones. So the real hypersurfaces inHP^m withAJ…?M† J…?M† are precisely the tubes around totally geodesicHP^k;k2 f0;. . .;mÿ1g, andCP^m.

In this paper we study the analogous question in the complex Grassmann manifold G2…C^m‡2† of all two-dimensional linear subspaces in C^m‡2. This Riemannian symmetric space has a remarkable geometrical structure. It is the unique compact, KaÈhler, quaternionic KaÈhler manifold with positive scalar curvature. So, inG2…C^m‡2†we have the two natural geometrical conditions for real hypersurfaces that J…?M† and J…?M† are invariant under the shape operator.

The main result of this paper is the classi®cation of all real hypersurfaces in G2…C^m‡2†satisfying both conditions.

Theorem 1.Let M be a connected real hypersurface in G2…C^m‡2†;m53. Then both J…?M†andJ…?M†are invariant under the shape operator of M if and only if (1) M is an open part of a tube around a totally geodesic G2…C^m‡1† in G2…C^m‡2†, or

(2) m is even, say mˆ2n, and M is an open part of a tube around a totally geodesicHPⁿ in G2…C^m‡2†:

Any tube aroundG2…C^m‡1†has four (resp. three for a particular radius) distinct constant principal curvatures and might also be regarded as a tube around the focal set ofG2…C^m‡1†in G2…C^m‡2†, which is a totally geodesicCP^m. Any tube around HPⁿ has ®ve distinct constant principal curvatures, and the other focal set of the tube is a complex hypersurface inG2…C^m‡2†which is a Riemannian homogeneous space isomorphic to Sp…n‡1†=…U…2† Sp…nÿ1††. The two families of tubes together with their focal sets are just the orbits of the isometric actions of the subgroups S…U…1† U…m‡1††and Sp…n‡1†of SU…m‡2†, respectively.

In [3] the ®rst author proved that any tube as in Theorem 1 satis®es the geometrical hypotheses in that theorem. So in this paper we are concerned with the proof of the converse statement, which is much more complicated. The paper is organized as follows. In Section 2 we recall basic facts aboutG2…C^m‡2†. In Section 3 we study thoroughly the Codazzi equation for real hypersurfaces inG2…C^m‡2†.

By using the hypotheses this leads to two cases which will be studied separately in Sections 4 and 5.

2. Riemannian Geometry ofG2…C^m‡2†

In this section we summarize basic material about G2…C^m‡2†, for details we refer to [3]. ByG2…C^m‡2†we denote the set of all complex two-dimensional linear

subspaces inC^m‡2. The special unitary groupGˆSU…m‡2†acts transitively on G2…C^m‡2† with stabilizer isomorphic to K ˆS…U…2† U…m†† G. Then G2…C^m‡2† can be identi®ed with the homogeneous space G=K, which we equip with the unique analytic structure for which the natural action ofGonG2…C^m‡2† becomes analytic. Denote bygandkthe Lie algebra ofGandK, respectively, and bymthe orthogonal complement ofkingwith respect to the Cartan-Killing form Bofg. Thengˆkmis an Ad(K)-invariant reductive decomposition of g. We put oˆeK and identify ToG2…C^m‡2† with m in the usual manner. Since B is negative de®nite ong, its negative restricted to mmyields a positive definite inner product onm. By Ad(K)-invariance ofBthis inner product can be extended to a G-invariant Riemannian metric g on G2…C^m‡2†. In this way G2…C^m‡2† becomes a Riemannian homogeneous space, even a Riemannian symmetric space.

For computational reasons we normalize g such that the maximal sectional curvature of …G2…C^m‡2†;g† is eight. Since G2…C³† is isometric to the two- dimensional complex projective space CP² with constant holomorphic sectional curvature eight we will assume m52 from now on. Note that the isomorphism Spin…6† ' SU…4† yields an isometry between G2…C⁴† and the real Grassmann manifoldG^‡₂…R⁶†of oriented two-dimensional linear subspaces ofR⁶.

