Advances in Applied Mathematics

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1 Advances in Applied Mathematics ) Contents lists available at SciVerse ScienceDirect Advances in Applied Mathematics Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians Young Jin Suh Kyungpook National University, Department of Mathematics, Taegu , Republic of Korea article info abstract Article history: Received 23 February 2012 Accepted 9 January 2013 Available online 23 January 2013 MSC: 53C40 53C15 We classify the real hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians SU 2,m /SU 2 U m ), m 2. Each can be described as a tube over a totally geodesic SU 2,m 1 /SU 2 U m 1 ) in SU 2,m /SU 2 U m ) or a horosphere whose center at infinity is singular Elsevier Inc. All rights reserved. Keywords: Real hypersurfaces Noncompact complex two-plane Grassmannians Isometric Reeb flow Totally geodesic Horosphere at infinity 0. Introduction The motivation of this paper is to classify all orientable real hypersurfaces M in almost Hermitian manifold M for which the Reeb flow is isometric. The almost Hermitian structure on almost Hermitian manifold M induces an almost contact metric structure on M. The corresponding unit tangent vector field on M is the Reeb vector field, and its flow is said to be the Reeb flow on M. A classical example is the anti-de Sitter sphere H 2m 1 1 in C m, where the orbits of the Reeb flow induce the Hopf foliation on H 2m 1 1 with principal S 1 -bundle of time-like totally geodesic fibres. It is well known that H 2m 1 1 is a principal S 1 -bundle over a complex hyperbolic space CH m with projection π : H 2m+1 1 CH m. Moreover, in a paper due to Montiel and Romero [10] it was proved This work was supported by grant Proj. Nos. NRF C00002 and BSRP , and partly by BSRP R1A2A2A from National Research Foundation. address: yjsuh@knu.ac.kr /$ see front matter 2013 Elsevier Inc. All rights reserved.

2 646 Y.J. Suh / Advances in Applied Mathematics ) that the second fundamental tensor A of a Lorentzian hypersurface in H 2m 1 1 is parallel if and only if ahypersurfaceinch m is with isometric Reeb flow, that is, φ A = Aφ, where π A = A, π A is called a pullback oftheshapeoperatora for a hypersurface in CH m by the projection π and φ denotes the structure tensor induced from the Kähler structure J of CH m. The classification of all real hypersurfaces in complex projective space CP m with isometric Reeb flow has been obtained by Okumura [11]. The corresponding classification in complex hyperbolic space CH m is due to Montiel and Romero [10] and in quaternionic projective space HP m due to Martinez and Pérez [9] respectively. Let us denote by SU 2,m /SU 2 U m ) the complex hyperbolic two-plane Grassmannian. It is a noncompact complex two-plane Grassmannian which consists of all complex two-dimensional linear subspaces in indefinite complex Euclidean space C m+2 2, where SU 2,m denotes the set of m + 2) m + 2) indefinite special unitary matrices, U 2 and U m the set of 2 2 and m m-unitary matrices respectively. Then it is known that SU 2,m /SU 2 U m ) has both a Kähler structure J and a quaternionic Kähler structure { J 1, J 2, J 3 }. Now the purpose of this paper is to classify all real hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmann manifold SU 2,m /SU 2 U m ) as follows: Main Theorem. Let M be a connected orientable real hypersurface in the complex hyperbolic two-plane Grassmannian SU 2,m /SU 2 U m ),m 3. Then the Reeb flow on M is isometric if and only if M is an open part of a tube around some totally geodesic SU 2,m 1 /SU 2 U m 1 ) in SU 2,m /SU 2 U m ) or a horosphere whose center at infinity is singular. A tube around SU 2,m 1 /SU 2 U m 1 ) in SU 2,m /SU 2 U m ) is a principal orbit of the isometric action of the maximal compact subgroup SU 1,m+1 of SU m+2, and the orbits of the Reeb flow corresponding to the orbits of the action of U 1. The action of SU 1,m+1 has two kinds of singular orbits. One is a totally geodesic SU 2,m 1 /SU 2 U m 1 ) in SU 2,m /SU 2 U m ) and the other is a totally geodesic CH m in SU 2,m /SU 2 U m ). A remarkable consequence of our Main Theorem is that a connected complete real hypersurface in SU 2,m /SU 2 U m ), m 3 with isometric Reeb flow is homogeneous. This was also true in complex two-plane Grassmannians G 2 C m+2 ), which could be identified with symmetric space of compact type SU m+2 /SU 2 U m ), as follows from the classification. It would be interesting to understand the actual reason for it see [1,2,12 14]). The basic idea for the non-trivial part of the proof is as follows. We first show that the maximal complex subbundle C of TM is invariant under the shape operator of M. Next, we prove that the Reeb flow is isometric if and only if the principal curvature spaces in C are complex subspaces of C. We then use the Codazzi equation to show that the principal curvatures of M are constant. Finally, we use focal set theory due to Berndt and the author [3], Eberlein[6], and Helgason [7,8] to show that M lies locally on a tube around a totally geodesic complex submanifold SU 2,m 1 /SU 2 U m 1 ) of SU 2,m /SU 2 U m ) or on a horosphere with its center at infinity. A discussion of the possibilities then leads to the result. This paper is organized as follows. In Section 1 we summarize some basic facts about the Riemannian geometry of SU 2,m /SU 2 U m ). In Section 2 we obtain some basic geometric equations for real hypersurfaces in SU 2,m /SU 2 U m ). In Section 3 we study real hypersurfaces in SU 2,m /SU 2 U m ) with geodesic Reeb flow, and in Section 4 those with isometric Reeb flow. Finally in Section 5 we use these results to derive our classification. 1. The complex hyperbolic two-plane Grassmannian SU 2,m /SU 2 U m ) In this section we summarize basic material about complex hyperbolic Grassmann manifolds SU 2,m /SU 2 U m ), for details we refer to [3 5,7,8]. The Riemannian symmetric space SU 2,m /SU 2 U m ), which consists of all complex two-dimensional linear subspaces in indefinite complex Euclidean space C m+2 2, becomes a connected, simply connected, irreducible Riemannian symmetric space of noncompact type and with rank two. Let G = SU 2,m and K = SU 2 U m ), and denote by g and k the corresponding Lie algebra of the Lie group G and K

