A Parametrization of Isometric Immersions between Hyperbolic Spaces

We parametrize the space of isometric immersions of the hyperbolic n -space into the hyperbolic ( n +1)-space by a family of properly chosen (at most) countable n -tuples of real-valued functions. This is an answer to the problem posed by Nomizu in 1973. We also construct a one-parameter family of deformable isometric immersions from the quotient spaces of the hyperbolic 2-plane modulo and some totally discontinuous subgroups of isometries.     

Characterization of totally umbilic hypersurfaces in a space form by circles

In this paper we characterize totally umbilic hypersurfaces in a space form by a property of the extrinsic shape of circles on hypersurfaces. This characterization corresponds to characterizations of isoparametric hypersurfaces in a space form by properties of the extrinsic shape of geodesics due to Kimura-Maeda.The first author was partially supported by Grant-in-Aid for Scientific Research (C) No. 14540075, Ministry of Education, Science, Sports and Culture. The second author was partially supported by Grant-in-Aid for Scientific Research (C) No. 14540080, Ministry of Education, Science, Sports and Culture.     

Conformally Flat Semi-Riemannian Manifolds with Nilpotent Ricci Operators and Affine Differential Geometry

We investigate the conformally flat semi-Riemannian manifolds withnilpotent Ricci operators. We construct a lot of complete orhomogeneous, conformally flat semi-Riemannian manifolds with nilpotentRicci operators. In this construction, we show interesting relationsbetween the semi-Riemannian geometry and the affine differentialgeometry of centro-affine hypersurfaces.     

Integral formulas for compact spacelike hypersurfaces in de sitter space and their applications to Goddard's conjecture

We establish integral formulas of Minkowski's type for compact spacelike hypersurfaces in de sitter spaceS ₁ ⁿ⁺¹ (1) and give their applications to the case of constantr-th mean curvature (r=1,2,…,n−1). Whenr=1 we recover Montiel's result. Li Haizhong is supported by NNSFC No.19701017 and Basic Science Research Foundation of Tsinghua University and Chen Weihua is supported by NNSFC No. 19571005     

Real hypersurfaces in complex space forms concerned with the local symmetry

This paper consists of two parts. In the first, we find some geometric conditions derived from the local symmetry of the inverse image by the Hopf fibration of a real hypersurface M in complex space form M _m(4ε). In the second, we give a complete classification of real hypersurfaces in M _m(4ε) which satisfy the above geometric facts. The second author was supported by DGICYT research project BFM 2001-2871-C04-01 and the first and the third authors were supported by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2006.     

Real hypersurfaces with constant totally real bisectional curvature in complex space forms

In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space form M _m (c), c ≠ 0 as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional curvature given in [11].     

Laguerre isoparametric hypersurfaces in ℝ n with two distinct non-zero principal curvatures

An umbilical free oriented hypersurfacex：M→Rnwith non-zero principal curvatures is called a Laguerre isoparametric hypersurface if its Laguerre form C=i Ciωi=iρ1（Ei（logρ）（r ri）Ei（r））ωi vanishes and Laguerre shape operator S=ρ1（S 1 rid）has constant eigenvalues.Hereρ=i（r ri）2,r=r1＋r2＋···＋rn 1n 1is the mean curvature radius andSis the shape operator ofx.{Ei}is a local basis for Laguerre metric g=ρ2III with dual basis{ωi}and III is the third fundamental form ofx.In this paper,we classify all Laguerre isoparametric hypersurfaces in Rn（n〉3）with two distinct non-zero principal curvatures up to Laguerre transformations.     

Hopf Hypersurfaces in Complex Two-Plane Grassmannians with Generalized Tanaka–Webster D-Parallel Shape Operator

In this paper, we consider a new notion of generalized Tanaka–Webster D-parallel shape operator for a real hypersurface in a complex two-plane Grassmannian and prove a non-existence theorem of a real hypersurface.     

Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

We classify real hypersurfaces with constant principal curvatures in the complex hyperbolic plane. It follows from this classification that all of them are open parts of homogeneous ones.