Pseudo-Einstein real hypersurfaces in complex hyperbolic two-plane Grassmannians

Advances in Applied Mathematics 76 (2016) 39–67

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Pseudo-Einstein real hypersurfaces in complex hyperbolic two-plane Grassmannians ✩ Young Jin Suh Department of Mathematics, Kyungpook National University, and Research Institute of Real & Complex Manifolds, Taegu 702-701, Republic of Korea

a r t i c l e

i n f o

Article history: Received 10 March 2015 Received in revised form 10 February 2016 Accepted 10 February 2016 Available online 20 February 2016 MSC: primary 53C40 secondary 53C15

a b s t r a c t In this paper we first introduce the full expression of the curvature tensor of a real hypersurface M in complex hyperbolic two-plane Grassmannians SU 2,m /S(U2 ·Um ), m ≥ 2 from the equations of Gauss. Next we derive a new formula for the Ricci tensor of M in SU 2,m /S(U2 ·Um ). As an application of this result we prove that there do not exist Einstein Hopf or Q⊥ -invariant Einstein real hypersurfaces in SU 2,m /S(U2 ·Um ). © 2016 Elsevier Inc. All rights reserved.

Keywords: Real hypersurfaces Complex hyperbolic two-plane Grassmannians Pseudo-Einstein hypersurface Q⊥ -invariant hypersurface

1. Introduction A Riemannian manifold M is said to be Einstein if the Ricci tensor S is a scalar multiple of the Riemannian metric g on M , that is, g(SX, Y ) = λg(X, Y ) for a smooth function λ and any vector fields X, Y on the tangent bundle of M . Classically, Einstein ✩ This work was supported by grant Proj. No. NRF-2015-R1A2A1A-01002459 from National Research Foundation of Korea. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.aam.2016.02.001 0196-8858/© 2016 Elsevier Inc. All rights reserved.

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Y.J. Suh / Advances in Applied Mathematics 76 (2016) 39–67

hypersurfaces in real space forms have been studied by many differential geometers (see Cartan [2] and Fialkow [5]). In complex space forms or in quaternionic space forms many differential geometers have discussed real Einstein hypersurfaces, complex Einstein hypersurfaces or more generally real hypersurfaces with parallel Ricci tensor, that is ∇S = 0, where ∇ denotes the Riemannian connection of M (see Cecil and Ryan [3], Kimura [11,10], Montiel and Romero [17], Romero [21,22] and Martinez and Pérez [15], Perez and Suh [19]). From such a view point Kon [13] has considered the notion of pseudo-Einstein real hypersurfaces M in complex projective space CP m , which are defined by SX = aX + bη(X)ξ, where a, b are constants, η(X) = g(ξ, X) and ξ = −JN for any tangent vector X and a unit normal vector N defined on M . In [13] he has also given a complete classification of pseudo-Einstein real hypersurfaces in CP m by using the work of Takagi [26] and proved that there do not exist Einstein real hypersurfaces in CP m , m ≥ 3. Moreover, Kon [14] has considered a new notion of the Ricci tensor Sˆ in the generalized Tanaka–Webster ˆ (k) . connection ∇ The notion of pseudo-Einstein was generalized by Cecil and Ryan [3] to any smooth functions a and b defined on M . By using the theory of tubes, Cecil and Ryan [3] have given a complete classification of such pseudo-Einstein real hypersurfaces and proved that there do not exist Einstein real hypersurfaces in CP m , m ≥ 3. On the other hand, Montiel [16] considered pseudo-Einstein real hypersurfaces in complex hyperbolic space CH m and gave a complete classification of such hypersurfaces and also proved that there do not exist Einstein real hypersurfaces in CH m , m ≥ 3. For real hypersurfaces in quaternionic projective space HP m the notion of pseudoEinstein was considered by Martinez and Pérez [15]. But in [18] Pérez proved that the unique Einstein real hypersurfaces in HP m are geodesic hyperspheres of radius r, 1 0 < r < π2 and cot2 r = 2m . Now let us denote by G2 (Cm+2 ) the set of all complex two-dimensional linear subspaces in Cm+2 . The situation mentioned above is not so simple if we consider a real hypersurface in the complex two-plane Grassmannian G2 (Cm+2 ). This Riemannian symmetric space has a remarkable geometrical structure. It is the unique compact irreducible Riemannian manifold equipped with both a Kähler structure J and a quaternionic Kähler structure J not containing J. In other words, G2 (Cm+2 ) is the unique compact, irreducible, Kähler, quaternionic Kähler manifold which is not a hyperkähler manifold. So, in G2 (Cm+2 ) we have two natural geometrical conditions for real hypersurfaces M : That [ξ] = Span{ξ} or Q⊥ = Span{ξ1 , ξ2 , ξ3 } is invariant under the shape operator, being ξ = −JN , ξi = −Ji N , i = 1, 2, 3, where N denotes a unit normal vector field on M in G2 (Cm+2 ) and {J1 , J2 , J3 } a local basis of J (see Suh [23]). A real hypersurface M in G2 (Cm+2 ) is said to be pseudo-Einstein if the Ricci tensor S of M satisfies

Y.J. Suh / Advances in Applied Mathematics 76 (2016) 39–67

SX = aX + bη(X)ξ + c

3 

ην (X)ξν

41

(∗)

ν=1

for any constants a, b and c on M . In a paper by Pérez, Suh and Watanabe [20] we have defined the notion of pseudo-Einstein hypersurfaces in G2 (Cm+2 ) with the assumption that b and c are non-vanishing constants. In this case the meaning of pseudo-Einstein is proper pseudo-Einstein. So in [20] we have given a complete classification of proper Hopf pseudo-Einstein real hypersurfaces as follows. Theorem A. Let M be a pseudo-Einstein Hopf real hypersurface in G2 (Cm+2 ). Then M is congruent to √ m+1 (a) a tube of radius r, cot2 2r = m−1 ), where a = 4m + 8, b + c = 2 , over G2 (C −2(m + 1), provided that c= − 4; √ 4m−3 (b) a tube of radius r, cot r = 1+2(m−1) , over HP m , m = 2n, where a = 8n + 6, b = −16n + 10, c = −2. For the real hypersurfaces of type (a) or of type (b) in Theorem A the constants b and c of pseudo-Einstein real hypersurfaces M in G2 (Cm+2 ) never vanish at the same time on M , that is, at least one of them is non-vanishing at any point of M . As a direct consequence of Theorem A, we have also asserted that there are no Einstein Hopf real hypersurfaces in G2 (Cm+2 ). Now let us denote by SU 2,m /S(U2 ·Um ) the set of all complex two-dimensional linear subspaces in the indefinite complex Euclidean spaces Cm+2 . Then the above situation is 2 not so simple if we consider a real hypersurface in the complex two-plane Grassmannians SU 2,m /S(U2 ·Um ). In a paper due to Suh [24] a complete classification of hypersurfaces in SU 2,m /S(U2 ·Um ) was given when the distributions C and Q are invariant by the shape operator A. Moreover, Suh [24] has given a complete classification of hypersurfaces in SU 2,m /S(U2 ·Um ) with isometric Reeb flow, that is Aφ = φA. The complex hyperbolic two-plane Grassmannian SU 2,m /S(U2 ·Um ) is the unique noncompact, irreducible, Kähler, quaternionic Kähler manifold which is not a hyperkähler manifold. So, we have considered two geometric conditions as for real hypersurfaces in G2 (Cm+2 ). By using such two geometric conditions and the results in Eberlein [4], Berndt and Suh [1] introduced the following: Theorem B. Let M be a connected hypersurface in SU 2,m /S(U2 Um ), m ≥ 2. Then the maximal complex subbundle C of T M and the maximal quaternionic subbundle Q of T M are both invariant under the shape operator of M if and only if M is congruent to an open part of one of the following hypersurfaces: (A) a tube around a totally geodesic SU 2,m−1 /S(U2 Um−1 ) in SU 2,m /S(U2 Um ); (B) a tube around a totally geodesic HH n in SU 2,2n /S(U2 U2n ), m = 2n;

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(C) a horosphere in SU 2,m /S(U2 Um ) whose center at infinity is singular; or the following exceptional case holds: (D) The normal bundle νM of M consists of singular tangent vectors of type JX ⊥ JX. Moreover, M has at least four distinct principal curvatures, three of which are given by α=

√ 2,

γ = 0,

1 λ= √ 2

with corresponding principal curvature spaces Tα = T M  (C ∩ Q),

Tγ = J(T M  Q),

Tλ ⊂ C ∩ Q ∩ JQ.