3 Y.J. Suh / Advances in Applied Mathematics ) respectively. Let B be the Killing form of g and denote by p the orthogonal complement of k in g with respect to B. The resulting decomposition g = k p is a Cartan decomposition of g. The Cartan involution θ Autg) on su 2,m is given by θa) = I 2,m AI 2,m, where ) I2 0 I 2,m = 2,m 0 m,2 I m I 2 and I m denote the identity 2 2)-matrix and m m)-matrix respectively. Then X, Y = BX,θY ) becomes a positive definite AdK )-invariant inner product on g. Itsrestrictiontop induces a metric g on SU 2,m /SU 2 U m ), which is also known as the Killing metric on SU 2,m /SU 2 U m ). Throughout this paper we consider SU 2,m /SU 2 U m ) together with this particular Riemannian metric g. The Lie algebra k decomposes orthogonally into k = su 2 su m u 1, where u 1 is the onedimensional center of k. The adjoint action of su 2 on p induces the quaternionic Kähler structure J on SU 2,m /SU 2 U m ), and the adjoint action of mi m+2 Z = I ) 2 0 2,m 2i 0 m,2 m+2 I u 1 m induces the Kähler structure J on SU 2,m /SU 2 U m ). By construction, J commutes with each almost Hermitian structure J ν in J for ν = 1, 2, 3. Recall that a canonical local basis J 1, J 2, J 3 of a quaternionic Kähler structure J consists of three almost Hermitian structures J 1, J 2, J 3 in J such that J ν J ν+1 = J ν+2 = J ν+1 J ν, where the index ν is to be taken modulo 3. The tensor field JJ ν, which is locally defined on SU 2,m /SU 2 U m ), is self-adjoint and satisfies JJ ν ) 2 = I and tr JJ ν ) = 0, where I is the identity transformation. For a non-zero tangent vector X we define RX ={λx λ R}, CX = RX R JX, and HX = RX JX. We identify the tangent space T o SU 2,m /SU 2 U m ) of SU 2,m /SU 2 U m ) at o with p in the usual way. Let a be a maximal abelian subspace of p. SinceSU 2,m /SU 2 U m ) has rank two, the dimension of any such subspace is two. Every non-zero tangent vector X T o SU 2,m /SU 2 U m ) = p is contained in some maximal abelian subspace of p. Generically this subspace is uniquely determined by X, in which case X is called regular. If there exists more than one maximal abelian subspaces of p containing X, then X is called singular. There is a simple and useful characterization of the singular tangent vectors: A non-zero tangent vector X p is singular if and only if JX JX or JX JX. Up to scaling there exists a unique SU2) Um))-invariant Riemannian metric g on SU 2,m / SU 2 U m ).EquippedwiththismetricSU 2,m /SU 2 U m ) is a Riemannian symmetric space of rank two which is both Kähler and quaternionic Kähler. For computational reasons we normalize g such that the minimal sectional curvature of SU 2,m /SU 2 U m ), g) is 4. The sectional curvature K of the noncompact symmetric space SU 2,m /SU 2 U m ) equipped with the Killing metric g is bounded by 4 K 0. The sectional curvature 4 is obtained for all 2-planes CX when X is a non-zero vector with JX JX. When m = 1, G 2 C3 ) = SU 1,2 /SU 1 U 2 ) is isometric to the two-dimensional complex hyperbolic space CH 2 with constant holomorphic sectional curvature 4. When m = 2, we note that the isomorphism SO4, 2) SU2, 2) yields an isometry between G 2 C4 ) = SU 2,2 /SU 2 U 2 ) and the indefinite real Grassmann manifold G 2 R6 2 ) of oriented two-dimensional linear subspaces of an indefinite Euclidean space R 6 2. For this reason we assume m 3 from now on, although many of the subsequent results also hold for m = 1, 2. The Riemannian curvature tensor R of SU2,m /SU 2 U m ) is locally given by RX, Y )Z = 1 2 [ gy, Z)X gx, Z)Y + g JY, Z) JX g JX, Z) JY 2g JX, Y ) JZ

