Semi-parallel real hypersurfaces in complex two-plane Grassmannians

We prove that there does not exist any semi-parallel real hypersurface in complex two-plane Grassmannians. With this result, the nonexistence of recurrent real hypersurfaces in complex two-plane Grassmannians can also be proved.     

Recurrent real hypersurfaces in complex two-plane Grassmannians

Summary We introduce the notion of recurrent shape operator for a real hypersurface M in the complex two-plane Grassmannians G₂(C^m+2) and give a non-existence property of real hypersurfaces in G₂(C^m+2) with the recurrent shape operator.     

Reeb parallel Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two‐plane Grassmannians

In this paper, we introduce the notion of Reeb parallel Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two‐plane Grassmannians which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. By using a new method of simultaneous diagonalizations, we give a complete classification for real hypersurfaces in complex hyperbolic two‐plane Grassmannians with the Reeb parallel Ricci tensor.     

Real hypersurfaces in complex hyperbolic two‐plane Grassmannians with parallel Ricci tensor

In this paper we prove that there does not exist any Hopf real hypersurface in complex hyperbolic two‐plane Grassmannians with parallel Ricci tensor.     

Homogeneous hypersurfaces in complex hyperbolic spaces

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.      

Pseudo anti‐commuting Ricci tensor and Ricci soliton real hypersurfaces in complex hyperbolic two‐plane Grassmannians

In this paper, first we introduce a new notion of pseudo anti commuting Ricci tensor for real hypersurfaces in complex hyperbolic two‐plane Grassmannians and prove a complete classification theorem that such a hypersurface must be a tube over a totally real totally geodesic , , a horosphere whose center at the infinity is singular or an exceptional case.     

Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator

We give a complete classification of -invariant real hypersurfaces in complex two-plane Grassmannians G ₂(C ^m+2) with commuting normal Jacobi operator . The first author was supported by MCYT-FEDER grant BFM 2001-2871-C04-01, the second author by grant Proj. No. KRF-2006-351-C00004 from Korea Research Foundation and the third author by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2006 and Proj. No. R17-2007-006-01000-0 from KOSEF.     

Real minimal hypersurfaces in complex two-plane Grassmannians

We give a pinching condition for compact minimal hypersurfaces in complex two-plane Grassmannians G ₂(? ^m+2) in terms of sectional curvature and the squared norm of the shape operator.     

Commuting structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians

We classify real hypersurfaces in complex two-plane Grassmannians whose structure Jacobi operator commutes either with any other Jacobi operator or with the normal Jacobi operator.     

Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting restricted normal Jacobi operators

We give a classification of Hopf real hypersurfaces in complex hyperbolic two-plane Grassmannians SU_2,m/S(U₂·U _m ) with commuting conditions between the restricted normal Jacobi operator \({\bar R_{N\varphi }}\) and the shape operator A (or the Ricci tensor S).     

Integral curves of characteristic vector fields of real hypersurfaces in nonflat complex space forms

In this paper, we study real hypersurfaces all of whose integral curves of characteristic vector fields are plane curves in a nonflat complex space form.      

On real hypersurfaces in a quaternionic hyperbolic space in terms of the derivative of the second fundamental tensor

The purpose of this paper is to give a characterization of real hypersurfaces of type A₀, A in a quaternionic hyperbolic space QH ^m by the covariant derivative of the second fundamental tensor. This revised version was published online in June 2006 with corrections to the Cover Date.     

Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition

In this paper, first we introduce a new notion of commuting condition that φφ ₁ A = A φ ₁ φ between the shape operator A and the structure tensors φ and φ ₁ for real hypersurfaces in G ₂(? ^m+2). Suprisingly, real hypersurfaces of type (A), that is, a tube over a totally geodesic G ₂(? ^m+1) in complex two plane Grassmannians G ₂(? ^m+2) satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in G ₂(? ^m+2) satisfying the commuting condition. Finally we get a characterization of Type (A) in terms of such commuting condition φφ ₁ A = A φ ₁ φ.     

Real hypersurfaces foliated by totally real totally geodesic submanifolds

We bridge between submanifold geometry and curve theory. In the first half of this paper we classify real hypersurfaces in a complex projective plane and a complex hyperbolic plane all of whose integral curves γ of the characteristic vector field are totally real circles of the same curvature which is independent of the choice of γ in these planes. In the latter half, we construct real hypersurfaces which are foliated by totally real (Lagrangian) totally geodesic submanifolds in a complex hyperbolic plane, which provide one of the examples obtained in the classification.     

Topological and metric rigidity teorems for hypersurfaces in a hyperbolic space

In this paper we study the topological and metric rigidity of hypersurfaces in ℍ ⁿ⁺¹, the (n + 1)-dimensional hyperbolic space of sectional curvature −1. We find conditions to ensure a complete connected oriented hypersurface in ℍ ⁿ⁺¹ to be diffeomorphic to a Euclidean sphere. We also give sufficient conditions for a complete connected oriented closed hypersurface with constant norm of the second fundamental form to be totally umbilic.     

Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians

We classify the real hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians _SU2,m/S(_U2⋅_Um)${\mathit{SU}}_{2,m}/S(U_2\cdot U_m)$, m?2$m?2$. Each can be described as a tube over a totally geodesic _SU2,m−1/S(_U2⋅_Um−1)${\mathit{SU}}_{2,m-1}/S(U_2\cdot U_{m-1})$ in _SU2,m/S(_U2⋅_Um)${\mathit{SU}}_{2,m}/S(U_2\cdot U_m)$ or a horosphere whose center at infinity is singular.     

Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II

Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces M of Type (A) in complex two plane Grassmannians G ₂(? ^m+2) with a commuting condition between the shape operator A and the structure tensors φ and φ ₁ for M in G ₂(? ^m+2). Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator A and a new operator φφ ₁ induced by two structure tensors φ and φ ₁. That is, this commuting shape operator is given by φφ ₁ A = A φφ ₁. Using this condition, we prove that M is locally congruent to a tube of radius r over a totally geodesic G ₂(? ^m+1) in G ₂(? ^m+2).     

Hopf hypersurfaces in pseudo-Riemannian complex and para-complex space forms

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by J be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field N is said to be Hopf if the tangent vector field JN is a principal direction. It is proved that if a hypersurface is Hopf, then the corresponding principal curvature (the Hopf curvature) is constant. It is also observed that in some cases a Hopf hypersurface must be, locally, a tube over a complex (or para-complex) submanifold, thus generalizing previous results of Cecil, Ryan and Montiel.