Real hypersurfaces in the complex quadric with parallel Ricci tensor

In this paper we consider a new notion of D ⊥ -parallel shape operator for real hypersurfaces in complex two-plane Grassmannians G2(C m+2 ) and give a non-existence theorem for a Hopf hypersurface in G2(C m+2 )w ithD ⊥ -parallel shape operator. Mathematics Subject Classification (2010). Primary 53C40; Secondary 53C15.

Real Hypersurfaces in Complex Two-plane Grassmannians with D ⊥ -Parallel Shape Operator

Generalized Einstein real hypersurfaces in complex two-plane Grassmannians

Real hypersurfaces in the complex quadric with harmonic curvature

In this paper we give some non-existence theorems for Hopf real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2) with &#x1d507;⊥-parallel structure Jacobi operator, where &#x1d507;⊥ = Span {ξ1, ξ2, ξ3}.

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH &#x1d507;⊥-PARALLEL STRUCTURE JACOBI OPERATOR

. In this paper, ﬂrst we introduce the full expression of thecurvature tensor of a real hypersurface M in complex two-plane Grass-mannians G 2 (C m +2 ) from the equation of Gauss and derive a new formulafor the Ricci tensor of M in G 2 (C m +2 ). Next we prove that there do notexist any Hopf real hypersurfaces in complex two-plane Grassmannians G 2 (C m +2 ) with parallel and commuting Ricci tensor. Finally we showthat there do not exist any Einstein Hopf hypersurfaces in G 2 (C m +2 ). IntroductionIn the geometry of real hypersurfaces in complex space forms or in quater-nionic space forms it can be easily checked that there do not exist any realhypersurfaces with parallel shape operator A by virtue of the equation of Co-dazzi.But if we consider a real hypersurface with parallel Ricci tensor S in suchspace forms, the proof of its non-existence is not so easy. In the class of Hopfhypersurfaces Kimura [7] has asserted that there do not exist any real hyper-surfaces in a complex projective space C

/pdf/the-ricci-tensor-of-real-hypersurfaces-in-complex-two-plane-40heaihv1c.pdf

The ricci tensor of real hypersurfaces in complex two-plane grassmannians

We know that there are no real hypersurfaces with parallel Ricci tensor or parallel structure Jacobi operator in a nonflat complex space form (See [4], [6], [10] and [11]) In this paper we investigate real hypersurfaces M in a nonflat complex space form Mn(c) under the condition that ∇ξS = 0 and ∇ξRξ = 0, where S and Rξ respectively denote the Ricci tensor and the structure Jacobi operator of M in Mn(c) 0 Introduction A Kaehler manifold of constant holomorphic sectional curvature c is called a complex space form, which is denoted by Mn(c) It is well known that complete and simply connected complex space forms are isometric to a complex projective space PnC, a complex Euclidean space C or a complex hyperbolic space HnC according as c > 0, c = 0 and c < 0 In this paper we consider a real hypersurface M in a complex space form Mn(c), c6=0 Then M has an almost contact metric structure (φ, ξ, η, g) induced from the complex structure J and the Kaehler metric of Mn(c) The structure vector field ξ is said to be principal if Aξ = αξ is satisfied, where A denotes the shape operator of M and α = η(Aξ) A real hypersurface is said to be a Hopf hypersurface if the structure vector field ξ of M is principal In the study of real hypersurfaces in PnC, Takagi [12] classified all homogeneous real hypersurfaces and Cecil and Ryan [2] showed that they can be regarded as the tubes of constant radius over Kaehler submanifolds when the structure vector field ξ is principal Such tubes can be divided into six kinds of type A1, A2, B, C, D and E Received August 8, 2005; Revised May 30, 2006 2000 Mathematics Subject Classification Primary 53C40; Secondary 53C15

http://koreascience.or.kr:80/article/JAKO200712242658582.pdf

Suh Young-Jin

Papers

The ricci tensor of real hypersurfaces in complex two-plane grassmannians

REAL HYPERSURFACES IN COMPLEX SPACE FORMS WITH ε-PARALLEL RICCI TENSOR AND STRUCTURE JACOBI OPERATOR