The Lie algebrak has the direct sum decomposition kˆsu…m† su…2† R, whereR is the center of k. Viewingk as the holonomy algebra ofG2…C^m‡2†, the center R induces a KaÈhler structure J and the su(2)-part a quaternionic KaÈhler structure J on G2…C^m‡2†. If J1 is any almost Hermitian structure in J, then J J1 ˆJ1J, and J J1 is a symmetric endomorphism with …J J1†²ˆI and tr…J J1† ˆ0. 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‡1 ˆJ‡2ˆ ÿJ‡1J, where the index is taken modulo three. SinceJis parallel with respect to the Riemannian connectionr of …G2…C^m‡2†;g†, there exist for any canonical local basisJ1;J2;J3 ofJthree local one-formsq1;q2;q3 such that

rXJ ˆq‡2…X†J‡1ÿq‡1…X†J‡2

for all vector ®eldsXonG2…C^m‡2†. Also this fact will be used frequently.

Let p2G2…C^m‡2† and W a subspace of TpG2…C^m‡2†. We say that W is a quaternionic subspace ofTpG2…C^m‡2†ifJ1W W for all J1 2 J_p. And we say thatW is a real subspace of TpG2…C^m‡2† ifJW ? W. A quaternionic resp. real submanifold of G2…C^m‡2† is a submanifold all of whose tangent spaces are quaternionic resp. real subspaces of the corresponding tangent spaces of G2…C^m‡2†.

The Riemannian curvature tensorR ofG2…C^m‡2†is locally given by R…X; Y†Zˆg…Y;Z†Xÿg…X;Z†Y‡g…JY;Z†JXÿg…JX;Z†JYÿ2g…JX;Y†JZ

‡X³

ˆ1

…g…JY;Z†JXÿg…JX;Z†JYÿ2g…JX;Y†JZ†

‡X³

ˆ1

…g…JJY;Z†JJXÿg…JJX;Z†JJY†;

whereJ1;J2;J3 is any canonical local basis ofJ. A non-zero tangent vectorXof G2…C^m‡2†is said to be singular ifXis tangent to more than one ¯at ofG2…C^m‡2†:

In G2…C^m‡2† there are two types of singular tangent vectors X which are characterized by the propertiesJX?JXandJX2JX. We will have to compute explicitly Jacobi vector ®elds along geodesics whose tangent vectors are all singular. For this we need the eigenvalues and eigenspaces of the Jacobi operator RX :ˆR…; X†X. LetXbe a unit vector tangent toG2…C^m‡2†. IfJX?JXthen the eigenvalues and eigenspaces ofRX are

0 RXJJX 1 …HCX†^? 4 RJXJX;

where HCXˆRXRJXJXJJX. If JX 2 JX, there exists an almost Hermitian structure J1 in J such that JXˆJ1X. Then the eigenvalues and eigenspaces ofRX are

0 RX fYjY ?HX;JYˆ ÿJ1Yg 2 C^?X fYjY?HX;JYˆJ1Yg 8 RJX;

whereCXandHXdenote the complex and quaternionic span ofX, respectively, andC^?X is the orthogonal complement ofCXin HX.

3. The Codazzi Equation for Real Hypersurfaces inG2…C^m‡2† In this section we derive some basic formulae from the Codazzi equation for a real hypersurface inG2…C^m‡2†.

LetMbe a real hypersurface ofG2…C^m‡2†, that is, a hypersurface ofG2…C^m‡2† with real codimension one. The induced Riemannian metric on M will also be denoted by g, and r denotes the Riemannian connection of …M;g†. Let N be a local unit normal ®eld ofMandAthe shape operator ofMwith respect toN. The KaÈhler structureJ ofG2…C^m‡2† induces onM an almost contact metric structure …; ; ;g†. Furthermore, letJ1;J2;J3 be a canonical local basis ofJ. Then eachJ

induces an almost contact metric structure…; ; ;g† onM. Using the above expression forR, the Codazzi equation becomes

…rXA†Yÿ …rYA†Xˆ…X†Yÿ…Y†Xÿ2g…X;Y†

‡X³

ˆ1

……X†Yÿ…Y†Xÿ2g…X;Y††

‡X³

ˆ1

……X†Yÿ…Y†X†

‡X³

ˆ1

……X†…Y† ÿ…Y†…X††:

The following identities can be proved in a straightforward manner and are used frequently in subsequent calculations:

‡1 ˆ ÿ‡2; ‡1ˆ‡2; ˆ ; …X† ˆ…X†:

LetDbe the maximal quaternionic subbundle of the tangent bundleTMofMand D^?the orthogonal complement ofDinTM. Now we assume thatAD D. Then D^?is also invariant underA, and by a suitable choice ofJ1;J2;J3the vector ®elds are principal curvature vectors everywhere, sayA ˆ forˆ1;2;3. The Codazzi equation then implies

2…X†…Y† ÿ2…Y†…X† ÿ2g…X;Y†…†

‡2‡1…X†‡2…Y† ÿ2‡1…Y†‡2…X† ÿ2g…X;Y†

‡2‡1…X†‡2…Y† ÿ2‡1…Y†‡2…X†

ˆg……rXA†Yÿ …rYA†X; †

ˆg……rXA†;Y† ÿg……rYA†;X†

ˆ …X†…Y† ÿ …Y†…X†

‡ …ÿ‡1†…q‡2…X†‡1…Y† ÿq‡2…Y†‡1…X††

ÿ …ÿ‡2†…q‡1…X†‡2…Y† ÿq‡1…Y†‡2…X††

‡g……A‡A†X;Y† ÿ2g…AAX;Y†:

PuttingXˆ in this equation yields

Y ˆ…†…Y† ‡ …ÿ‡1†q‡2…†‡1…Y† ÿ …ÿ‡2†q‡1…†‡2…Y†

ÿ4…†…Y† ‡2…‡1†‡1…Y† ‡2…‡2†‡2…Y†:

Inserting this and the corresponding equation forXinto the previous one implies 2…X†…Y† ÿ2…Y†…X† ÿ2g…X;Y†…†

‡2‡1…X†‡2…Y† ÿ2‡1…Y†‡2…X† ÿ2g…X;Y†

‡2‡1…X†‡2…Y† ÿ2‡1…Y†‡2…X†

ˆ2…‡1†…‡1…X†…Y† ÿ‡1…Y†…X††

‡2…‡2†…‡2…X†…Y† ÿ‡2…Y†…X††

‡4…†……Y†…X† ÿ…X†…Y††

‡ …ÿ‡1†…q‡2…†…‡1…X†…Y† ÿ‡1…Y†…X††

‡q‡2…X†‡1…Y† ÿq‡2…Y†‡1…X††

ÿ …ÿ‡2†…q‡1…†…‡2…X†…Y† ÿ‡2…Y†…X††

‡q‡1…X†‡2…Y† ÿq‡1…Y†‡2…X††

‡g……A‡A†X;Y† ÿ2g…AAX;Y†:

This implies

Lemma 1. If A ˆ and X 2 D with AXˆX, then 0ˆ…2ÿ†AXÿ …2‡†X

ÿ …2‡1…†‡1…X† ‡2‡2…†‡2…X† ÿ4…†…X††

ÿ …ÿ‡1†q‡2…X†‡1‡ …ÿ‡2†q‡1…X†‡2

ÿ2…†Xÿ2…X†ÿ2…X†‡2‡2…X†‡1ÿ2‡1…X†‡2: Now we assume that Aˆ instead of AD D and denote by ^? the orthogonal complement of the real span ofin TM. Taking inner product of the Codazzi equation withyields

ÿ2g…X;Y† ‡2X³

ˆ1

……X†…Y† ÿ…Y†…X† ÿg…X;Y†…††

ˆg……rXA†Yÿ …rYA†X; †

ˆg……rXA†;Y† ÿg……rYA†;X†

ˆ …X†…Y† ÿ …Y†…X† ‡g……A‡A†X;Y† ÿ2g…AAX;Y†:

PuttingXˆ implies

Yˆ …†…Y† ÿ4X³

ˆ1

…†…Y†:

Inserting this and the corresponding equation for X into the previous equation gives

ÿ2g…X;Y† ‡2X³

ˆ1

……X†…Y† ÿ…Y†…X† ÿg…X;Y†…††

ˆ4X³

ˆ1

……X†…Y† ÿ…Y†…X††…† ‡g……A‡A†X;Y†

ÿ2g…AAX;Y†:

From this we easily derive

Lemma 2. If Aˆand X 2^? with AXˆX, then 0ˆ…2ÿ†AXÿ …2‡†X

‡2X³

ˆ1

…2…†…X†ÿ…X†ÿ…X† ÿ…†X†:

Next, we assume again AD D. The ‡1-component of the equation in Lemma 1 gives

0ˆ …ÿ‡1†q‡2…X† ‡4…†‡1…X† ‡2…‡1†…X† ÿ2…‡2†…X†;

whereas the‡2-component leads to

0ˆ …ÿ‡2†q‡1…X† ÿ4…†‡2…X† ÿ2…‡2†…X† ÿ2…‡1†…X†:

Both equations hold for any index. Raising the index of the second equation by one and then combining with the ®rst equation yields

2…‡2†…X† ˆ…†‡1…X† ÿ…‡1†…X†:

We now assume in addition thatis a principal curvature vector ofMeverywhere, sayAˆ. We may write

ˆ…X†X‡…Z†Z

with suitable unit vector ®eldsX2DandZ2D^?. SinceDandD^? are invariant underA, we have AX2DandAZ2D^?. Thus

…X†X‡…Z†ZˆˆAˆ…X†AX‡…Z†AZ

implies …X† ˆ0, …Z† ˆ0, or …AXˆX and AZ ˆZ†. Suppose that the last possibility holds. Without loss of generality we may assume that the canonical local basis is chosen in such a way thatZˆ3. Then…1† ˆ0ˆ…2†, and we get

2…Z†…X† ˆ2…3†…X† ˆ…1†2…X† ÿ…2†1…X† ˆ0:

Therefore we necessarily have…X† ˆ0 or…Z† ˆ0. But this just means thatis either a vector ®eld inD^? or inD. We summarize this in

Proposition 1.Let M be a real hypersurface of G2…C^m‡2†;m53. Suppose that AD Dand Aˆ. Then is tangent toDor to D^? everywhere.

From now on we assume that M is a connected real hypersurface of G2…C^m‡2†;m53, withAD DandAˆ. According to Proposition 1 we have to consider two cases. This will be done separately in the next two sections.

4. The Case:Is Tangent toD

In this section we assume that is tangent toD. Then the unit normalN is a singular tangent vector ofG2…C^m‡2†of typeJN ?JN and the vectors; 1; 2; 3; 1; 2; 3 are orthonormal.

Putting Xˆ in Lemma 2, yields 0ˆ …2ÿ†Aÿ …4‡†. If 2 ˆ;then 0ˆ4‡ ˆ4‡2², which is impossible. Hence 2ÿ6ˆ0.

InsertingXˆin Lemma 1 gives

0ˆ …2ÿ†Aÿ …4‡†ÿ …ÿ‡1†q‡2…†‡1

‡ …ÿ‡2†q‡1…†‡2:

Sinceˆ is perpendicular to‡1 and‡2, we may now conclude Lemma 3. For each we have

A ˆ with ˆ4‡

2ÿˆ4‡

2ÿ: In particular, we have either4‡ ˆ0orˆ:

Next, from Lemma 1, we obtain withXˆˆthe equation 0ˆ…2ÿ†Aÿ …2‡†

ÿ …ÿ‡1†q‡2…†‡1‡ …ÿ‡2†q‡1…†‡2

‡2;ÿ2‡2;‡1‡2‡1;‡2:

Putting ˆ this implies 0ˆ …2ÿ†Aÿ …4‡†. If ˆ2, then 0ˆ4‡ ˆ4‡2², which is impossible. Thus, comparing withAˆwe get

Lemma 4. For each we have

ˆ4‡

2ÿ:

Taking as indexˆ‡1 we obtain 0ˆ …2‡1ÿ†A‡2ÿ‡1‡2: If ˆ2‡1, then 0ˆ‡1 and therefore ˆ0. Lemma 3 and ˆ0 impliesˆÿ=2 and henceˆ0, which contradictsˆ2= 6ˆ0 in Lemma 4.