If μ is another (possibly nonconstant) principal curvature function, then we have Tμ ⊂ C ∩ Q ∩ JQ, JTμ ⊂ Tλ and JTμ ⊂ Tλ . In the proof of Theorem B we have used focal set theory due to Eberlein [4], and Helgason [6,7] to show that M lies locally in a tube around a totally geodesic complex submanifold SU 2,m−1 /S(U2 ·Um−1 ), quaternionic hyperbolic space HH n in SU 2,m /S(U2 ·Um ) or on a horosphere with its center at infinity. A tube around SU 2,m−1 /S(U2 ·Um−1 ) in SU 2,m /S(U2 ·Um ) 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 U1 . The action of SU 1,m+1 has two kinds of singular orbits. One is a totally geodesic SU 2,m−1 /S(U2 ·Um−1 ) in SU 2,m /S(U2 ·Um ) and the other is a totally geodesic CH m in SU 2,m /S(U2 ·Um ). Now in this paper let us consider the notion of pseudo-Einstein defined by (∗) for real hypersurfaces in SU 2,m /S(U2 ·Um ). Then, motivated by Theorem B mentioned above, we give a complete classification of pseudo-Einstein real hypersurfaces in SU 2,m /S(U2 ·Um ) as follows: Theorem 1.1. Let M be a pseudo-Einstein Hopf real hypersurface in complex hyperbolic two-plane Grassmannians SU 2,m /S(U2 Um ), m ≥ 2. Then M is locally congruent to a √ real hypersurface with four distinct constant principal curvatures 2, 0, λ = √12 and μ = such that p +q = 4(m −2), where p and q denote the multiplicities of the principal curvatures λ and μ respectively. In this case M becomes a proper pseudo-Einstein real hypersurface with a = − 12 (4m + 5), b = c = 32 . q−4m+3 √ q 2

By virtue of this theorem naturally we can classify Einstein Hopf hypersurfaces in SU 2,m /S(U2 ·Um ). In our main theorem, there only exist proper pseudo-Einstein hypersurfaces in SU 2,m /S(U2 ·Um ). From this we know that there do not exist any Einstein Hopf real hypersurfaces in SU 2,m /S(U2 ·Um ), m ≥ 3.

Y.J. Suh / Advances in Applied Mathematics 76 (2016) 39–67

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Now let us consider Q⊥ -invariant Einstein hypersurfaces in SU 2,m /S(U2 ·Um ). Then from the Q⊥ -invariant property we can prove that the Reeb vector field ξ belongs to the maximal quaternionic subbundle Q or Q⊥ of M in G2 (Cm+2 ). Then each case yields that M is Hopf. So by using the classification of pseudo-Einstein Hopf hypersurfaces in our theorem, we assert the following: Theorem 1.2. There does not exist any Q⊥ -invariant Einstein real hypersurface in SU 2,m /S(U2 ·Um ). In section 2 we recall the Riemannian geometry of the complex hyperbolic two-plane Grassmannian SU 2,m /S(U2 ·Um ) and in section 3 we will show some fundamental properties of real hypersurfaces in SU 2,m /S(U2 ·Um ). The formula for the Ricci tensor S and its covariant derivative ∇S will be shown explicitly in this section. In sections 4 and 5 we will give a complete proof of Theorem 1.1. In section 6 we prove Theorem 1.2. 2. The complex hyperbolic two-plane Grassmannian SU 2,m /S(U2 ·Um ) In this section we summarize basic material about complex hyperbolic Grassmann manifolds SU 2,m /S(U2 ·Um ), for details we refer to [24] and [25]. The Riemannian symmetric space SU 2,m /S(U2 ·Um ), which consists of all complex two-dimensional linear subspaces in indefinite complex Euclidean space Cm+2 , is a con2 nected, simply connected, irreducible Riemannian symmetric space of noncompact type and with rank two. Let G = SU 2,m and K = S(U2 ·Um ), and denote by g and k the corresponding Lie algebras of the Lie groups G and K 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 ∈ Aut(g) on su2,m is given by θ(A) = I2,m AI2,m , where  I2,m =

−I2 0m,2

02,m Im



I2 and Im denote the identity (2 × 2)-matrix and (m × m)-matrix respectively. Then < X, Y > = −B(X, θY ) becomes a positive definite Ad(K)-invariant inner product on g. Its restriction to p induces a metric g on SU 2,m /S(U2 ·Um ), which is also known as the Killing metric on SU 2,m /S(U2 ·Um ). Throughout this paper we consider SU 2,m /S(U2 ·Um ) together with this particular Riemannian metric g. Then by such a construction, the manifold SU 2,m /S(U2 ·Um ) becomes an orbit SU 2,m ·o, so it can be regarded as a homogeneous space and moreover, an Hermitian symmetric space with rank 2 of noncompact type. The Lie algebra k decomposes orthogonally into k = su2 ⊕ sum ⊕ u1 , where u1 is the one-dimensional center of k. The adjoint action of su2 on p induces the quaternionic Kähler structure J on SU 2,m /S(U2 ·Um ), and the adjoint action of

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Y.J. Suh / Advances in Applied Mathematics 76 (2016) 39–67

 Z=

mi m+2 I2

0m,2

02,m −2i m+2 Im

 ∈ u1

induces the Kähler structure J on SU 2,m /S(U2 ·Um ). By construction, J commutes with each almost Hermitian structure Jν in J for ν = 1, 2, 3. Recall that a canonical local basis J1 , J2 , J3 of a quaternionic Kähler structure J consists of three almost Hermitian structures J1 , J2 , J3 in J such that Jν Jν+1 = Jν+2 = −Jν+1 Jν , where the index ν is to be taken modulo 3. The tensor field JJν , which is locally defined on SU 2,m /S(U2 ·Um ), is self-adjoint and satisfies (JJν )2 = I and Tr(JJν ) = 0, where I is the identity transformation. For a nonzero tangent vector X we define RX = {λX|λ ∈ R}, CX = RX ⊕ RJX, and HX = RX ⊕ JX. We identify the tangent space To SU 2,m /S(U2 ·Um ) of SU 2,m /S(U2 ·Um ) at o with p in the usual way. Let a be a maximal Abelian subspace of p. Since SU 2,m /S(U2 ·Um ) has rank two, the dimension of any such subspace is two. Every nonzero tangent vector X ∈ To SU 2,m /S(U2 ·Um ) ∼ = 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 exist 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 nonzero tangent vector X ∈ p is singular if and only if JX ∈ JX or JX ⊥ JX. Up to scaling there exists a unique S(U2 · Um )-invariant Riemannian metric g on SU 2,m /S(U2 ·Um ). Equipped with this metric SU 2,m /S(U2 ·Um ) 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 /S(U2 ·Um ), g) is −4. The sectional curvature K of the noncompact symmetric space SU 2,m /S(U2 ·Um ) 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 /S(U1 ·U2 ) 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 SO(4, 2)  SU (2, 2) yields an isometry between G∗2 (C4 ) = SU 2,2 /S(U2 ·U2 ) and the indefinite real Grassmann manifold G∗2 (R62 ) of oriented two-dimensional linear subspaces of an indefinite Euclidean space R62 . For this reason we assume m ≥ 3 from now on, although many of the subsequent results also hold for m = 1, 2. ¯ of SU 2,m /S(U2 ·Um ) is locally given by The Riemannian curvature tensor R 1 ¯ R(X, Y )Z = − g(Y, Z)X − g(X, Z)Y + g(JY, Z)JX 2 − g(JX, Z)JY − 2g(JX, Y )JZ +