4 648 Y.J. Suh / Advances in Applied Mathematics ) g Jν Y, Z) J ν X g J ν X, Z) J ν Y 2g J ν X, Y ) } J ν Z g Jν JY, Z) J ν JX g J ν JX, Z) }] J ν JY, 1.1) where J 1, J 2, J 3 is any canonical local basis of J. Recall that a maximal flat in a Riemannian symmetric space M is a connected complete flat totally geodesic submanifold of maximal dimension. A non-zero tangent vector X of M is singular if X is tangent to more than one maximal flat in M, otherwisex is regular. The singular tangent vectors of SU 2,m /SU 2 U m ) are precisely the eigenvectors and the asymptotic vectors of the self-adjoint endomorphisms JJ 1, where J 1 is any almost Hermitian structure in J. In other words, a tangent vector X to SU 2,m /SU 2 U m ) is singular if and only if JX JX or JX JX. Now we want to focus on a singular vector X of type JX JX. In this paper, we will have to compute explicitly Jacobi vector fields along geodesics whose tangent vectors are all singular vectors with the property JX JX. For this we need the eigenvalues and eigenspaces of the Jacobi operator R X := R., X)X. LetX be a singular unit vector tangent to SU2,m /SU 2 U m ) of type JX JX. Then there exists an almost Hermitian structure J 1 in J such that JX = J 1 X and the eigenvalues, eigenspaces and multiplicities of R X are respectively given by 0 RX {Y Y HX, JY = J 1 Y } 2m 1 1 HX CX {Y Y HX, JY = J 1 Y } 2m 4 R JX 1 where RX, CX and HX denote the real, complex and quaternionic span of X, respectively, and C X the orthogonal complement of CX in HX. The maximal totally geodesic submanifolds in complex hyperbolic two-plane Grassmannian SU 2,m /SU 2 U m ) are SU 2,m 1 /SU 2 U m 1 ), CH m, CH k CH m k 1 k [m/2]), G 2 Rm+2 ) and HH n if m = 2n) see [3,5,7,8]). The first three are complex submanifolds and the other two are real submanifolds with respect to the Kähler structure J. The tangent spaces of the totally geodesic CH m are precisely the maximal linear subspaces of the form {X JX= J 1 X} with some fixed almost Hermitian structure J 1 J. 2. Real hypersurfaces in SU 2,m /SU 2 U m ) Let M be a real hypersurface in SU 2,m /SU 2 U m ), that is, a hypersurface in SU 2,m /SU 2 U m ) with real codimension one. The induced Riemannian metric on M will also be denoted by g, and denotes the Levi Civita covariant derivative of M, g). WedenotebyC and Q the maximal complex and quaternionic subbundle of the tangent bundle TM of M, respectively. Now let us put JX= φ X + ηx)n, J ν X = φ ν X + η ν X)N 2.1) for any tangent vector field X of a real hypersurface M in SU 2,m /SU 2 U m ), where φ X denotes the tangential component of JX and N a unit normal vector field of M in SU 2,m /SU 2 U m ). From the Kähler structure J of SU 2,m /SU 2 U m ) there exists an almost contact metric structure φ, ξ, η, g) induced on M in such a way that φ 2 X = X + ηx)ξ, ηξ) = 1, φξ = 0, and ηx) = gx,ξ) 2.2) for any vector field X on M and ξ = JN.

5 Y.J. Suh / Advances in Applied Mathematics ) If M is orientable, then the vector field ξ is globally defined and said to be the induced Reeb vector field on M. Furthermore,let J 1, J 2, J 3 be a canonical local basis of J. Then each J ν induces a local almost contact metric structure φ ν,ξ ν, η ν, g), ν = 1, 2, 3, on M. Locally, C is the orthogonal complement in TM of the real span of ξ, and Q the orthogonal complement in TM of the real span of {ξ 1,ξ 2,ξ 3 }. Furthermore, let { J 1, J 2, J 3 } be a canonical local basis of J. Then the quaternionic Kähler structure J ν of SU 2,m /SU 2 U m ), together with the condition J ν J ν+1 = J ν+2 = J ν+1 J ν in Section 1, induces an almost contact metric 3-structure φ ν,ξ ν, η ν, g) on M as follows: φ 2 ν X = X + η νξ ν ), φ ν ξ ν = 0, η ν ξ ν ) = 1 φ ν+1 ξ ν = ξ ν+2, φ ν ξ ν+1 = ξ ν+2, φ ν φ ν+1 X = φ ν+2 X + η ν+1 X)ξ ν, φ ν+1 φ ν X = φ ν+2 X + η ν X)ξ ν+1 2.3) for any vector field X tangent to M. The tangential and normal component of the commuting identity JJ ν X = J ν JX give φφ ν X φ ν φ X = η ν X)ξ ηx)ξ ν and η ν φ X) = ηφ ν X). 2.4) The last equation implies φ ν ξ = φξ ν. The tangential and normal component of J ν J ν+1 X = J ν+2 X = J ν+1 J ν X give and φ ν φ ν+1 X η ν+1 X)ξ ν = φ ν+2 X = φ ν+1 φ ν X + η ν X)ξ ν+1 2.5) η ν φ ν+1 X) = η ν+2 X) = η ν+1 φ ν X). 2.6) Putting X = ξ ν and X = ξ ν+1 into the first of these two equations yields φ ν+2 ξ ν = ξ ν+1 and φ ν+2 ξ ν+1 = ξ ν respectively. Using the Gauss and Weingarten formulas, the tangential and normal component of the Kähler condition X J)Y = 0 give X φ)y = ηy )AX gax, Y )ξ and X η)y = gφ AX, Y ). The last equation implies X ξ = φ AX. Finally, using the explicit expression for the Riemannian curvature tensor R of SU2,m /SU 2 U m ) in [3] the Codazzi equation takes the form [ X A)Y Y A)X = 1 2 ηx)φy ηy )φ X 2gφ X, Y )ξ } + ην X)φ ν Y η ν Y )φ ν X 2gφ ν X, Y )ξ ν + ην φ X)φ ν φy η ν φy )φ ν φ } X + ηx)ην φy ) ηy )η ν φ X) } ] ξ ν. 2.7)