Thus, we have 2‡1ÿ 6ˆ0 and therefore Lemma 5. For each we have

‡2ˆ ‡1

2‡1ÿ:

Using successively Lemma 3, Lemma 5, and again Lemma 3, we derive 4‡‡2

2‡2ÿˆ‡2 ˆ ‡1

2‡1ÿ ˆ 4‡‡1

8‡‡2‡1ÿ2‡1

and therefore

…²ÿ8†…‡1‡‡2† ‡2²‡1‡2

ˆ4‡1‡2ÿ8…‡‡1‡‡2† ÿ32:

This formula holds for any index. Subtracting the corresponding equation obtained by replacingby‡1 gives 0ˆ …²‡8†…‡1ÿ†‡2. As above,‡2ˆ0 leads to a contradiction. Therefore, 1; 2; 3 must be equal, say 1ˆ2 ˆ3 ˆ:. Lemma 3 shows that also 1; 2; 3 are equal, say 1ˆ2ˆ3ˆ:. From Lemma 5 we then get…ÿ† ˆ0. First we suppose thatˆ. According to Lemma 3 we have eitherˆ or 4‡ˆ0. In the

®rst case we haveˆˆ, which contradicts Lemma 4. In the second case we get ˆˆ0 from Lemma 3, which contradicts 4‡ˆ0. Therefore 6ˆ

and henceˆ0:We summarize this in

Lemma 6. For each we have A ˆ and A ˆ0, where is determined by4‡ˆ0:

Next, letXbe a principal curvature vector ofMorthogonal toRJJJ, sayAXˆX. Note thatXorthogonal toRJJJjust meansX 2 …HC†^?. Then Lemma 1 implies for eachthe equation 0ˆ …2ÿ†AXÿ…2‡†X:

If ˆ2, then 0ˆ2‡ˆ2‡2², which is impossible. Hence 2ÿ6ˆ0

and we obtainAjJXˆidJX with ˆ^2‡_2ÿ. ReplacingX by, for instance, 1X yields AjJ1Xˆ^2‡_2ÿidJ1X. Since 2X2JX\J1X, we get ˆ^2‡_2ÿ; and sinceX2J1Xthis implies ˆand henceˆ^2‡_2ÿ:From this we conclude

Lemma 7.Aj…HC†^? has at most two distinct eigenvalues, each of which is a solution of x²ÿxÿ1ˆ0. The corresponding eigenspaces are quaternionic.

For X as above we derive from Lemma 2 the equation 0ˆ …2ÿ†AXÿ ÿ…2‡†X. One easily sees that 2ÿ6ˆ0, whence AXˆX with ˆ^2‡_2ÿ:By means of Lemma 7 we know that²ÿÿ1ˆ0. Ifˆ, then the preceding formula implies²ÿÿ1ˆ0, and so6ˆ0 andˆ, which contradicts 0ˆ4‡in Lemma 6. Therefore we must have6ˆ, and together with Lemma 7 we conclude

Lemma 8. Aj…HC†^? has two distinct eigenvalues and , which are the solutions of x²ÿxÿ1ˆ0. The corresponding eigenspaces T and T are quaternionic and real, that is,

JTˆT; JTˆT; JTˆT:

In particular, the quaternionic dimension m of G2…C^m‡2†is even, say mˆ2n.

Since is tangent to D, we know from a formula obtained for the proof of Lemma 2 that

gradˆ …†:

So for the Hessian ofwe get

hess…X;Y† ˆg…rXgrad;Y† ˆg…rX……††;Y†

ˆ…X…††…Y† ‡ …†g…rX;Y†

ˆ…X…††…Y† ‡ …†g…AX;Y†:

By symmetry of the Hessian this implies

0ˆ …X…††…Y† ÿ …Y…††…X† ‡ …†g……A‡A†X;Y†:

For Xˆ this gives Y…† ˆ ……††…Y†. Inserting this and the corresponding equation forXinto the previous one yields

0ˆ …†g……A‡A†X;Y†:

Suppose thatA‡Avanishes at some pointpofM. Then, atp, forX 2 …HC†^? withAXˆXwe getAXˆ ÿX. But, sinceis a solution ofx²ÿxÿ1ˆ0 and 6ˆ0, the number ÿ cannot be a solution of that equation also, which contradicts Lemma 8. Hence we must haveA‡A6ˆ0 at each point ofM. This yieldsˆ0 and hence grad ˆ0, that is, is constant onM. It follows from Lemmata 6 and 8 that all principal curvatures ofMare constant. Since 0ˆ4‡ we have 6ˆ0. Without loss of generality we may assume >0. Then there exists some r 2Š0; =4‰ so that ˆ2 cot…2r†. Then 0ˆ4‡ gives ˆ ÿ2 tan…2r†. The solutions of x²ÿxÿ1ˆ0 are then ˆ cot…r† and ˆ ÿtan…r†. We summarize this discussion in

