Ricci曲率平行的一类Riemann流形的C∞紧性

本文证明,在Gromov-Hausdorff拓扑下,Ricci曲率平行,截面曲率和单一半径有下界,体积有上界的Riemann流形的集合是c^∞紧的.作为应用,我们证明一个pinching结果,即在某些条件下,Ricci平坦的流形必定平坦.     

k维环面上坐标自映射下拓扑熵的一个下界

给出了ｋ维环面上坐标自映射下拓扑熵的一个下界，最后，还指出了ｋ维环面上渐近Ｒｅｉｄｅｍｅｉｓｔｅｒ数严格大于渐近Ｎｉｅｌｓｅｎ数的情形，并说明了文（３）（或文（４）中引理１为该文的一个特例。     

小Excess与具负下界Ricci曲率开流形的拓扑

In this paper,we study the relation between the excess of open manifolds and their topology by using the methods of comparison geometry.We prove that a complete open Riemmannian manifold with Ricci curvature negatively lower bounded is of finite topological type provided that the conjugate radius is bounded from below by a positive constant and its Excess is bounded by some function of its conjugate radius,which improves some results in [4].     

Excess函数的应用

在本文中，我们研究了曲率有下界的开流形的拓扑，并推广了文[7]中的结果，证明了截曲率有下界的开流形如果它的excess函数被它的临界半径的某个函数所界定时，它就具有有限拓扑型或者微分同胚于R^n．     

关于Lapleca算子的大特征值

在两种情下，给出了Laplace算子大特征值的的上界估计，改进了两个已有的结果。     

一般Hirsch问题

Hirsch问题设UR3,VR2为开集,如果fU→V是C1满映射,则f必须有正则点吗?更一般地,张敦穆[8]提出了一般的Hirsch问题设N,P为cm流形,dimN=n,dimp=P,n＞p,fN→P是C映射(1≤r≤n).当1≤r≤n-P时f必须有正则点吗?本文讨论了这个问题,应用[8]中方法和Norton[5]和Bates[1]的估计,我们得到了定理1和定理2.它们部分地推广了[8]中主要定理.     

具有非负Ricci曲率和大体积增长的开流形

In this paper,we prove that a complete n-dimensional Riemannian manifold with n0nnegative kth-Ricci curvature,large volume growth has finite topological type provided that lim{((vol[B(p,r))]/(ω_nr~n)-αM)r(k(n-1))/(k 1)(1-α/2)}＜=εfor some constantε＞0.We also prove that a complete Riemannian manifold with nonnegative kth-Ricci curvature and under some pinching conditions is diffeomorphic to R~n.     

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.     

球面Sn+p(c)中子流形的拓扑

研究了球面Sn+p(c)子流形Mn的Pinching定理,证明了当s＜2√n-1c时,Mn(n＞3)与n维球面同胚.