6 650 Y.J. Suh / Advances in Applied Mathematics ) Hereafter, unless otherwise stated, we want to use these basic equations mentioned above frequently without referring to them explicitly. 3. Real hypersurfaces with geodesic Reeb flow It can be easily seen that the integral curves of a unit Killing vector field are geodesics. By virtue of this fact, in this section we study real hypersurfaces with geodesic Reeb flow, thatis,realhypersurfaces for which the integral curves of the Reeb flow are geodesics. Now we introduce some equivalent conditions of this property given in Berndt and Suh [2] as follows: Proposition 3.1. Let M be a connected orientable real hypersurface in a Kähler manifold M. The following statements are equivalent: a) The Reeb flow on M is geodesic; b) The Reeb vector field ξ is a principal curvature vector of M everywhere; c) The maximal complex subbundle C of T M is invariant under the shape operator A of M. Proof. We denote by φ, ξ, η, g) the induced almost contact metric structure on M. The equivalence of b) and c) is obvious since ξ and C are perpendicular and their sum spans the tangent space at each point. The integral curves of ξ are geodesics in M if and only if ξ ξ = 0 holds everywhere. The Kähler equation J = 0implies ξ ξ = φ Aξ. Thus the integral curves of ξ are geodesics in M if and only if φ Aξ = 0 holds everywhere. The latter condition means that Aξ is in the kernel of φ everywhere, which is just the real span of ξ. This shows that a) and b) are equivalent. We now assume that the Reeb flow on M in SU 2,m /SU 2 U m ) is geodesic. Then, according to Proposition 3.1, there exists a smooth function α on M so that Aξ = αξ. Taking inner product of the right-hand side of the Codazzi equation 2.7) with ξ we get gφ X, Y ) ην X)η ν φy ) η ν Y )η ν φ X) gφ ν X, Y )η ν ξ) } = g X A)Y Y A)X,ξ ) = g X A)ξ, ) Y g Y A)ξ, ) X = Xα)ηY ) Y α)ηx) + αg Aφ + φ A)X, ) Y 2gAφ AX, Y ). 3.1) Substituting X = ξ yields Y α = ξα)ηy ) η ν ξ)η ν φy ), and inserting this equation and the corresponding one for Xα into the previous equation implies Proposition 3.2. If M is a connected orientable real hypersurface in complex hyperbolic two-plane Grassmannian SU 2,m /SU 2 U m ) with geodesic Reeb flow, then 2gAφ AX, Y ) αg Aφ + φ A)X, ) Y + gφ X, Y ) = ην X)η ν φy ) η ν Y )η ν φ X) gφ ν X, Y )η ν ξ) 2ηX)η ν φy )η ν ξ) + 2ηY )η ν φ X)η ν ξ) }. 3.2) From this proposition we easily obtain

7 Y.J. Suh / Advances in Applied Mathematics ) Corollary 3.1. Let M be a connected orientable real hypersurface in complex hyperbolic two-plane Grassmannian SU 2,m /SU 2 U m ) with geodesic Reeb flow. If Aξ = αξ and X C with A X = λx, then 2λ α)aφ X + 1 αλ)φ X = 2ην ξ)η ν φ X)ξ η ν X)φ ν ξ η ν φ X)ξ ν η ν ξ)φ } ν X. 3.3) This corollary will be useful for us to get certain relations among the principal curvatures of M. 4. Real hypersurfaces with isometric Reeb flow In this section we study real hypersurfaces for which the Reeb flow is isometric. The following proposition also given in [2] provides some equivalent characterizations of this property. Proposition 4.1. Let M be a connected orientable real hypersurface in a Kähler manifold M. The following statements are equivalent: a) The Reeb flow on M is isometric; b) The shape operator A commutes with the structure tensor field φ; c) The Reeb vector field ξ is a principal curvature vector of M everywhere and the principal curvature spaces contained in the maximal complex subbundle C of T M are complex subspaces. Proof. The equation J = 0impliesg X ξ,y ) + gx, Y ξ) = gaφ φ A)X, Y ) for all X, Y TM. Since ξ is a Killing vector field if and only if ξ is a skew-symmetric tensor field, the equivalence of a) and b) follows. Next, suppose that Aφ = φ A. Then0= Aφξ = φ Aξ.Thisgivesthatξ belongs to the kernel of φ and hence a multiple of ξ everywhere, which means that ξ is a principal curvature vector everywhere. The orthogonal complement of the span of ξ is C, and on C both φ and J coincide. Thus Aφ = φ A on C means that the eigenspaces of A restricted to C are J -invariant and hence complex. Thus b) implies c). The converse can be shown by a similar argument. We now investigate real hypersurfaces M in complex hyperbolic two-plane Grassmannian SU 2,m /SU 2 U m ) with isometric Reeb flow. The equation in the next result encodes crucial information about the principal curvatures. According to Proposition 4.1, we have Aφ φ A = 0. Differentiating this equation covariantly and using the formulas in Section 2, we obtain 0 = X A)φY + ηy )A 2 X αgax, Y )ξ φ X A)Y ηay)ax + gax, AY)ξ. Taking inner product with Z gives g ) X A)Y,φZ + g ) X A)Z,φY = αηy )gax, Z) + αηz)gax, Y ) ηy )gax, AZ) ηz)gax, AY), which implies g ) X A)Y,φZ + g ) X A)Z,φY + g ) Y A)Z,φX + g ) Y A)X,φZ g ) Z A)X,φY g ) Z A)Y,φX = 2αηZ)gAX, Y ) 2ηZ)gAX, AY). 4.1)