3   ν=1

g(Jν Y, Z)Jν X − g(Jν X, Z)Jν Y − 2g(Jν X, Y )Jν Z



Y.J. Suh / Advances in Applied Mathematics 76 (2016) 39–67

+

3  

g(Jν JY, Z)Jν JX − g(Jν JX, Z)Jν JY

45

 ,

(2.1)

ν=1

where J1 , J2 , J3 is any canonical local basis of J. ¯ is a connected comRecall that a maximal flat in a Riemannian symmetric space M plete flat totally geodesic submanifold of maximal dimension. A non-zero tangent vector ¯ is singular if X is tangent to more than one maximal flat in M ¯ , otherwise X is X of M regular. The singular tangent vectors of SU 2,m /S(U2 ·Um ) are precisely the eigenvectors and the asymptotic vectors of the self-adjoint endomorphisms JJ1 , where J1 is any almost Hermitian structure in J. In other words, a tangent vector X to SU 2,m /S(U2 ·Um ) 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 of type JX ∈ JX. For this we need the eigenvalues and eigenspaces ¯ X := R(., ¯ X)X. Let X be a singular unit vector tangent to of the Jacobi operator R SU 2,m /S(U2 ·Um ) of type JX ∈ JX. Then there exists an almost Hermitian structure ¯X J1 in J such that JX = J1 X and the eigenvalues, eigenspaces and multiplicities of R are respectively given by Principal curvature

Eigenspace

Multiplicity

0 −1 −4

RX ⊕ {Y |Y ⊥ HX, JY = −J1 Y } HX  CX ⊕ {Y |Y ⊥ HX, JY = J1 Y } RJX

2m − 1 2m 1

where RX, CX and HX denote the real, complex and quaternionic span of X, respectively, and C⊥ X is the orthogonal complement of CX in HX. The maximal totally geodesic submanifolds in complex hyperbolic two-plane Grassmannian SU 2,m /S(U2 ·Um ) are SU 2,m−1 /S(U2 ·Um−1 ), CH m , CH k × CH m−k (1 ≤ k ≤ [m/2]), G∗2 (Rm+2 ) and HH n (if m = 2n) (see Helgason [6] and Knapp [12]). 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 = J1 X} with some fixed almost Hermitian structure J1 ∈ J. 3. Real hypersurfaces in SU 2,m /S(U2 ·Um ) Let M be a real hypersurface in SU 2,m /S(U2 ·Um ), that is, a submanifold in SU 2,m /S(U2 ·Um ) 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). We denote by C and Q the maximal complex and quaternionic subbundle of the tangent bundle T M of M , respectively. Now let us put

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Y.J. Suh / Advances in Applied Mathematics 76 (2016) 39–67

JX = φX + η(X)N,

Jν X = φν X + ην (X)N

(3.1)

for any tangent vector field X of a real hypersurface M in SU 2,m /S(U2 ·Um ), where φX denotes the tangential component of JX and N a unit normal vector field of M in SU 2,m /S(U2 ·Um ). From the Kähler structure J of SU 2,m /S(U2 ·Um ) there exists an almost contact metric structure (φ, ξ, η, g) induced on M in such a way that φ2 X = −X + η(X)ξ,

η(ξ) = 1,

φξ = 0 and η(X) = g(X, ξ)

(3.2)

for any vector field X on M and ξ = −JN . 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 {J1 , J2 , J3 } be a canonical local basis of J. Then the quaternionic Kähler structure Jν of SU 2,m /S(U2 ·Um ), together with the condition Jν Jν+1 = Jν+2 = −Jν+1 Jν in section 2, induced an almost contact metric 3-structure (φν , ξν , ην , g) on M as follows ([8] and [9]): φ2ν X = −X + ην (X)ξν , φν+1 ξν = −ξν+2 ,

φν ξν = 0,

ην (ξν ) = 1,

φν ξν+1 = ξν+2 ,

φν φν+1 X = φν+2 X + ην+1 (X)ξν , φν+1 φν X = −φν+2 X + ην (X)ξν+1

(3.3)

for any vector field X tangent to M . Locally, C is the orthogonal complement in T M of the real span of ξ, and Q is the orthogonal complement in T M of the real span of {ξ1 , ξ2 , ξ3 }. The tangential and normal components of the commuting identity JJν X = Jν JX give φφν X − φν φX = ην (X)ξ − η(X)ξν

and ην (φX) = η(φν X).

(3.4)

The last equation implies φν ξ = φξν . The tangential and normal components of Jν Jν+1 X = Jν+2 X = −Jν+1 Jν X give φν φν+1 X − ην+1 (X)ξν = φν+2 X = −φν+1 φν X + ην (X)ξν+1

(3.5)

ην (φν+1 X) = ην+2 (X) = −ην+1 (φν X).

(3.6)

and

Putting X = ξν and X = ξν+1 into the first of these two equations yields φν+2 ξν = ξν+1 and φν+2 ξν+1 = −ξν respectively. Using the Gauss and Weingarten formulas, the

Y.J. Suh / Advances in Applied Mathematics 76 (2016) 39–67

47

¯ X J)Y = 0 give (∇X φ)Y = tangential and normal components of the Kähler condition (∇ η(Y )AX − g(AX, Y )ξ and (∇X η)Y = g(φAX, Y ). The last equation implies ∇X ξ = ¯ of φAX. Finally, using the explicit expression for the Riemannian curvature tensor R SU 2,m /S(U2 ·Um ) in [1] the Codazzi equation takes the form 1 η(X)φY − η(Y )φX − 2g(φX, Y )ξ 2 3    ην (X)φν Y − ην (Y )φν X − 2g(φν X, Y )ξν +

(∇X A)Y − (∇Y A)X = −

ν=1

+

3  

ην (φX)φν φY − ην (φY )φν φX



ν=1

+

  η(X)ην (φY ) − η(Y )ην (φX) ξν

3  

(3.7)

ν=1

for any vector fields X and Y on M . Moreover, by the expression of the curvature tensor (2.1), we have the equation of Gauss as follows: 1 g(Y, Z)X − g(X, Z)Y 2 + g(φY, Z)φX − g(φX, Z)φY − 2g(φX, Y )φZ

R(X, Y )Z = −

+

3 

{g(φν Y, Z)φν X − g(φν X, Z)φν Y − 2g(φν X, Y )φν Z}

ν=1

+

3 

{g(φν φY, Z)φν φX − g(φν φX, Z)φν φY }

ν=1

−

3 

{η(Y )ην (Z)φν φX − η(X)ην (Z)φν φY }

ν=1

−

3 

{η(X)g(φν φY, Z) − η(Y )g(φν φX, Z)} ξν



ν=1

+ g(AY, Z)AX − g(AX, Z)AY

(3.8)

for any vector fields X, Y , Z and W on M . Hereafter, unless otherwise stated, we will use these basic equations frequently without referring to them explicitly. 4. Preliminaries The purpose of this section is to give some important formulas about the Ricci tensor S for real hypersurfaces in SU 2,m /S(U2 ·Um ). These formulas will be used in sections 4 and 5. Moreover, together with the derivative formula of the Ricci tensor S

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48

for pseudo-Einstein hypersurfaces in SU 2,m /S(U2 ·Um ), we give a lemma which will play an important role in the classification of pseudo-Einstein or η-Einstein hypersurfaces in SU 2,m /S(U2 ·Um ) in section 5. By using the Gauss equation for the curvature tensor in section 2, let us contract Y and Z. Then the Ricci tensor S of a real hypersurface M in SU 2,m /S(U2 ·Um ) is given by SX =

