The constancy of principal curvatures of curvature-adapted submanifolds in symmetric spaces

In this paper, we investigate complete curvature-adapted submanifolds with maximal flat section and trivial normal holonomy group in symmetric spaces of compact type or non-compact type under a certain condition, and derive the constancy of the principal curvatures of such submanifolds. As a result, we derive that such submanifolds are isoparametric.     

On spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

The purpose of this paper is to study compact or complete spacelike hypersurfaces with constant normalized scalar curvature in a locally symmetric Lorentz space satisfying some curvature conditions. We give an optimal estimate of the squared norm of the second fundamental form of such hypersurfaces. Furthermore, the totally umbilical hypersurfaces are characterized.     

Ruled Real Hypersurfaces in a Nonflat Quaternionic Space Form

In this paper we construct many ruled real hypersurfaces in a nonflat quaternionic space form systematically, and in particular give an example of a homogeneous ruled real hypersurface in a quaternionic hyperbolic space. In the second half of this paper we characterize them by investigating the extrinsic shape of their geodesics. We also characterize curvature-adapted real hypersurfaces in nonflat quaternionic space forms from the same viewpoint.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.     

The classification of golden shaped hypersurfaces in Lorentz space forms

In this paper, we will study the golden shaped hypersurfaces in Lorentz space forms. Based on the classification of isoparametric hypersurfaces, we obtain the whole families of the golden shaped hypersurfaces in Minkowski space, de Sitter space and anti-de Sitter space, respectively.     

Periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces

This paper discusses the existence and multiplicity of periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces. We prove that each such energy hypersurface carries at least one symmetric periodic orbit. Under some suitable pinching conditions, we also get an existence result of multiple symmetric periodic orbits.     

On Ruled Real Hypersurfaces in Complex Space Forms

Ruled real hypersurfaces of complex space forms are investigated by using the fact that such hypersurfaces can be constructed by moving a 1-codimensional complex totally geodesic submanifold of the ambient space along a curve. Among other results, a classification of minimal ruled real hypersurfaces and an example of a homogeneous ruled real hypersurface are given.     

Quasi—Einstein Hypersurfaces in a Hyperbolic Space

§1.　IntroductionLetRijbethecomponentsofRiccitensorofann-dimensionalRiemannianmanifoldM.IfRij=Agij Bξiξj,　(i,j=1,2,…,n)(1.1)whereξisanunitvectorfield,thenMiscalledaquasi-EinsteinmanifoldanddenotedbyQE(ξ).Ifξisanisotropicvectorfield,thenMiscalledageneralizedquasi-Einsteinmanifold.Intheequality(1.1),AandBarescalarfunctions.WeknowQE(ξ)manifoldisEinsteinwhenB≡0.Especially,if〈ξ,ξ〉=e=±1,thenQE(ξ)iscalledanormalquasi-Einsteinmani-fold.Itiseasytoknowfrom[1]and[2]:Rij=R-Tn-1…     

Einstein hypersurfaces of the Cayley projective plane

We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a certain radius; this completes the classification of Einstein hypersurfaces in rank-one symmetric spaces.     

Stable minimal hypersurfaces in locally symmetric spaces

LetM be a complete non‐compact stable minimal hypersurface in a locally symmetric space N of nonnegative Ricci curvature. We prove that if the integral of square norm of the second fundamental form is finite, i.e., ∫_M |A |² dv < ∞, then M must be totally geodesic. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

局部对称黎曼流形中的超曲面

研究了局部对称黎曼流形中具有常平均曲率超曲面的几何性质,得到了关于其第二基本形式模长平方的拼挤定理.     

Generalized Cartan Identities on Isoparametric Manifolds

We first introduce the concept of an isoparametric manifold, which is the natural generalization of the concept of an isoparametric hypersurface in a real space form and that of a space-like isoparametric hypersurface in a Lorentzian space form. Finally, we establish generalized Cartan identities for isoparametric manifolds. This revised version was published online in June 2006 with corrections to the Cover Date.     

Symmetry of Stationary Hypersurfaces in Hyperbolic Space

We deposit a prescribed amount of liquid on an umbilical hypersurface Π of the hyperbolic space . Under the presence of a uniform gravity vector field directed towards Π, we seek the shape of such a liquid drop in a state of equilibrium of the mechanical system. The liquid-air interface in then modeled by a hypersurface under the condition that its mean curvature is a function of the distance from Π, together with the fact that the angle that makes with Π along its boundary is constant. We show that the hypersurface is rotational symmetric with respect to a geodesic orthogonal to Π. We extend this result to other configurations, for example, liquid bridges trapped between two umbilical hypersurfaces. Finally, we obtain a result which says that, under some assumptions on the mean curvature, an embedded hypersurface inherits a certain symmetry from its boundary.Partially supported by MEC-FEDER Grant no. MTM2004-00109.     

ON A CURVATURE CHARACTERIZATION OF RICCI-PSEUDOSYMMETRIC HYPERSURFACES

The author presents a curvature characterization of Ricci-pseudosymmetric hypersurfaces in semi-Riemannian space forms Ns^n 1(c), n≥4.     

A Ricci-semi-symmetric hypersurface of Euclidean space which is not semi-symmetric

We construct the first explicit example of a Ricci-semi-symmetric hypersurface of Euclidean space which is not semi-symmetric.

     

A NOTE ON THE MEAN CURVATURE FLOW IN RIEMANNIAN MANIFOLDS

Under the hypothesis of mean curvature flows of hypersurfaces, we prove that the limit of the smooth rescaling of the singularity is weakly convex. It is a generalization of the result due to G.Huisken and C. Sinestrari in. These apriori bounds are satisfied for mean convex hypersurfaces in locally symmetric Riemannian manifolds with nonnegative sectional curvature.     

Pseudosymmetric Lightlike Hypersurfaces in Indefinite Sasakian Space Forms

We study pseudosymmetric lightlike hypersurfaces of an indefinite Sasakian space form, tangent to the structure vector field. We obtain sufficient conditions for a lightlike hypersurface to be pseudosymmetric, pseudoparallel and Ricci-pseudosymmetric in an indefinite Sasakian space form. We also find certain conditions for a pseudosymmetric lightlike hypersurface of an indefinite Sasakian space form to be totally geodesic and check the effect of Weyl projective pseudosymmetry conditions on the geometry of a lightlike hypersurface of an indefinite Sasakian space form. Moreover we give some physical interpretations of pseudo-symmetry conditions.     

Normal connections induced by a framed hypersurface in the conformal space

We find two normal connections induced by the normal framing of a hypersurface V _{n ? 1} in the conformal space C _n , and establish relationship between these connections and a Weyl connection which is also induced by the normal framing of V _{n ? 1}. We study two normal connections induced by a complete framing of a hypersurface V _{n ? 1} in C _n . We establish relationship between geometries of a framed hypersurface V _{n ? 1} of the conformal space C _n and a quadratic hyperband of the projective space P _{n + 1} associated with V _{n ? 1}.     

De Sitter空间的2-调和超曲面

设M是de sitter空间S1^n 1(1)的紧致2－调和类空超曲面，获得了关于M的第二基本形式模长平方的Pinching结果。     

局部对称空间中具有常数量曲率的超曲面

利用自伴算子研究局部对称空间中具有常数量曲率的紧致超曲面,得到了这类超曲面中的某些刚性定理,推广了已有的结果.