8 652 Y.J. Suh / Advances in Applied Mathematics ) The left-hand side of this equation can be written as 4αηZ)gAX, Y ) + 4ηZ)gAX, AY) = 4g ) X A)Y,φZ [ 2ηY )gφ X,φZ) + ηx)ην φy ) 2ηY )η ν φ X) 2gφ ν X, Y ) } η ν φ Z) + ην X)gφ ν Y,φZ) η ν Y )gφ ν X,φZ) + 2η ν φ X)gφ ν φy,φz) }] [ + 2ηZ)gφY,φX) + ηy )ην φ Z) 2ηZ)η ν φy ) 2gφ ν Y, Z) } η ν φ X) + ην Y )gφ ν Z,φX) η ν Z)gφ ν Y,φX) }] ηz)ην φ X) ηx)η ν φ Z) 2gφ ν Z, X) } η ν φy ) [ { ηz)ην φ X) ηx)η ν φ Z) 2gφ ν Z, X) } η ν φy ) + ην Z)gφ ν X,φY ) η ν X)gφ ν Z,φY ) }] = 4g ) X A)Y,φZ + 2ηY )gφ X,φZ) + 2 ηy )ην φ X) + gφ ν X, Y ) } η ν φ Z) ην X)ηY )η ν Z) η ν X)η ν Y )ηz) } 2 η ν Y )gφ ν φ X, Z) η ν Y )η ν X)ηZ) + η ν Y )ηx)η ν Z) 2 η ν φ X) { gφ ν Y, Z) ηz)η ν φy ) + ηy )η ν φ Z) } 2ηZ)gφY,φX) 2 ηz)ην φy ) + gφ ν Y, Z) } η ν φ X) + 2 η ν Z)gφ ν φ X, Y ) + ην Z)η ν X)ηY ) η ν Z)ηX)η ν Y ) } + 2 gφ ν Z, X)η ν φy ), 4.2)

9 Y.J. Suh / Advances in Applied Mathematics ) where we have used the equation of Codazzi and the following formulas gφ ν φy,φz) = g φφ ν Y + ηy )ξ ν η ν Y )ξ, φ Z ) = gφ ν Y, Z) ηz)η ν φy ) + ηy )η ν φ Z), and gφφ ν Y, Z) gφ ν φy, Z) = η ν Y )ηz) η ν Z)ηY ). Then 4.2) can be written as follows: 4αηZ)gAX, Y ) + 4ηZ)gAX, AY) = 4g ) X A)Y,φZ + 2ηY )gφ X,φZ) + 2 gφ ν X, Y )η ν φ Z) 2 η ν Y )gφ ν φ X, Z) 4 η ν φ X)gφ ν Y, Z) 2ηZ)gφY,φX) + 2 η ν Z)gφ ν φ X, Y ) + 2 gφ ν Z, X)η ν φy ). Then for any hypersurface M in SU 2,m /SU 2 U m ) with isometric Reeb flow we have g X A)Y,φZ ) = αηz)gax, Y ) ηz)gax, AY) { ηy )gx, Z) ηz)gy, X) } 1 ην Y )gφ ν φ X, Z) η ν Z)gφ ν φ X, Y ) } gφν X, Y )η ν φ Z) 2gφ ν Y, Z)η ν φ X) + gφ ν Z, X)η ν φy ) }. 4.3) 2 From this, replacing Z by φ Z the left-hand side of the previous equation becomes g X A)Y,φZ ) = g X A)Y, Z ) + ηz)g X A)Y,ξ ) = Xα)ηY )ηz) + αηz)gφ AX, Y ) ηz)gaφ AX, Y ) g X A)Y, Z ). Replacing Z by φ Z also on the right-hand side of 3.3) weeventuallyget