4m−1 

R(X, ei )ei

i=1

=−

3  1 (4m + 10)X − 3η(X)ξ − 3 ην (X)ξν 2 ν=1

+

3 

{(Trφν φ)φν φX − (φν φ) X} − 2

ν=1

−

3 

3 

{ην (ξ)φν φX − η(X)φν φξν }

ν=1

 {(Tr φν φ)η(X) − η(φν φX)}ξν + hAX − A2 X,

(4.1)

ν=1

where h denotes the trace of the shape operator A of M in SU 2,m /S(U2 ·Um ). From the formula JJν = Jν J, TrJJν = 0, ν = 1, 2, 3 we calculate the following for any basis {e1 , · · · , e4m−1 , N } of the tangent space of SU 2,m /S(U2 ·Um ) 0 = Tr JJν =

4m−1 

g(JJν ek , ek ) + g(JJν N, N )

k=1

= Tr φφν − ην (ξ) − g(Jν N, JN ) = Tr φφν − 2ην (ξ)

(4.2)

and

(φν φ)2 X = φν φ φφν X − ην (X)ξ + η(X)ξν   = φν − φν X + η(φν X)ξ + η(X)φ2ν ξ   = X − ην (X)ξν + η(φν X)φν ξ + η(X) − ξ + ην (ξ)ξν .

(4.3)

From these formulas (4.1), (4.2) and (4.3), we assert the following. Lemma 4.1. Let M be a real hypersurface in SU 2,m /S(U2 ·Um ). Then the Ricci tensor of M is given by 3  1 SX = − (4m + 7)X − 3η(X)ξ − 3 ην (X)ξν 2 ν=1

Y.J. Suh / Advances in Applied Mathematics 76 (2016) 39–67 3 

+

49



{ην (ξ)φν φX − η(φν X)φν ξ − η(X)ην (ξ)ξν }

ν=1

+ hAX − A2 X

(4.4)

for any vector field X tangent to M . Now the covariant derivative of (4.4) is given by 3 3 3 3 3 3 (∇Y S)X = (∇Y η)(X)ξ + η(X)∇Y ξ + (∇Y ην )(X)ξν + ην (X)∇Y ξν 2 2 2 ν=1 2 ν=1

1   Y (ην (ξ))φν φX + ην (ξ)(∇Y φν )φX + ην (ξ)φν (∇Y φ)X 2 ν=1 3

−

− (∇Y η)(φν X)φν ξ − η((∇Y φν )X)φν ξ − η(φν X)∇Y (φν ξ) − (∇Y η)(X)ην (ξ)ξν − η(X)∇Y (ην (ξ))ξν − η(X)ην (ξ)∇Y ξν



+ (Y h)AX + h(∇Y A)X − (∇Y A2 )X

(4.5)

for any vector fields X and Y tangent to M . Then from (4.5), together with the formulas in section 3, we have (∇Y S)X =

3 3 g(φAY, X)ξ + η(X)φAY 2 2 3 3 + {qν+2 (Y )ην+1 (X) − qν+1 (Y )ην+2 (X) + g(φν AY, X)}ξν 2 ν=1 3 + ην (X){qν+2 (Y )ξν+1 − qν+1 (Y )ξν+2 + φν AY } 2 ν=1 3

1  Y (ην (ξ))φν φX + ην (ξ){−qν+1 (Y )φν+2 φX 2 ν=1 3

−

+ qν+2 (Y )φν+1 φX + ην (φX)AY − g(AY, φX)ξν } + ην (ξ){η(X)φν AY − g(AY, X)φν ξ} − g(φAY, φν X)φν ξ + {qν+1 (Y )η(φν+2 X) − qν+2 (Y )η(φν+1 X) − ην (X)η(AY ) + η(ξν )g(AY, X)}φν ξ − η(φν X){qν+2 (Y )φν+1 ξ − qν+1 (Y )φν+2 ξ + φν φAY − η(AY )ξν + η(ξν )AY } − g(φAY, X)ην (ξ)ξν − η(X)Y (ην (ξ))ξν − η(X)ην (ξ)∇Y ξν + (Y h)AX + h(∇Y A)X − (∇Y A2 )X.

 (4.6)

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50

Now let M be a pseudo-Einstein real hypersurface in SU 2,m /S(U2 ·Um ). That is, M satisfies the formula (∗) in the introduction. Then by the formulas in section 4 the covariant derivative of the Ricci tensor S satisfies (∇Y S)X = ∇Y (SX) − S(∇Y X) = b(∇Y η)(X)ξ + bη(X)∇Y ξ + c

3 

(∇Y ην )(X)ξν + c

ν=1

3 

ην (X)∇Y ξν

(∗∗)

ν=1

for any constants b and c on M . From this, together with (4.6), we have  3  (b − ) g(φAY, X)ξ + η(X)φAY 2 3 3  + (c − ) {qν+2 (Y )ην+1 (X) − qν+1 (Y )ην+2 (X) + g(φν AY, X)}ξν 2 ν=1 +

3 



ην (X){qν+2 (Y )ξν+1 − qν+1 (Y )ξν+2 + φν AY }

ν=1

1  + g(qν+2 (Y )φν+1 ξ − qν+1 (Y )φν+2 ξ + φν φAY, X)φν ξ 2 ν=1 3

− g(η(AY )ξν − η(ξν )AY, X)φν ξ + g(φν ξ, X){qν+2 (Y )φν+1 ξ − qν+1 (Y )φν+2 ξ  + φν φAY − η(AY )ξν + η(ξν )AY } 1  Y (ην (ξ))φν φX + ην (ξ){−qν+1 (Y )φν+2 φX + qν+2 (Y )φν+1 φX 2 ν=1 3

+

+ ην (φX)AY − g(AY, φX)ξν } + ην (ξ){η(X)φν AY − g(AY, X)φν ξ}  − g(φAY, X)ην (ξ)ξν − η(X)Y (ην (ξ))ξν − η(X)ην (ξ)∇Y ξν − (Y h)AX − (hI − A)(∇Y A)X + (∇Y A)AX = 0.

(4.7)

From these formulas, we have the following lemma which will be used in section 5. Lemma 4.2. Let M be a pseudo-Einstein real hypersurface in SU 2,m /S(U2 ·Um ). Then we have

bφAξ + c

3 

φν Aξν = 0.

ν=1

(4.8)

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51

Proof. Now contracting X and Y in (4.7) and using the formulas mentioned below Y (ην (ξ)) = Y (g(ξν , ξ)) = g(∇Y ξν , ξ) + g(ξν , ∇Y ξ) = qν+2 (Y )η(ξν+1 ) − qν+1 (Y )η(ξν+2 ) − 2g(AY, φν ξ), 3  

qν+2 (Y )η(ξν+1 )Trφν φ − qν+1 (Y )η(ξν+2 )Trφν φ

ν=1

− qν+1 (Y )ην (ξ)Trφν+2 φ + qν+2 (Y )ην (ξ)Trφν+1 φ = 0,

and 2

3 

{qν+2 (Y )η(ξν+1 ) − qν+1 (Y )η(ξν+2 )} = 0,

ν=1

then we have the following −(Y h)h + Tr(∇Y A)(2A − hI) = 0.

