局部共形对称黎曼流形的孤立现象

讨论了局部共形对称的封闭黎曼流形,证明了黎曼曲率张量模长的一个拼挤定理.当M是局部共形平坦流形时,得到了曲率张量模长的最佳拼挤常数,并确定了达到该值的黎曼流形.     

单位球面中极小子流形的曲率拼挤

本文证明了单位球面中极小子流形的一些拼挤定理,特别注意到单位球面中的极小超曲面、给出了截曲率的拼挤常数,我们也改进了由Ｎ．Ｅｊｉｒｉ得到的Ｒｉｃｃｉ曲率拼挤常数。     

局部对称共形平坦黎曼流形中紧致子流形的一个刚性定理

本文研究局部对称共形平坦黎曼流形N^n p(p≥2）中具有平行平均曲率向量的紧致子流形M^n的余维可约性问题，在n≥8的条件下得到了最佳拼挤常数。     

李奇曲率平行的黎曼流形的孤立现象

本文研究李奇曲率平行的封闭黎曼流形，证明了黎曼曲率平方的一个拚挤定理。     

局部对称共形平坦黎曼流形中紧致子流形的一个刚性定理

研究局部对称共形平坦黎曼流形N^n＋p（p≥2）中具有平等平均曲率向量的紧致子流形M^n的余维可约性问题，在n≥8的条件下得到了量佳拼挤常数．     

关于局部对称空间中的极小子流形

本文研究局部对称完备黎曼流形中的紧致极小流形，得到了这类子流形的第二基本形式模长平方的一个拼挤定理，推广了［１］中的结论．     

局部对称共形平坦黎曼流形中的紧致子流形

本文讨论局部对称共形平坦黎曼流形中紧子流形问题．改进了［１］的结果并将［２］中关于球面子流形的一个结果推广到局部对称共形平坦黎曼流形子流形．     

径向截面曲率有下界黎曼流形的微分同胚定理

研究了径向截面曲率以一类旋转模曲面的Gauss曲率为下界的非紧完备黎曼流形的拓扑,得到了该类黎曼流形与欧氏空间微分同胚的一个合理的充分条件,推广了径向截面曲率有常数下界完备黎曼流形的微分同胚定理.     

李奇曲率平行的黎曼流形到欧氏空间的等距浸入

设ｆ：Ｍｎ→Ｒｎ＋ｐ为具平行李奇曲率的黎曼流形到欧氏空间的等距浸入．对ｐ＝１，本文给出了极小条件下以及平均曲率处处非零条件下该浸入的分类     

黎曼流形上的Nash不等式

本文通过对满足Nash不等式的黎曼流形的研究,证明了对任一完备的Ricci曲率非负的n维黎曼流形,若它满足Nash不等式,且Nash常数大于最佳Nash常数,则它微分同胚于Rn．     

拟常曲率流形中全脐点子流形的拼嵌定理

We study the global umbilic submanifolds with parallel mean curvature vector fields in a Riemannian manifold with quasi constant curvature and get a local pinching theorem about the length of the second fundamental form.     

Finsler Manifolds with Positive Constant Flag Curvature

It is shown that a Finsler metric with positive constant flag curvature and vanishing mean tangent curvature must be Riemannian. As applications, we also discuss the case of Cheng's maximal diameter theorem and Green's maximal conjugate radius theorem in Finsler manifolds.     

常曲率空间中具平行平均曲率向量的子流形

本文利用第二基本形式的长度平方和平均曲率的关系研究常曲率空间中具平行平均曲率向量的子流形为全脐的ｐｉｎｃｈｉｎｇ问题，获得了一定条件下的最佳ｐｉｎｃｈｉｎｇ区间，并确定了ｐｈｉｎｃｎｉｎｇ区间端点处对应非全脐子流形的分类．     

Riemannian Manifold in Which the Skew-Symmetric Curvature Operator has Pointwise Constant Eigenvalues

We study manifolds where the natural skew-symmetric curvature operator has pointwise constant eigenvalues. We give a local classification (up to isometry) of such manifolds in dimension 4. In dimension 3, we describe such manifolds up to a classification of three - dimensional Riemannian manifolds with principal Ricci curvatures r₁ = r₂ = 0, r₃- arbitrary. We give examples of such manifolds in all dimensions which do not have constant sectional curvature; these manifolds are not pointwise Osserman manifolds in general.     

Lie Superalgebras Associated with Constant Sectional Curvature

This note describes an observation connecting Riemannian manifolds of constant sectional curvature with a particular class of Lie superalgebras. Specifically, it is shown that the structural equations of a space M with constant sectional curvature, of one variety or another, nearly coincide with some identities satisfied by tensors which can be used to construct some specific families of Lie superalgebras. In particular, one obtains either osp(n,2), spl(n,2), or osp(4,2n) if the Riemannian manifold has constant curvature, constant holomorphic curvature or constant quaternion-holomorphic curvature, respectively.Mathematics Subject Classiffications (2000). 17A70, 53C29, 53C99, 57Rxx     

Rigidity of complete manifolds with parallel Cotton tensor

The aim of this paper is to show some rigidity results for complete Riemannian manifolds with parallel Cotton tensor. In particular, we prove that any compact manifold of dimension \(n\ge 3\) with parallel Cotton tensor and positive constant scalar curvature is isometric to a finite quotient of \({\mathbb {S}}^n\) under a pointwise or integral pinching condition. Moreover, a rigidity theorem for stochastically complete manifolds with parallel Cotton tensor is also given. The proofs rely mainly on curvature elliptic estimates and the weak maximum principle.     

具有两个不同Ricci主曲率的局部共形平坦Riemann流形

得到了具有常m阶Schouten曲率与两个不同Schouten主曲率(或者等价地,两个不同Ricci主曲率)的完备局部共形平坦Riemann流形的分类结果.作为应用,得到了若干Schouten张量的pinching性质.     

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.