Proposition 2.Let M be a connected real hypersurface of G2…C^m‡2†. Suppose that AD D, Aˆ, andis tangent toD. Then the quaternionic dimension m of G2…C^m‡2† is even, say mˆ2n; and M has ®ve distinct constant principal curvatures

ˆ ÿ2 tan…2r†; ˆ2 cot…2r†; ˆ0; ˆcot…r†; ˆ ÿtan…r†

with some r2Š0; =4‰. The corresponding multiplicities are

m…† ˆ1;m…† ˆ3ˆm…†;m…† ˆ4nÿ4ˆm…† and the corresponding eigenspaces are

TˆR ;TˆJJ ;TˆJ ;T;T; where

TTˆ …HC†^?; JTˆT;JTˆT; JT ˆT:

For p 2 M denote by cp the geodesic in G2…C^m‡2† with cp…0† ˆp and _

cp…0† ˆNp, and byFthe smooth map

F:M!G2…C^m‡2†; p7!cp…r†:

Geometrically,Fis the displacement ofMat distancerin direction of the normal

®eldN. For eachp2Mthe differentialdpFofFatpcan be computed by means of Jacobi vector ®elds by

dpF…X† ˆZX…r†:

Here, ZX is the Jacobi vector ®eld along cp with initial values ZX…0† ˆX and Z_X⁰…0† ˆ ÿAX. Using the explicit descriptions of the Jacobi operator RN for the caseJN ?JN in Section 2 and of the shape operatorAofMin Proposition 2 we get

ZX…r† ˆ cos…2r† ÿ 2 sin…2r†

EX…r†; ifX 2 Tand 2 f; g …cos…r† ÿsin…r††EX…r†; ifX 2 Tand 2 f; g

EX…r†; ifX 2 T;

8>

<

>:

whereEXdenotes the parallel vector ®eld alongcpwith theEX…0† ˆX. This shows that the kernel of dF is TT ˆJNT and that F is of constant rank dim…TTT† ˆ4n. So, locally, F is a submersion onto a 4n-dimensional submanifoldBofG2…C^m‡2†. Moreover, the tangent space ofBatF(p) is obtained by parallel translation of …TTT†…p†ˆ…HT†…p†, which is a quater- nionic and real subspace ofTpG2…C^m‡2†. Since bothJandJare parallel alongcp, also TF…p†B is a quaternionic and real subspace of TF…p†G2…C^m‡2†. Thus B is a quaternionic and real submanifold ofG2…C^m‡2†. SinceBis quaternionic, it is totally geodesic inG2…C^m‡2†[1]. The only quaternionic totally geodesic submanifolds of G2…C^m‡2†;mˆ2n54, of half dimension are G2…C^n‡2† and HPⁿ [3]. But only HPⁿis embedded inG2…C^m‡2†as a real submanifold. So we conclude thatBis an open part of a totally geodesic HPⁿ in G2…C^m‡2†. Rigidity of totally geodesic

submanifolds ®nally implies thatMis an open part of the tube with radiusraround a totally geodesicHPⁿ in G2…C^m‡2†. Thus we have proved

Theorem 1A.Let M be a connected real hypersurface of G2…C^m‡2†;m53. If AD D; Aˆ, andis tangent toD, then mˆ2n and M is an open part of a tube around a totally geodesicHPⁿ in G2…C^m‡2†.

5. The Case: Is Tangent toD^?

We now assume thatis tangent toD^?. Then the unit normalN is a singular tangent vector ofG2…C^m‡2†of typeJN 2 JN. So there exists an almost Hermitian structureJ1 2 J such thatJN ˆJ1N. Then we have

ˆ1; ˆ1; 2 ˆ ÿ3; 3ˆ2; D D:

In the following the indices andwill be either 2 or 3 and distinct from each other.