(4.9)

Now let us take the inner product of (4.7) with any vector field W and use the equation of Codazzi for the final terms of the left side of (4.7). From this, contracting Y and W , we have 3 (b − )g(φAξ, X) 2 3 3  + (c − ) {qν+2 (ξν )ην+1 (X) − qν+1 (ξν )ην+2 (X) + g(φν Aξν , X)} 2 ν=1  + ην (X){qν+2 (ξν+1 ) − qν+1 (ξν+2 )} 1  qν+2 (φν ξ)g(φν+1 ξ, X) − qν+2 (φν ξ)g(φν+2 ξ, X) 2 ν=1  + g(φν φAφν ξ, X) + η(ξν )g(Aφν ξ, X) − η(Aφν ξ)ην (X)   + g(φν ξ, X) qν+2 (φν+1 ξ) − qν+1 (φν+2 ξ) + Trφν φA − η(Aξν ) + η(ξν )TrA 3

+

 1  (φν φX)(ην (ξ)) + ην (ξ) − qν+1 (φν+2 φX) 2 ν=1  + qν+2 (φν+1 φX) − g(Aξν , φX) + ην (φX)TrA 3

+

+ ην (ξ)η(X)Trφν A − ην (ξ)g(Aφν ξ, X) − g(φAξν , X)ην (ξ) − η(X)ξν (ην (ξ))  − η(X)ην (ξ){qν+2 (ξν+1 ) − qν+1 (ξν+2 )}

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52

3 3 3 − g(Aξ, φX) − g(Aφν X, ξν ) − (AXh) + Tr(∇AX A) 2 2 ν=1

+

1 {g((A − hI)φξν , φν φX) + ην (φX)TrAφν φ − hην (φX)Trφν φ} 2 ν=1

−

1 {g((A − hI)ξ, ξν )ην (φX) + η(X)g(Aφξν , ξν )} 2 ν=1

+

1 {g(φν φAX, φξν ) + ην (φAX)Trφν φ − η(ξν )ην (φAX)} 2 ν=1

3

3

3

= 0.

(4.10)

Now we calculate more explicitly the terms in (4.10) as follows: φν φX(ην (ξ)) = g(∇φν φX ξ, ξν ) + g(ξ, ∇φν φX ξν ) = 2g(φAφν φX, ξν ) + η(ξν+1 )qν+2 (φν φX) − η(ξν+2 )qν+1 (φν φX),

(4.11)

η(X)ξν (ην (ξ)) = η(X){g(∇ξν ξ, ξν ) + g(ξ, ∇ξν ξν )} = 2η(X)g(φAξν , ξν ) + η(X){η(ξν+1 )qν+2 (ξν ) − η(ξν+2 )qν+1 (ξν )}.

(4.12)

Now let us substitute (4.11) and (4.12) into (4.10). From this, together with the formulas g(φν φAφν ξ, X) = −g(ξν , φAφν φX) + ην (X)g(Aφν ξ, ξ) − η(X)g(Aφν ξ, ξν ) and 3 

g((A − hI)φξν , φν φX) = −

ν=1

3 

g(φAφν φX, ξν ) + h

3 

ν=1

ην (φX)ην (ξ),

ν=1

we have bg(φAξ, X) + c

3 

g(φν Aξν , X)

ν=1

1  − g(ξν , φAφν φX) + ην (X)g(Aφν ξ, ξ) − η(X)g(Aφν ξ, ξν ) 2 ν=1  − η(Aφν ξ)ην (X) − η(Aξν )g(φν ξ, X) 3

+

1  + 2g(φAφν φX, ξν ) + η(ξν+1 )qν+2 (φν φX) − η(ξν+2 )qν+1 (φν φX) 2 ν=1 3

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53



− 2η(X)g(φAξν , ξν ) − η(X){η(ξν+1 )qν+2 (ξν ) − η(ξν+2 )qν+1 (ξν )ξ} +

1  − g(φAφν φX, ξν ) + hην (φX)ην (ξ) − hην (φX)Trφν φ 2 ν=1

−

1  g((A − hI)ξ, ξν )ην (φX) + η(X)g(Aφξν , ξν ) , 2 ν=1

3

3

(4.13)

where we have used Tr(∇Y A)A = (Y h)h and Trφν φ = 2ην (ξ). Then this equation implies bφAξ + c

3 

φν Aξν = 0

ν=1

for any constants b and c. This gives our assertion. 2 When b = c = 0 in Lemma 4.2, the Ricci tensor S is parallel. Recently, in [25] we have proved that there do not exist any real hypersurfaces in complex hyperbolic two-plane Grassmannians SU 2,m /S(U2 ·Um ) with parallel Ricci tensor. Hereafter, unless otherwise stated, we say a real hypersurface M in SU 2,m /S(U2 ·Um ) is proper pseudo-Einstein if

3 the Ricci tensor S satisfies SX = aX + bη(X)ξ + c ν=1 ην (X)ξν for any non-vanishing constants a, b and c on M . Related to Lemma 4.2, we can arrange some remarks as follows: Remark 4.3. When b = c = 0 in Lemma 4.2, the formulas (∗) and (∗∗) in the introduction respectively give that M is Einstein and the Ricci tensor S of M is parallel. Then the Ricci tensor commutes with the structure tensor. But recently Suh and Woo [25] have proved that there do not exist Hopf real hypersurfaces in SU 2,m /S(U2 ·Um ) with parallel Ricci tensor. So this case does not occur for pseudo-Einstein Hopf real hypersurfaces in SU 2,m /S(U2 ·Um ). Remark 4.4. When g(AQ, Q⊥ ) = 0, where Q denotes the maximal quaternionic subbundle of T M in SU 2,m /S(U2 ·Um ) spanned by {ξ1 , ξ2 , ξ3 }, Aξν = βν ξν for ν = 1, 2, 3. So Lemma 4.2 gives that M is Hopf, provided that b = 0. Then by these two remarks and using Theorem B in the introduction we assert the following: Theorem 4.5. Let M be a connected proper pseudo-Einstein real hypersurface in SU 2,m /S(U2 Um ), m ≥ 2, with Q⊥ -invariant shape operator. Then M is congruent to an open part of one of the following hypersurfaces: (A) a tube around a totally geodesic SU 2,m−1 /S(U2 Um−1 ) in SU 2,m /S(U2 Um ); (B) a tube around a totally geodesic HH n in SU 2,2n /S(U2 U2n ), m = 2n; (C) a horosphere in SU 2,m /S(U2 Um ) whose center at infinity is singular;

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or the following exceptional case holds: (D) The normal bundle νM of M consists of singular tangent vectors of type JX ⊥ JX. Moreover, M has at least four distinct principal curvatures, three of which are given by α=

√ 1 2, γ = 0, λ = √ 2

with corresponding principal curvature spaces Tα = T M  (C ∩ Q), Tγ = J(T M  Q), Tλ ⊂ C ∩ Q ∩ JQ. If μ is another (possibly nonconstant) principal curvature function, then we have Tμ ⊂ C ∩ Q ∩ JQ, JTμ ⊂ Tλ and JTμ ⊂ Tλ . 5. Pseudo-Einstein Hopf real hypersurfaces In this section, let M be a pseudo-Einstein Hopf real hypersurface in G2 (Cm+2 ). Then its Ricci tensor is given by

SX = aX + bη(X)ξ + c

3 

ην (X)ξν

ν=1

for any constants a, b and c = 2 on M . Then from this, together with expression of the Ricci tensor (4.4) in Lemma 4.1 we have

aX + bη(X)ξ + c

3 

ην (X)ξν = −

ν=1

3  1 (4m + 7)X − 3η(X)ξ − 3 ην (X)ξν 2 ν=1

+

3 



{ην (ξ)φν φX − η(φν X)φν ξ − η(X)ην (ξ)ξ}

ν=1

+ hAX − A2 X.

(5.1)

On the other hand, by (5.1) we have −2aX − 2bη(X)ξ − 2c

3 

3   ην (X)ξν = (4m + 7)X − 3η(X)ξ − 3 ην (X)ξν

ν=1

ν=1

+

3 



ην (ξ)φν φX − η(φν X)φν ξ − η(X)ην (ξ)ξν

ν=1

− 2hAX + 2A2 X.



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55

So it follows that 2A2 X − 2hAX + (4m + 7 + 2a)X =−

3  

ην (ξ)φν φX − η(φν X)φν ξ − η(X)ην (ξ)ξν



ν=1

+ (3 − 2b)η(X)ξ + (3 − 2c)

3 

ην (X)ξν .

