On Einsteinian Manifolds Admitting Orthogonal Families of Totally Umbilical Hypersurfaces

The aim of the present paper is to study globally the Riemannian manifold admitting two or more mutually orthogonal families of totally umbilieal hypersurfaces of which each is Einsteinian. This paper consists of four parts: (i) to establish anew the canonical form of the metric of (M,g)admitting p (p≥2) families of mutually orthogonal totally umbilical hypersurfaces from the standpoint of global differential geometry; (ii) to prove in a n-dimensional (n>2) Einsteinian manifold E_n of nonvanishing scalar curvature there doesn't exist one family of compact totally geodesic Einsteinian hypersurfaces (Theorem I); (iii) to prove in a n-dimensional (n≥5) Einsteinian manifold E, of nonnegative scalar curvature there don't exist two orthogonal families of totally umbilical but not geodesic complete Einsteinian hypersurfaces (Theorem II); (iv) to show that a n-dimensional (n≥5) Riemannian manifold of negative constant scalar curvature admitting p (p≥3) mutually orthogonal families of compact, totally umbili     

常曲率空间关于直交全脐超曲面系的特征性质

Let V_n be Riemannian space of genernal constant curvature.In this paper, we have proved following;Theorem I If a V_n(n≥5 ) admits three mutually orthogonal families oftotally numbilical hypersurfaces such that they are of constant curvature and Einsteinian and of general constant curvature respectively, then V_n is space with constant curvature.Theorem 2 If a V_n ( n ≥ 5 ) admits three mutually orthogonal famities of totally umbilical hypersurfaces, of which one is conformally flat and other two are Einsteinian and of constant curvature respectively, and latter either is of constant meam curvature, then V_n is of constant curvature.     

Translating Solutions of the Nonparametric Mean Curvature Flow with Nonzero Neumann Boundary Data in Product Manifold Mn × R*

In this paper, the authors can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold Mⁿ × R, where Mⁿ is an n-dimensional (n ≥ 2) complete Riemannian manifold with nonnegative Ricci curvature, and R is the Euclidean 1-space.     

某些黎曼流形中的全脐子流形

The main result obtaind in this paper is that :Let M be a totally umbilical submanifolds in Riemannian manifold N. If the Weyl conformal curvature tensor for N satisfies the following condition: ▽_xC=ω(X)C, for some 1-form ω and any vector field X in M, then M is con-formally flat or it is totally geodesic .     

具有调和曲率与有限$L^p$曲率的完备流形

Let(M~n, g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R?m the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R?m goes to zero uniformly at infinity if for p ≥ n, the L~p-norm of R?m is finite.As applications, we prove that(M~n, g) is compact if the L~p-norm of R?m is finite and R is positive, and(M~n, g) is scalar flat if(M~n, g) is a complete noncompact manifold with nonnegative scalar curvature and finite L~p-norm of R?m. We prove that(M~n, g) is isometric to a spherical space form if for p ≥n/2, the L~p-norm of R?m is sufficiently small and R is positive.In particular, we prove that(M~n, g) is isometric to a spherical space form if for p ≥ n, R is positive and the L~p-norm of R?m is pinched in [0, C), where C is an explicit positive constant depending only on n, p, R and the Yamabe constant.     

Rigidity of closed submanifolds in a locally symmetric Riemannian manifold

Let M~n(n ≥ 4) be an oriented closed submanifold with parallel mean curvature in an(n + p)-dimensional locally symmetric Riemannian manifold N~(n+p). We prove that if the sectional curvature of N is positively pinched in [δ, 1], and the Ricci curvature of M satisfies a pinching condition, then M is either a totally umbilical submanifold, or δ = 1, and N is of constant curvature. This result generalizes the geometric rigidity theorem due to Xu and Gu[15].     

Generalized Einstein tensor for a Weyl manifold and its applications

It is well known that the Einstein tensor G for a Riemannian manifold defined by G βα = R βα 1/2 Rδβα , R βα = g βγ R γα where R γα and R are respectively the Ricci tensor and the scalar curvature of the manifold, plays an important part in Einstein's theory of gravitation as well as in proving some theorems in Riemannian geometry. In this work, we first obtain the generalized Einstein tensor for a Weyl manifold. Then, after studying some properties of generalized Einstein tensor, we prove that the conformal invariance of the generalized Einstein tensor implies the conformal invariance of the curvature tensor of the Weyl manifold and conversely. Moreover, we show that such Weyl manifolds admit a one-parameter family of hypersurfaces the orthogonal trajectories of which are geodesics. Finally, a necessary and sufficient condition in order that the generalized circles of a Weyl manifold be preserved by a conformal mapping is stated in terms of generalized Einstein tensors at corresponding points.     

A RIGIDITY THEOREM FOR NON-NEGATIVELY IMMERSED SUBMANIFOLDS

The paper is to generalize the rigidity theorem that the special Weingarten surface isthe sphere to the case of submanifolds.It is proved that a non-negatively immersedcompact submaifnold in space form of constant curvature is a Riemannian product ofseveral totally umbilical submanifolds if the mean curvature and the scalar curvature ofthe submanifold satisfy a certain function relation.     

HYPERSURFACES IN SPACE FORMS WITH SCALAR CURVATURE CONDITIONS

Let f : M^n→S^n 1真包含于R^n 2 be an n-dimensional complete oriented Riemannian manifold minimally immersed in an (n 1)-dimensional unit sphere S^n 1. Denote by S^n 1 the upper closed hemisphere. If f(M^n)包含于S ^n 1, then under some curvature conditions the authors can get that the isometric immersion is a totally embedding. They also generalize a theorem of Li Hai Zhong on hypersurface of space form with costant scalar curvature.     

RANDERS SPACES WITH SCALAR FLAG CURVATURE

Let(M, F) be an n-dimensional Randers space with scalar flag curvature. In this paper, we will introduce the definition of a weak Einstein manifold. We can prove that if(M, F) is a weak Einstein manifold, then the flag curvature is constant.