Inserting Xˆ in the equation of Lemma 2 yields …2 ÿ†A ˆ

ˆ …4‡†: Since 2ÿˆ0 implies 0ˆ4‡ ˆ4‡2², which is impossible, we get 2ÿ6ˆ0 and A ˆ^4‡_2ÿ. Using Aˆ; 2ˆ ÿ3 and3 ˆ2 we obtainˆ^4‡_2ÿ and hence

Lemma 9. 223ÿ…2‡3† ÿ4ˆ0:

Next, letX 2 DwithA XˆX. TheD-component of the equation in Lemma 1 is 0ˆ …2ÿ†AXÿ …2‡†X. If 2ÿ ˆ0, then 0ˆ2‡ ˆ

ˆ2‡2², which is impossible. Thus we get Lemma 10. If X 2 Dwith AXˆX, then

2ÿ 6ˆ0 and AXˆX with ˆ2‡

2ÿ:

We now replaceXby2X. Then theD-component of the equation in Lemma 1 with index one is 0ˆ …223ÿ…2‡3† ÿ2†3Xÿ22X. Applying 2 to this equation yields 0ˆ …223ÿ…2‡3† ÿ2†1X‡2X. Thus we obtain

Lemma 11. If X 2 Dwith AXˆX, then either

Xˆ1X and 223ÿ…2‡3† ˆ0 or

Xˆ ÿ1X and 223ÿ…2‡3† ÿ4ˆ0:

Now supposeXsatis®esXˆ1X. Then theD-component of the equation in Lemma 1 with index one is…2ÿ†A1Xˆ …4‡†1X. If 2ÿˆ0, then 0ˆ4‡ˆ4‡2², which is impossible. Hence we obtain

Lemma 12. If X 2 Dwith AXˆX andXˆ1X, then 2ÿ6ˆ0 and A1Xˆ4‡

2ÿ1X:

From Lemma 10 we derive

A1XˆA23Xˆ …3†₂23Xˆ…4‡23† ‡2…2ÿ3† …4‡23† ÿ2…2ÿ3†1X and

A1Xˆ ÿA32Xˆ ÿ…2†₃32Xˆ…4‡23† ÿ2…2ÿ3† …4‡23† ‡2…2ÿ3†1X:

Comparing these two equations leads to 4‡23 ˆ0 or2 ˆ3. We ®rst assume 4‡23 ˆ0. Then Lemma 9 shows that 6ˆ0, and 2 and 3 are the two solutions of the quadratic equation x²‡12xÿ4ˆ0. If X satis®es Xˆ ÿ1X, then 2X satis®es 2Xˆ12X. Thus we may choose X so that Xˆ1X. From Lemma 12 and the subsequent equation we derive

ÿ1

1XˆA1Xˆ4‡ 2ÿ1X:

Therefore,is a solution of the quadratic equationx²‡6xÿˆ0. It follows that 2is a solution ofx²‡12xÿ4ˆ0 and must therefore coincide with2or 3, which is a contradiction to Lemma 10. Thus we have proved

Lemma 13. 2 ˆ3 ˆ::

Lemma 9 implies that is a solution of the quadratic equation x²ÿxÿ2ˆ0. Let X 2 D with AX ˆX. The equation after Lemma 12 shows thatA1Xˆ1X. So the eigenspaces ofAjDare1-invariant. Moreover, if X satis®es Xˆ1X, then Lemma 12 yields that is also a solution of x²ÿxÿ2ˆ0. On the other hand, from Lemmata 10 and 13 it follows2 ˆ3: Thus we may put:ˆ2ˆ3. According to Lemma 11,satis®es…ÿ† ˆ0 and hence 2 f0; g. If ˆ, then Lemma 10 implies that is a solution of x²ÿx‡2ˆ0, which is impossible forˆ0 and contradicts²ÿÿ2ˆ0 forˆ. Therefore,andare the distinct solutions ofx²ÿxÿ2ˆ0, that is

; 2 1

2 

²‡8

p

; 6ˆ:

Soˆ ÿ2, and using again Lemma 10 proves thatˆ0. We summarize this in Lemma 14.AjDhas exactly two distinct eigenvalues andwith the same multiplicities2mÿ2, and the corresponding eigenspaces T and T satisfy

T fXjX ? H ;JXˆJ1Xg;

T fXjX ? H ;JXˆ ÿJ1Xg:

We have ˆ0, and and are the distinct solutions of the equation x²ÿxÿ2ˆ0.