(5.2)

ν=1

By putting X = ξ into (5.2), we have 2(m + 1)ξ − 2

3 

ην (ξ)ξν − hAξ + A2 ξ = −(a + b)ξ − c

ν=1

3 

ην (ξ)ξν .

(5.3)

ν=1

If M is a Hopf hypersurface, (5.3) gives the following {2(m + 1) − hα + α2 + (a + b)}ξ = (2 − c)

3 

ην (ξ)ξν .

(5.4)

ν=1

From this, taking the scalar product with ξμ , we get {2m − hα + α2 + a + b + c}ημ (ξ) = 0.

(5.5)

So we consider two cases. Then first we consider Case I: 2m − hα + α2 + a + b + c = 0. Then from (5.5) we know that ημ (ξ) = 0, that is, ξ ∈ Q. Now let us put T = A2 − hA. Then by (5.2) we have the following T ξ1 = (A2 − hA)ξ1 = {2 + 2m + c + a}ξ1 , T ξ2 = (A2 − hA)ξ1 = {2 + 2m + c + a}ξ2 , T ξ3 = (A2 − hA)ξ1 = {2 + 2m + c + a}ξ3 . Moreover, using the fact T A = AT and g(T Q, Q⊥ ) = 0 we know that there exists a basis X1 , X2 , X3 of Q⊥ such that AXi = λi Xi and T Xi = μi Xi , i = 1, 2, 3, in such a way that ⎛ ⎞ ⎛ ⎞ X1 ξ1 ⎝ X2 ⎠ = SO(3) ⎝ ξ2 ⎠ X3 ξ3 where SO(3) denotes a 3 × 3 special orthogonal matrix. Then Span{X1 , X2 , X3 } = Span{ξ1 , ξ2 , ξ3 }. Accordingly, we conclude that g(AQ, Q⊥ ) = 0.

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Case II: 2m − hα + α2 + a + b + c = 0. Subcase II.1: ην (ξ) = 0 for ν = 1, 2, 3. Then by (5.3) we know 2(m + 1) − hα + α2 + (a + b) = 0. From this, together with the assumption, we have c = 2, which makes a contradiction. So this case does not occur. Subcase II.2: ην (ξ)=0 for some ν ∈ {1, 2, 3}.

3 When c = 2, the Reeb vector field ξ in (5.4) becomes ξ = ν=1 ην (ξ)ξν ∈ Q⊥ . So we may put ξ = ξ1 . Then (5.2) gives the following T ξ1 = −{2m + a + b + c}ξ1 ,

(5.6)

T ξ2 = −{2m + 3 + a + c}ξ2 ,

(5.7)

T ξ3 = −{2m + 3 + a + c}ξ3 .

(5.8)

Then we know that g(T Q, Q⊥ ) = 0. Moreover, as in Case I we know that T A = AT . Then there exists a basis {X1 , X2 , X3 } in Q⊥ such that T Xi = λi Xi and AXi = μi Xi , where Span{X1 , X2 , X3 } = Span{ξ1 , ξ2 , ξ3 }. This means that the maximal quaternionic subbundle Q of M in SU 2,m /S(U2 ·Um ) is invariant by the shape operator A, that is, g(AQ, Q⊥ ) = 0. Summing up the above two cases and using Theorem B in the introduction, we assert that M is congruent to a tube over a totally geodesic SU 2,m−1 /S(U2 Um−1 ), HH n = Sp1,n /Sp1 ·Spn , a horosphere or an exceptional case. First of all, let us check the Case II with c = 2 and ξ ∈ Q⊥ , that is, JN ∈ JN . This case corresponds to (A), (C) with ξ ∈ Q⊥ in Theorem B. So let us introduce a proposition due to Suh [24] which is needed here as follows: Proposition A. Let M be a connected hypersurface in SU 2,m /S(U2 Um ), m ≥ 2. Assume that the maximal complex subbundle C of T M and the maximal quaternionic subbundle Q of T M are both invariant under the shape operator of M . If JN ∈ JN , then one the following statements holds: (i) M has exactly four distinct constant principal curvatures α = 2 coth(2r), β = coth(r), λ1 = tanh(r), λ2 = 0,

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57

and the corresponding principal curvature spaces are Tα = T M  C, Tβ = C  Q, Tλ1 = E−1 , Tλ2 = E+1 . The principal curvature spaces Tλ1 and Tλ2 are complex (with respect to J) and totally complex (with respect to J). (ii) M has exactly three distinct constant principal curvatures α = 2, β = 1, λ = 0 with corresponding principal curvature spaces Tα = T M  C, Tβ = (C  Q) ⊕ E−1 , Tλ = E+1 . Then by Proposition A we consider that the pseudo-Einstein real hypersurface is congruent to the case (i). Then for any X ∈ Tλ1 and Y ∈ Tλ2 we know that SX = aX = −

1 (4m + 7)X + φ1 φX + hAX − A2 X 2

SY = aY = −

1 (4m + 7)Y + φ1 φY + hAY − A2 Y. 2

and

From this, using φ1 φX = −X and φ1 φY = Y respectively we know that 1 1 − (4m + 6) + h tanh(r) − tanh2 (r) = − (4m + 8). 2 2 This gives h = tanh(r) − coth(r). From this, together with the trace of h = TrA, which is given by h = TrA = 2 coth(2r) + 3 coth(r) + 2(m − 1) tanh(r) = 4 coth(r) + (2m − 1) tanh(r),

(5.9)

it follows that (2m − 3) tanh2 r = −5, which gives a contradiction. So the case (i) cannot hold. Next let us consider that the pseudo-Einstein hypersurface is locally congruent to the case (ii). Note that on Q we have (φφ1 )2 = I and tr(φφ1 ) = 0. Let E+1 and E−1 be the

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eigenbundles of φφ1 |Q with respect to the eigenvalues +1 and −1, respectively. Then the maximal quaternionic subbundle Q of T M decomposes into the Whitney direct sum Q = E+1 ⊕E−1 , and the rank of both eigenbundles E±1 is equal to 2m − 2. Then we have X ∈ E+1 if and only if φX = −φ1 X and X ∈ E−1 if and only if φX = φ1 X. By using these facts, for X ∈ E−1 and Y ∈ E+1 we have respectively SX = aX  1 = − (4m + 7)X − X + hAX − A2 X 2 1 = − (4m + 6)X + hX − X 2 and SY = aY  1 = − (4m + 7)Y + Y + hAX − A2 X 2 = −2(m + 2)Y. Then these two equations give the trace of the shape operator h = 0. But the trace h becomes h = 2 + 3β + 2(m − 1)β = 2m + 3, which gives us a contradiction. So the case (ii) for our pseudo-Einstein hypersurface cannot occur. Next let us consider the case that the Reeb vector field ξ belongs to the maximal quaternionic subbundle Q of M in SU 2,m /S(U2 ·Um ). Then we introduce a proposition due to Suh [24] as follows: Proposition B. Let M be a connected hypersurface in SU 2,m /S(U2 Um ), m ≥ 2. Assume that the maximal complex subbundle C of T M and the maximal quaternionic subbundle Q of T M are both invariant under the shape operator of M . If JN ⊥ JN , then one of the following statements holds: (i) M has five (four for r = principal curvatures

√

√ 2Arctanh(1/ 3) in which case α = λ2 ) distinct constant

√ √ √ √ 2 tanh( 2r), β = 2 coth( 2r), γ = 0, 1 1 1 1 λ1 = √ tanh( √ r), λ2 = √ coth( √ r), 2 2 2 2 α=

and the corresponding principal curvature spaces are Tα = T M  C, Tβ = T M  Q, Tγ = J(T M  Q) = JTβ .