10 654 Y.J. Suh / Advances in Applied Mathematics ) X A)Y = { Xα)ηY ) + αgaφ X, Y ) g A 2 φ X, )} Y ξ + 1 ηy )φ X } ην Y )φφ ν φ X gφ ν X, Y )φξ ν [ gφν X, Y ) { ξ ν + η ν ξ)ξ } + 2η ν φ X)φφ ν Y + η ν φy )φ ] ν X. 4.4) Putting Y = ξ and using X A)ξ = Xα)ξ + αφ AX A 2 φ X then yields the following proposition. Proposition 4.2. Let M be a real hypersurface in complex hyperbolic two-plane Grassmannian SU 2,m / SU 2 U m ) with isometric Reeb flow. Then we have 2α Aφ X 2A 2 φ X φ X } = ην ξ)φ ν X η ν X)φξ ν 4η ν ξ)ηφ ν X)ξ + 3η ν φ X)ξ ν. 4.5) An important step towards the classification of real hypersurfaces with isometric Reeb flow is the following: Proposition 4.3. Let M be a connected orientable real hypersurface in complex hyperbolic two-plane Grassmannian SU 2,m /SU 2 U m ) with isometric Reeb flow. Then the maximal quaternionic subbundle Q of T M is contained in the maximal complex subbundle C of T M. In particular, the Reeb vector field ξ is perpendicular to Q everywhere. Proof. Let X C. From Proposition 4.1 we get αgaφ X,φX) g A 2 φ ) X,φX = 1 2 gφ X,φX) + 1 η ν ξ)gφ ν X,φX) 1 η ν X) η ν φ X) ) On the other hand, by Proposition 3.2, togetherwithaφ = φ A, wehavefory = φ X, X C 2gAφ AX,φX) 2αgAφ X,φX) + gφ X,φX) = ην X) 2 + η ν φ X) 2 + η ν ξ)gφ ν X,φX) }. 4.7) Comparing the previous two equations 4.6) and 4.7) shows that 2 3 η ν X) 2 = 2 3 η ν φ X) 2 for all X C. Consequently,φ X C Q for all X C Q. This shows that C Q is φ-invariant. Since ξ is perpendicular to C, and so also to C Q, thesubspacec Q has even dimension. As dimc Q) = dim C + dim Q dimc + Q) and dim C = 4m 2, dim Q = 4m 4, and dimc + Q) {4m 2, 4m 1} we therefore must have dimc + Q) = 4m 2. But this means that Q C, which shows that ξ is perpendicular to Q everywhere. We will now prove that the principal curvatures of a real hypersurface in SU 2,m /SU 2 U m ) with isometric Reeb flow are constant.

11 Y.J. Suh / Advances in Applied Mathematics ) Proposition 4.4. Let M be a connected orientable real hypersurface in complex hyperbolic two-plane Grassmannian SU 2,m /SU 2 U m ) with isometric Reeb flow. We choose a canonical local basis in J so that ξ 1 = ξ see Proposition 4.3). The principal curvature function α given by Aξ = αξ is constant and for all X C with AX = λx one of the following two statements holds: 1) λλ α) = 0, X Q and φ X = φ 1 X; 2) λ 2 αλ + 1 = 0 and φqx = φ 1 QX, where QX denotes the orthogonal projection of X onto Q. In particular, all principal curvatures of M are constant. Proof. Inserting X = ξ in 3.1) yields Y α = ξα)ηy ) + 2 η ν ξ)η ν φy ). Using the expression for Y α and the assumption ξ 1 = ξ, we calculated in the proof of Proposition 3.2 we see that grad M α = ξα)ξ and hence X grad M α ) = Xξα) ) ξ + ξα)φ AX, where grad M α denotes the gradient of the function α on M. The symmetry of the Hessian of α, and using the identity Aφ=φ A, implies that 2ξα)gAφ X, Y )= Xξα)ηY ) Y ξα)ηx) for all X, Y TM. Putting X = ξ implies Y ξα) = ξξα)ηy ), forally TM, and inserting this and the corresponding expression for Xξα) into the previous equation implies ξα)gaφ X, Y ) = 0forallX, Y TM. Thus either ξα = 0, which implies that grad M α = 0 and hence that α is constant, or Aφ = 0. But the latter equation implies that AX = αηx)ξ, that is, the shape operator A restricted to C vanishes. By Proposition 4.2 we have for ξ = ξ 1 Q φ X = φ 1 X η 2 X)φξ 2 η 3 X)φξ 3 + 3η 2 φ X)ξ 2 + 3η 3 φ X)ξ 3. From this, putting X = ξ 2,wehave φξ 2 = φ 1 ξ 2 + 3η 2 φξ 2 )ξ 2 + 3η 3 φξ 2 )ξ 3. This gives ξ 3 = 2ξ 3, which makes a contradiction. Thus α must be constant on M. Now let X Q with AX = λx. First,wehaveAφ X = φ AX = λφ X. Thus, from Corollary 3.1 we get 2λ 2 2αλ + 1 ) φ X + φ 1 X = 2η 2 X)ξ 3 2η 3 X)ξ 2. Replacing X by φ X we obtain 2λ 2 2αλ + 1 ) X φ 1 φ X = 2η 3 X)ξ 3 2η 2 X)ξ 2. We now decompose X into X = QX + η 2 X)ξ 2 + η 3 X)ξ 3 and insert this expression into the previous equation. Then it follows that 2λ 2 2αλ + 1 ) QX φ 1 φqx + 2λ 2 2αλ + 2 ) η 2 X)ξ 2 + 2λ 2 2αλ + 2 ) η 3 X)ξ 3 = )