The above inclusions are in fact equalities, because all these sets are vector spaces of the same dimension. As in the case whereis tangent toDwe deduce in an analogous way

gradˆ …† and 0ˆ …†g……A‡A†X;Y†:

Suppose that A‡Aˆ0 at some point p 2 M. Then, at p, we use the 1- invariance of T to derive XˆAXˆ ÿAXˆ ÿX. This implies ˆ0, which is impossible by means of Lemma 14. Thus A‡A is non-zero everywhere and we conclude that ˆ0 everywhere. Thus is constant onM, and Lemma 14 implies thatM has constant principal curvatures.

From ˆ ÿ2 we see thatand have different sign. We may choose the unit normalNin such a way thatis positive, sayˆ 

p2

cot… 

p2

r†with some r 2Š0; = 

p8

‰. We use Lemma 14 to compute ˆ 

p8

cot… 

p8

r† and ˆ ÿ2=ˆ ÿ 

p2

tan… 

p2

r†. Altogether we have thus proved

Proposition 3.Let M be a connected real hypersurface of G2…C^m‡2†. Suppose that AD D;Aˆ, and is tangent to D^?. Let J1 2 J be the almost Hermitian structure such that JNˆJ1N. Then M has three…if rˆ=2 

p8

†or four (otherwise) distinct constant principal curvatures

ˆ 

p8

cot… 

p8

r†; ˆ 

p2

cot… 

p2

r†; ˆ ÿ 

p2

tan… 

p2

r†; ˆ0 with some r 2Š0; = 

p8

‰. The corresponding multiplicities are m…† ˆ1; m…† ˆ2;m…† ˆ2mÿ2ˆm…† and for the corresponding eigenspaces we have

TˆRˆRJN; T ˆC^?ˆC^?N;

TˆfXjX?H ;JXˆJ1Xg;

TˆfXjX?H ;JXˆ ÿJ1Xg:

We de®necp,F,ZXandEXas in the preceding section. In the present situation we get

ZX…r† ˆ

cos… 

p8

r† ÿ

8 p sin… 

p8 r†

EX…r†; ifX 2 T

cos… 

p2

r† ÿ 

p2sin… 

p2 r†

EX…r†; ifX 2 T and 2 f; g

EX…r†; ifX 2 T;

8>

>>

><

>>

>>

:

So the kernel of dF is T TˆRJN C^?NˆJN. Hence F is of constant rank dim…T T† ˆ4mÿ4 and locally a submersion onto a submanifoldBof G2…C^m‡2†. As T Tˆ …HN†^? is quaternionic, Bis a quaternionic hypersur- face ofG2…C^m‡2†and hence totally geodesic. The only quaternionic hypersurfaces in G2…C^m‡2†;m53, are open parts of a totally geodesic G2…C^m‡1† in G2…C^m‡2† [3]. As in the preceding section we may now conclude thatMis an open part of the tube with radiusraround a totally geodesicG2…C^m‡1†inG2…C^m‡2†, and we have proved

Theorem 1B. Let M be a connected real hypersurface of G2…C^m‡2†;m53. If AD D; Aˆ, and is tangent to D^?, then M is an open part of a tube around a totally geodesic G2…C^m‡1†in G2…C^m‡2†.

Theorem 1 now follows from Theorems 1A and 1B and Proposition 1.

References

[1] ALEKSEEVSKIIDV (1968) Compact quaternion spaces. Funct Anal Appl2: 106±114

[2] B^ERNDTJ (1991) Real hypersurfaces in quaternionic space forms. J Reine Angew Math419: 9±26 [3] BERNDTJ (1997) Riemannian geometry of complex two-plane Grassmannians. Rend Sem Mat

Univ Politec Torino55: 19±83

[4] CECILTE, RYANPJ (1982) Focal sets and real hypersurfaces in complex projective space. Trans Amer Math Soc269: 481±499

J. BERNDT Y. J. SUH

Department of Mathematics Department of Mathematics

University of Hull Kyungpook National University

Hull HU6 7RX Taegu 702-701

United Kingdom South Korea