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59

The principal curvature spaces Tλ1 and Tλ2 are invariant under J and are mapped onto each other by J. In particular, the quaternionic dimension of SU 2,m /S(U2 Um ) must be even. (ii) M has exactly three distinct constant principal curvatures α=β=

√ 1 2, γ = 0, λ = √ 2

with corresponding principal curvature spaces Tα = T M  (C ∩ Q), Tγ = J(T M  Q), Tλ = C ∩ Q ∩ JQ. (iii) M has at least four distinct principal curvatures, three of which are given by α=β=

√ 1 2, γ = 0, λ = √ 2

with corresponding principal curvature spaces Tα = T M  (C ∩ Q), Tγ = J(T M  Q), Tλ ⊂ C ∩ Q ∩ JQ. If μ is another (possibly nonconstant) principal curvature function, then JTμ ⊂ Tλ and JTμ ⊂ Tλ . Now we check for case (i) in Proposition B. Then for any X ∈ Tλ1 and Y ∈ Tλ2 we have respectively for pseudo-Einstein real hypersurfaces in SU 2,m /S(U2 ·Um ) as follows: 1 SX = aX = − (4m + 7)X + (hλ1 − λ21 )X 2 and 1 SY = aY = − (4m + 7)Y + (hλ2 − λ22 )Y. 2 From this we know that h = λ1 + λ2 . On the other hand, the trace of the shape operator in case (i) becomes h = α + 3β + 3γ + 4(m − 1)λ1 + 4(m − 1)λ2 . Then these two equations for the trace h give α + 3β + (4m − 5)h = 0.

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From this, together with the fact that 2 coth

√

2r = tanh √r2 + coth √r2 it follows that

√ √ √ (4m − 2) 2 coth2 ( 2r) + 2 = 0, which gives a contradiction. So case (i) cannot appear. Now let us check the case (ii) in Proposition B. Then for any X ∈ Tλ and φ1 ξ ∈ Tγ such that AX = √12 X and Aφ1 ξ = 0 we have respectively the following 1 1 1 aX = SX = − (4m + 7)X + √ hX − X 2 2 2 and 1 aφ1 ξ = Sφ1 ξ = − (4m + 5)φ1 ξ. 2 So it follows that 1 1 1 1 − (4m + 7) + √ h − = − (4m + 5). 2 2 2 2 This gives that h =

3 2

√

2. But the trace h becomes for case (ii) h = α + 3β + 3γ + 4(m − 2)λ √ 1 = 4 2 + 4(m − 2) √ . 2

(5.10)

Then two equations for the trace h become √52 + 4(m − 2) √12 = 0, which gives a contradiction. Finally let us check case (iii) in Proposition B. By the similar method used in case (ii), √ we know that h = 32 2. On the other hand, the trace h becomes h = α + 3β + 3γ + pλ + qμ √ 1 = 4 2 + p √ + qμ 2 √ 3 = 2, 2

(5.11)

where p and q denote the multiplicity for the principal curvatures λ and μ such that p + q = 4(m − 2). Then (5.11) implies that √ 8 + p + qμ 2 = 3

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61

√ which gives the principal curvature μ is constant and μ = q−4m+3 . Now let us check for q 2 a pseudo-Einstein hypersurface in SU 2,m /S(U2 ·Um ) satisfying the case (iii) in Proposition B. In order to do this, we put X = ξ in (5.1). Then by using ξ ∈ Q, the left and the right side of (5.1) from the expression of the Ricci tensor give respectively the following

Sξ = (a + b)ξ, and  1 (4m + 7)ξ − 3ξ + hAξ − A2 ξ 2 = −(2m + 2)ξ + (hα − α2 )ξ √ = −(2m + 2)ξ + ( 2h − 2)ξ

Sξ = −

= −(2m + 1)ξ.

(5.12)

Then a + b = −(2m + 1). Moreover, by putting X = φ1 ξ in (5.1), we have known that a = − 12 (4m + 5). On the other hand, by putting X = ξ1 ∈ Q⊥ and using ξ ∈ Q, we know that √ √ √ Sξ1 = (a + c)ξ1 . Moreover, we know that Aξ1 = 2ξ1 , Aξ2 = 2ξ2 , Aξ3 = 2ξ3 . So it follows that for ξ ∈ Q  1 (4m + 7)ξ1 − 3ξ1 + hAξ1 − A2 ξ1 2 √ 1 = {− (4m + 4) + 2h − 2}ξ1 2 = −(2m + 1)ξ1 .

Sξ1 = −

(5.13)

So it gives a + c = −(2m + 1). So it follows that a = − 12 (4m + 5), b = c = 32 for a pseudoEinstein hypersurface M in SU 2,m /S(U2 ·Um ). Summing up all the statements mentioned above, we give a complete proof of our Theorem 1.1 mentioned in the introduction. Now let us consider an η-Einstein real hypersurface M in SU 2,m /S(U2 ·Um ) defined by SX = aX + bη(X)ξ for any constants a and b, and any vector field X on M . Then by (4.8) in Lemma 4.2 with c = 0 we know the following bφAξ = 0. From this we obtain the following. Lemma 5.1. Let M be an η-Einstein real hypersurface in SU 2,m /S(U2 ·Um ). Then M is either Einstein or a Hopf real hypersurface.

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We have proved a non-existence property for Einstein–Hopf real hypersurfaces in SU 2,m /S(U2 ·Um ) in Theorem 1.1. Now let us assume that M is an η-Einstein and at the same time a Hopf real hypersurface in SU 2,m /S(U2 ·Um ). Then by Theorem 1.1 we do not find any η-Einstein Hopf hypersurface, because c = 32 . Then we assert the following Theorem 5.2. There does not exist any η-Einstein Hopf real hypersurface in SU 2,m / S(U2 ·Um ). 6. Q⊥ -invariant Einstein real hypersurfaces We say that a real hypersurface M in SU 2,m /S(U2 ·Um ) is said to be Q⊥ -invariant if the distribution Q⊥ is invariant by the shape operator A, that is, AQ⊥ ⊂Q⊥ . This is equivalent to AQ⊂Q. As mentioned in Theorem A in the introduction, Pérez, Suh and Watanabe [20] gave a classification of proper pseudo-Einstein real hypersurface in G2 (Cm+2 ). Here the meaning of proper is that the constants b and c given in (∗) are non-vanishing. By virtue of this theorem, we have verified the non-existence of Einstein Hopf real hypersurfaces in G2 (Cm+2 ). Also in Theorem 1.1 and Theorem 5.2 of section 5, we have shown that there do not exist Einstein Hopf real hypersurfaces in SU 2,m /S(U2 ·Um ). Motivated by this result, in this section we want to prove another non-existence theorem for Q⊥ -invariant Einstein hypersurfaces in SU 2,m /S(U2 ·Um ). Let M be a Q⊥ -invariant Einstein real hypersurface in SU 2,m /S(U2 ·Um ). By the definition of the Ricci tensor S of M in SU 2,m /S(U2 ·Um ) and the notion of Einstein real hypersurfaces, we have 1  {ην (ξ)φν φX − η(φν X)φν ξ − η(X)ην (ξ)ξν } 2 ν=1 3

A2 X − hAX + ρX = −

− 3η(X)ξ − 3

3 

 ην (X)ξν ,

(6.1)

ν=1

where ρ = a + 12 (4m + 7). Now first of all, let us show the following. Lemma 6.1. Let M be a Q⊥ -invariant Einstein real hypersurface in a complex hyperbolic two-plane Grassmannian SU 2,m /S(U2 ·Um ). Then the structure vector ξ either belongs to the distribution Q or the distribution Q⊥ . Proof. Since T A = AT , we know that Q-invariance by the shape operator A is equivalent to Q-invariance by the operator T = hA−A2 , that is, g(T Q, Q⊥ ) = 0. From this, together with the assumption of Q-invariance by A and (6.1), we have the following

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3 3   1  ην (ξ)φν φξ1 − η(φν ξ1 )φν ξ − η(ξ1 )ην (ξ)ξν − ρξ1 − 3η(ξ1 )ξ − 3 ην (ξ1 )ξν 2 ν=1 ν=1

= A2 ξ1 − hAξ1 = T ξ1 ∈ Q⊥ .