12 656 Y.J. Suh / Advances in Applied Mathematics ) It is clear that φ 1, and hence also φ, leavesq-invariant. Thus we have φ 1 φqx Q, and therefore the previous equation splits into three equations: 2λ 2 2αλ + 1 ) QX φ 1 φqx = 0, 4.9) λ 2 αλ + 1 ) η 2 X)ξ 2 = 0, 4.10) and λ 2 αλ + 1 ) η 3 X)ξ 3 = ) If λ 2 αλ + 1 = 0 then the first equation implies φ 1 φqx = QX. On the other hand, if λ 2 αλ + 1 0, then the last two equations imply that X Q. The first equation shows that φ 1 φqx and QX are proportional. In fact, since both φ 1 and φ act orthogonally on Q we must have φ 1 φqx =±QX. Ifφ 1 φqx = QX then the first equation yields λ 2 αλ + 1 = 0, which is a contradiction. Hence we must have φ 1 φqx = QX, and the first equation implies λ 2 αλ = 0. This shows that either 1) or 2) holds. The constancy of the principal curvatures then follows from the constancy of α. 5. Proof of Main Theorem We will now prove our Main Theorem. The geometry of the tubes around the totally geodesic SU 2,m 1 /SU 2 U m 1 ) in complex hyperbolic two-plane Grassmannian SU 2,m /SU 2 U m ) has been thoroughly studied by Berndt and the author in [3]. In particular, it was shown there see [3, Proposition 6.18]) that the Reeb vector field ξ on any such tube is a principal curvature vector everywhere and that the principal curvature spaces in Q are complex. From Proposition 4.1 we thus see that the Reeb flow on any such tube is isometric. Conversely, let M be a connected orientable real hypersurface in complex hyperbolic two-plane Grassmannian SU 2,m /SU 2 U m ) and assume that the Reeb flow on M is isometric. We use the notations as in the previous sections. From Proposition 4.4 we see that M has constant principal curvatures, and the number of different principal curvatures is at most four. We choose a real number r with 0 < r < such that α = 2coth2r). Then, from Proposition 4.4, the other possible principal curvatures are β = cothr), λ 1 = tanhr), and λ 2 = 0, with corresponding principal spaces T α, T β, T λ1 and T λ2, where C = T α T β T λ1 T λ2. Note that β and λ 1 are the two different solutions of the quadratic equation x 2 αx + 1 = 0, where 2 coth2r) = cothr) + tanhr). Forρ {α,β,λ 1,λ 2 } we define T ρ = {X C AX = ρ X}. Thenwehave C = T α T β T λ1 T λ2, and, if T ρ is not trivial, T ρ is the subbundle of TM consisting of all principal curvature vectors of M with respect to ρ which are perpendicular to ξ. According to Proposition 3.1, eacht ρ is a complex subbundle of TM. Note that on Q we have φφ 1 ) 2 = I and trφφ 1 ) = 0. Let E +1 and E 1 be the eigenbundles of φφ 1 Q with respect to the eigenvalues +1 and 1 respectively. Then the maximal quaternionic subbundle Q of TM decomposes orthogonally into the Whitney sum Q = E +1 E 1, and the rank of both eigenbundles E ±1 is equal to 2m + 2. We have X E +1 if and only if φ X = φ 1 X and X E 1 if and only if φ X = φ 1 X. And it can be easily checked that T λ1 = E 1 and T λ2 = E +1.

13 Y.J. Suh / Advances in Applied Mathematics ) For p M we denote by γ p the geodesic in SU 2,m /SU 2 U m ) with γ p 0) = p and γ p 0) = N p, and by F the smooth map F : M SU 2,m /SU 2 U m ), p γ p r). Geometrically, F is the displacement of M at distance r in direction of the normal vector field N. For each p M the differential d p F of F at p can be computed using Jacobi vector fields by means of d p F X) = Z X r). Here, Z X is the Jacobi vector field along γ p with initial values Z X 0) = X and Z X 0) = AX.Since JN = ξ = ξ 1 = J 1 N we see that N is a singular tangent vector of type JX JX at each point. Using the explicit description of the Jacobi operator R N for the case JN JN in Section 1 and of the shape operator A of M in Proposition 4.4 we see that T α T λ2 is contained in the 0-eigenspace of R N and T β T λ1 in the 1-eigenspace of R N. For the Jacobi vector fields along γ p we thus get the expressions {cosh2r) α 2 sinh2r)}e Xr), if X Rξ, Z X r) = {coshr) ρ sinhr)}e X r), if X T ρ and ρ {β,λ 1 }, 1 ρr)e X r), if X T ρ and ρ {α,λ 2 }, where E X denotes the parallel vector field along γ p with E X 0) = X. This shows that the Ker df = Rξ T β and that F is of constant rank dimt α T λ1 T λ2 ). So, locally, F is a submersion onto a submanifold P of SU 2,m /SU 2 U m ). Moreover, the tangent space of P at F p) is obtained by parallel translation of T α T λ1 T λ2 )p), which is a complex subspace of T p SU 2,m /SU 2 U m )). Since J is parallel along γ p,alsot F p) P is a complex subspace of T F p) SU 2,m /SU 2 U m )). Byvirtueofthese geometric properties, eventually we conclude that the submanifold P is a complex submanifold of SU 2,m /SU 2 U m ). It is clear that η p = γ p r) is a unit normal vector of P at F p). The shape operator B ηp of P with respect to η p is given by B ηp Z X r) = Z X r), where Z X is the Jacobi vector field along γ p as defined above. From this we immediately get that for each ρ {α,λ 1,λ 2 } the parallel translate of T ρ p) along c p from p to F p) is a principal curvature space of P with respect to ρ, provided that T ρ p) is non-trivial. Moreover, the Kähler structure J is parallel along γ p. The vectors of the form η q, q F 1 {p}), form an open subset of the unit sphere in the normal space of P at F p). SinceB ηq vanishes for all η q it now follows that P is totally geodesic in SU 2,m /SU 2 U m ). Then by using the rigidity property of totally geodesic submanifolds in complex hyperbolic two-plane Grassmannians, we assert that the entire submanifold M is congruent to an open part of a tube with radius r around some connected, complete, totally geodesic, complex submanifold P of SU 2,m /SU 2 U m ). The vector space {X T p SU 2,m /SU 2 U m )) JX= J 1 X} is the +1-eigenspace of the self-adjoint endomorphism JJ 1 = J 1 J. Using the fact that JJ 1 ) 2 = I and tr JJ 1 ) = 0 we easily see that this eigenspace is a complex vector space of complex dimension m. Thus T α p) T μ p) = { X Qp) JX= J 1 X } is a complex vector space of complex dimension m 1, and the above arguments show that the parallel translate of this complex vector space along γ p from p to F p) lies in T F p) P.Sinceboth J and J

