de Sitter空间中具有常数量曲率的完备类空子流形的间隙现象

设Mn是de Sitter空间Snp p(c)中具有常数量曲率R(≤c)的完备类空子流形.得到了Mn关于其第二基本形式模长平方‖h‖2的间隙性定理,如果n(c-R)≤‖h‖2≤2√n-1c,那么,或者‖h‖2=n(c-R)且Mn是全脐点子流形,或者‖h‖2=2√n-1c且Mn是全脐的或是双曲柱面Sn-1(c-tanh2r)×H1(c-coth2r).     

de Sitter空间中第二基本形式模长平方为常数的类空超曲面

设Mn是de Sitter空间S1n+1+1(c)中具有第二基本形式模长平方‖h‖2是常数的类空超曲面,利用极大值原理得到了Mn是全脐超曲面的三个充分条件.     

正曲率空间形式中超曲面的全脐性与高阶平均曲率

本文研究正曲率空间形式S~(n+1)(c)(c0)中紧致的闭的等距浸入超曲面M~n的全脐性质和高阶平均曲率,所得结果改进和推广了这方面最近有关定理.     

de Sitter空间中法联络平坦的完备类空子流形

该文证明了de Sitter空间中具有平行平均曲率向量的常数量曲率完备类空子流形,如果其法联络是平坦的,且M的截面曲率小于0,或M的第二基本形式模长平方‖σ‖相似文献   

de Sitter空间中具平行平均曲率向量的完备类空子流形(Ⅱ)

本文证明了de Sitter空间中具平行平均曲率向量的完备类空子流形在其第二基本形式模长平方S＜2(n-1c)的平方根时是全脐子流形。     

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．     

共形空间${\mathbb Q}^n_s$中的正则Blaschke拟全脐子流形

[Nie C X,Wu C X,Regular submanifolds in the conformal space Q_p~n,ChinAnn Math,2012,33B(5):695-714]中研究了共形空间Q_s~n中的正则子流形,并引入了共形空间Q_s~n中的子流形理论.本文作者将分类共形空间Q_s~n中的Blaschke拟全脐子流形,证明伪Riemann空间形式中具有常数量曲率和平行的平均曲率向量场的正则子流形是共形空间中的Blaschke拟全脐子流形;反之,共形空间中的Blaschke拟全脐子流形共形等价于伪Riemann空间形式中具有常数量曲率和平行的平均曲率向量场的正则子流形.这一结论可看作是共形空间Q_s~n中共形迷向子流形分类定理的推广.     

双曲空间形式中的全脐超曲面与高阶平均曲率

本文研究了双曲空间形式中等距浸入的紧致无边超曲面的全脐性质和高阶平均曲率.利用高阶平均曲率积分估计的方法,获得了一个新的定理,改进了这个研究方向上有关的最近结果.     

拟常曲率空间中具有常平均曲率的完备超曲面

主要研究了拟常曲率空间中具有常平均曲率的完备超曲面,得到了这类超曲面全脐的一个结果.即若Nn+1的生成元η∈TM,且a-2|b|=c(常数)>0,则当S<2 n-1~(1/2)(a-2|b|)时,M为全脐超曲面.     

de Siteer空间中拟紧致超曲面的一个全脐条件

在这篇文章中，讨论了具有常数量曲率的拟紧致超曲面，并给出了它是全脐子流形的一个全脐条件。     

关于复射影空间中的全实伪脐子流形

设$M^n$是复射影空间${\bf C}P^{n+p}$中的全实子流形. 本文研究$M^n$的平行脐性法向量场在法丛中的位置. 在$p>0$的情形通过选取合适的标架场, 得到具有平行平均曲率向量的全实伪脐子流形关于第二基本形式模长平方的一个Pinching定理.     

The Topoligical Sphere Theorem for Complete Submanifolds

A topological sphere theorem is obtained from the point of view of submanifold geometry. An important scalar is defined by the mean curvature and the squared norm of the second fundamental form of an oriented complete submanifold Mn in a space form of nonnegative sectional curvature. If the infimum of this scalar is negative, we then prove that the Ricci curvature of Mn has a positive lower bound. Making use of the Lawson–Simons formula for the nonexistence of stable k-currents, we eliminate Hk (Mn, Z) for all 1 ` k ` n – 1.We then observe that the fundamental group of Mn is trivial. It should be emphasized that our result is optimal.     

SUBMANIFOLDS WITH NONNEGATIVE SECTIONAL CURVATURE

This paper gives some sufficient conditions for a compact submanifold with nonnegative sectional curvature in a space form to be totally umbilical. In particular, for a compact submanifold M with flat normal bundle, if the scalar curvature is proportional to the mean curvature everywhere, then M is totally umbilical or the Riemannian product of several totally umbilical constantly curved submanifolds.     

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.     

关于四元射影空间的正曲率全复子流形

本文讨论四元射影空间的全复子流形，证明了四元射影空间的正截面曲率紧致全复子流形一定是全测地的。     

Submanifolds with Flat Normal Bundle

Let M be a compact orientable submanifold immersed in a Riemannian manifold of constant curvature with flat normal bundle. This paper gives intrinsic conditions for M to be totally umbilical or a local product of several totally umbilical submanifolds. It is proved especially that a compact hypersurface in the Euclidean space with constant scalar curvature and nonnegative Ricci curvature is a sphere.     

关于复射影空间的三维全实极小子流形

本文给出复射影空间中三维紧致全实极小子流形的Ricci曲率和数量曲率的鞭些拼挤定理.特别是证得:若M³是CP³的紧致全实极小子流形且它的Ricci曲率大于1/6,则M³是全测地的.