(6.2)

Now we consider ξ = X1 + X2 for some X1 ∈ Q⊥ and X2 ∈ Q. Then by using (6.2) we get   1  ην (ξ)φν φ1 X2 − η(φν ξ1 )φν X2 − 3η(ξ1 )X2 ∈ D ∩ D⊥ = {0}. 2 ν=1 3

−

From this, by taking the inner product with X2 , we have η(ξ1 )X2 2 = 0. Similarly, we see η(ξν )X2 2 = 0 for ν = 2, 3. This means X2  = 0 or η(ξν ) = 0 for ν = 1, 2, 3. Thus either ξ ∈ Q or ξ ∈ Q⊥ . This completes the proof. 2 By Lemma 6.1 we consider the following two cases: Case 1: ξ ∈ Q. Then (6.1) gives A2 X − hAX + ρX =

3 3   1  η(φν X)φν ξ + 3η(X)ξ + 3 ην (X)ξν . 2 ν=1 ν=1

(6.3)

Now let us put Aξ = αξ + βU . We want to see that M is Hopf, that is, the function β identically vanishes on M . In order to do this, we put X = ξ in (6.3). Then it follows that A2 ξ − hAξ + {a + 2(m + 1)}ξ = 0. From this, by taking the inner product with ξ we have β 2 = hα − α2 − {a + 2(m + 1)}.

(6.4)

On the other hand, we know A2 ξ = αAξ + βAU = α2 ξ + αβU + βAU = hAξ − {a + 2(m + 1)}ξ   = hα − {a + 2(m + 1)} ξ + hβU,

(6.5)

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if we assume that the function β = 0, (6.4) and (6.5) give the following AU = (h − α)U + βξ.

(6.6)

On the other hand, by putting X = U in (6.3) and using U ∈ Q which is orthogonal to ξ, we have 1 η(φν U )φν ξ. 2 ν=1 3

A2 U − hAU + ρU =

(6.7)

Moreover, from (6.6) we know A2 U = (h − α)AU + βAξ = {(h − α)2 + β 2 }U + hβξ.

(6.8)

Substituting (6.6) and this formula into (6.7), we have 1 1 {(h − α)2 + β 2 − h(h − α) + a + (4m + 7)}U = η(φν U )φν ξ. 2 2 ν=1 3

(6.9)

This yields 1 1 {(h − α)2 + β 2 − h(h − α) + a + (4m + 7)}φν U = − η(φν U )ξ. 2 2 Then we have η(φν U ) = 0, because 1 1 (h − α)2 + β 2 − h(h − α) + a + (4m + 7) + = 2. 2 2 Now substituting η(φν U ) = 0 for ν = 1, 2, 3 into (6.9) and using (6.4), we have 3 1 0 = {(h − α)2 + β 2 − h(h − α) + a + (4m + 7)}U = U. 2 2 This yields a contradiction. Thus we must have β = 0. So the structure vector ξ is principal, that is, M is Einstein and Hopf. Moreover, M is assumed to be Q⊥ -invariant. Then by virtue of Theorem B in the introduction, we conclude that M is locally congruent to real hypersurfaces of type (A) or of type (B) in SU 2,m /S(U2 ·Um ). But there are no Einstein real hypersurfaces. Accordingly, we conclude the proof in this case.

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Case II: ξ ∈ Q⊥ . Now let us take ξ = ξ1 in (5.1). Then by formulas in section 2 it follows 1 φ2 φX + φ3 φX − η(φ2 X)φ3 ξ − η(φ3 X)φ2 ξ 2 3  − 4η(X)ξ − 3 ην (X)ξν

A2 X − hAX + ρX = −

=−

1 2

ν=1

φ2 φX + φ3 φX − 2η3 (X)ξ3 − 2η2 (X)ξ2 − 7η(X)ξ . (6.10)

Now putting X = ξ into (6.10), we have A2 ξ − hAξ + (a + 2m)ξ = 0.

(6.11)

Now let us also write Aξ = αξ + βU . From here we want to show β = 0. Taking the inner product of (6.11) with ξ, we have β 2 = hα − α2 − (2m + a).

(6.12)

Moreover, from (6.9) we know that βAU = {hα − (2m + a) − α2 }ξ + β(h − α)U. This gives AU = βξ + (h − α)U.

(6.13)

From this, by the Q⊥ -invariance we have U ∈ Q⊥ . Now substituting this formula into (6.10) gives A2 U − hAU + ρU = −

1 φ2 φU + φ3 φU − 2η3 (U )ξ3 − 2η2 (U )ξ2 . 2

(6.14)

On the other hand, by applying A to (6.13) and using (6.13) itself, we have A2 U = {(h − α)2 + β 2 }U + hβξ. Then substituting (6.15) and (6.13) into (6.14), we obtain 1 {(h − α)2 + β 2 − h(h − α) + a + (4m + 7)}U 2 1 = − φ2 φU + φ3 φU − 2η3 (U )ξ3 − 2η2 (U )ξ2 . 2

(6.15)

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From this, applying φ to both sides, we have 1 {(h − α)2 + β 2 − h(h − α) + a + (4m + 7)}φU 2 1 = φ2 U − η3 (U )ξ + φ3 U + η2 (U )ξ + 2η3 (U )ξ2 − 2η2 (U )ξ3 . 2 Then by taking its inner product with ξ2 and using (6.12), we get 1 0 = {(h − α)2 + β 2 − h(h − α) + a + (4m + 7) − 1}η3 (U ) 2 5 = η3 (U ). 2 This means η3 (U ) = 0 and also by using a similar method we have η2 (U ) = 0, which implies U = ±ξ1 . But Q⊥ is invariant by A, U belongs to Q⊥ and is orthogonal to ξ, because we have put Aξ = αξ + βU . This gives a contradiction. So we must have β = 0, that is, M is Hopf. But in section 5 we have already shown that there does not exist any Hopf–Einstein real hypersurface in SU 2,m /S(U2 ·Um ). This completes the proof of Theorem 1.2 in the introduction. Acknowledgment The present author would like to express his deep gratitude to the referee for his careful reading of our article and valuable suggestions to improve the first version of this manuscript. References [1] J. Berndt, Y.J. Suh, Hypersurfaces in noncompact complex Grassmannians of rank two, Internat. J. Math. 23 (2012) 1250103. [2] É. Cartan, Families de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl. (4) 17 (1938) 177–191. [3] T.E. Cecil, P.J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982) 481–499. [4] P.B. Eberlein, Geometry of Nonpositively Curved Manifolds, vol. 7, University of Chicago Press, Chicago, London, 1996. [5] A. Fialkow, Hypersurfaces of spaces of constant curvature, Ann. of Math. 39 (1938) 762–785. [6] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Grad. Stud. Math., vol. 34, Amer. Math. Soc., 2001. [7] S. Helgason, Geometric Analysis on Symmetric Spaces, 2nd edition, Math. Surveys Monogr., vol. 39, Amer. Math. Soc., 2008. [8] I. Jeong, C.J.G. Machado, J.D. Pérez, Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with D⊥ -parallel structure Jacobi operator, Internat. J. Math. 22 (2011) 655–673. [9] I. Jeong, Y.J. Suh, Real hypersurfaces of type A in complex two-plane Grassmannians related to commuting shape operator, Forum Math. 26 (2013) 179–192. [10] M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986) 137–149. [11] M. Kimura, Some real hypersurfaces of a complex projective space, Saitama Math. J. 5 (1987) 1–5. [12] A.W. Knap, Lie Groups beyond an Introduction, Progr. Math., Birkhäuser, 2002.

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