14 658 Y.J. Suh / Advances in Applied Mathematics ) are parallel along γ p it follows that T F p) P contains an m 1)-dimensional complex subspace of the form {X JX= J 1 X} for some fixed almost Hermitian structure J 1 J. Any such subspace is invariant under the curvature tensor R and hence there exists a connected, complete, totally geodesic, complex submanifold Q of SU 2,m /SU 2 U m ) with F p) Q and T F p) Q equal to that curvature-invariant complex subspace. Since that curvature-invariant subspace consists entirely of singular tangent vectors of this special type we see that Q is a totally geodesic CH m 1. We thus conclude that P contains a totally geodesic CH m 1. The classification of totally geodesic submanifolds in SU 2,m /SU 2 U m ) shows that the only totally geodesic submanifolds in SU 2,m /SU 2 U m ) containing a totally geodesic CH m 1 are CH m 1, CH m, CH m 1 CH 1 and SU 2,m 1 /SU 2 U m 1 ) see [3,5,7,8]). The normal spaces of both CH m 1 and CH m 1 CH 1 contain regular tangent vectors of SU 2,m /SU 2 U m ). Emanating along the geodesic in direction of such a regular tangent vector would give a normal vector to M which is a regular. But this contradicts the fact that the normal vector field N of M is singular everywhere. Thus P is either CH m or SU 2,m 1 /SU 2 U m 1 ). Then we finally see that M is an open part of the tube with radius r around a totally geodesic SU 2,m 1 /SU 2 U m 1 ) in SU 2,m /SU 2 U m ). This finishes the proof of our Main Theorem for the case that the radius r > 0. On the other hand, for the limiting case we have three principal curvatures α = 2, β = 1, and λ = 0. Asabovewedefinec p, F, X Z, E X, and we get E X t), if X T λ, Z X t) = exp t)e X t), if X T β, exp 2t)E X t), if X T α for all t R. Now consider a geodesic variation in SU 2,m /SU 2 U m ) consisting of geodesics γ p.the corresponding Jacobi field is a linear combination of the three types of the Jacobi fields Z X listed above, and hence its length remains bounded when t. This shows that all geodesics γ p in SU 2,m /SU 2 U m ) are asymptotic to each other and hence determine a singular point z SU 2,m /SU 2 U m ) ) at infinity see [3] and [6]). Therefore M is an integral manifold of the distribution on SU 2,m /SU 2 U m ) given by the orthogonal complements of the tangent vectors of the geodesics in the asymptotic class z. This distribution is integrable and the maximal leaves are the horospheres in SU 2,m /SU 2 U m ) whose center at infinity is z. Uniqueness of integral manifolds of integrable distributions finally implies that M becomes an open part of a horosphere in SU 2,m /SU 2 U m ) whose center is the singular point z at infinity. Acknowledgment The author would like to express his deep gratitude to the referee for his careful reading of the manuscript and many valuable suggestions. References [1] J. Berndt, Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monatsh. Math ) [2] J. Berndt, Y.J. Suh, Real hypersurfaces with isometric flow in complex two-plane Grassmannians, Monatsh. Math ) [3] J. Berndt, Y.J. Suh, Hypersurfaces in noncompact complex Grassmannians of rank two, Internat. J. Math ) , 35 pp. [4] J. Berndt, H. Tamaru, Homogeneous codimension one foliations on noncompact symmetric spaces, J. Differential Geom ) [5] J. Berndt, S. Console, C. Olmos, Submanifolds and Holonomy, Chapman & Hall/CRC, Boca Raton, FL, [6] P.B. Eberlein, Geometry of Nonpositively Curved Manifolds, University of Chicago Press, Chicago, London, [7] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Grad. Stud. Math., vol. 34, Amer. Math. Soc., 2001.

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