欧氏空间中子流形的Pinching现象

本文研究了欧氏空间中紧致子流形的Ｐｉｎｃｈｉｎｇ现象,得到了一些公式,并证明了一些几何量的Ｐｉｎｃｈｉｎｇ定理     

欧氏空间中子流形的Pinching现象

本文研究了欧氏空间中紧致子流形的Pinching现象,得到了一些公式,并证明了一些几何量的Pinching定理.     

常曲率空间中的正曲率子流形

本文研究了常曲率空间子流表余维数减少的问题,说明了在一定条件下,余维数可以减少到１。     

闵可夫斯基空间中子流形的若干性质

本文研究了Ｆｉｎｓｌｅｒ流形中的子流形,特别地,我们给出了闵可夫斯基空间中超球面的一个特征,同时,我们讨论了闵可夫斯基空间中超曲面的第二基本形式。     

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。     

欧氏空间中紧致子流形的积分公式(英文)

本文研究了(n+p)维欧氏空间R^n+p中n维定向紧致无边子流形Mn的积分公式的问题.首先定义了Mⁿ沿其单位平均曲率向量场ξ方向的高阶平均曲率H^r(0≤r≤n);然后,利用活动标架与外微分法,获得了关于Mn的一个新的积分公式.新公式推广了余维数p=1即超曲面情况下的经典积分公式.     

具有平行平均曲率向量场的子流形

本文改进了Ｓ．Ｔ．Ｙａｕ（文［１］）中关于单位球面中具有平行平均曲率向量场的子流形的一个结果。然而从截面曲率这一角度出发，给出了空间形式Ｒ＾ｎ＋ｐ（ｃ）（ｎ＞１，ｐ＞１）中具有平行平均曲率向量场的可定向闭子流形Ｍ＾ｎ的有关结果和积分不等式。     

常曲率空间中平均曲率为常数的迷向子流形

设M~n是n维黎曼流形,S~(n+p)(e)是n+p维截面曲率为常数c的黎曼流形,设fM~n→S~(n+p)(c)是等距浸入,我们分别用和表示f(M~n)和S~(n+p)(c)的协变微分,那么浸入f的第二基本形式A为 A(X,Y)=x~Y-x~Y     

常曲率空间中的具有共形第二基本形式的子流形

以把调和态射看作等距浸入的单位法投影的问题为背景,研究了具有共形第二基本形式的子流形,论证了具有共形第二基本形式的高维子流形,一般不是由极小点和全脐点构成,这和曲面的情形形成了鲜明的对照。也给出了常曲率空间中具有平行中曲率的奇数维子流形的一个完全分类。     

常曲率空间中的具有共形第二基本形式的子流形

以把调和态射看作等距浸入的单位法投影的问题为背景,研究了具有共形第二基本形式的子流形,论证了具有共形第二基本形式的高维子流形,一般不是由极小点和全脐点构成.这和曲面的情形形成了鲜明的对照.也给出了常曲率空间中具有平行中曲率的奇数维子流形的一个完全分类.     

Biharmonic properly immersed submanifolds in Euclidean spaces

We consider a complete biharmonic immersed submanifold M in a Euclidean space ${\mathbb{E}^N}$ . Assume that the immersion is proper, that is, the preimage of every compact set in ${\mathbb{E}^N}$ is also compact in M. Then, we prove that M is minimal. It is considered as an affirmative answer to the global version of Chen’s conjecture for biharmonic submanifolds.     

Diophantine vectors in analytic submanifolds of Euclidean spaces

Let M be an m-dimensional analytic manifold in R~n.In this paper,we prove that almost all vectors in M (in the sense of Lebesgue measure) are Diophantine if there exists one Diophantine vector in M.     

Non-existence of stable currents in submanifolds of the Euclidean spaces

We prove the non-existence theorems of stable integral currents for certain classes of hypersurfaces or higher codimensional submanifolds in the Euclidean spaces.     

On singularities of submanifolds of higher dimensional Euclidean spaces

Summary Generalizations of principle axes are found for surfaces in E⁴. The singularities generalize umbilics. The generic indicies are computed. For these computations the Thom Transversality Theorem as applied by Feldman to geometry is used. Hower we ? reduce the group ? rendering the calculations more tractible. Also we show that a torus or sphere cannot be immersed in E⁴ with everywhere nonzero curvature of the normal bundle. Entrata in Redazione il 19 novembre 1968.     

An extremal class of conformally flat submanifolds in Euclidean spaces

Summary Let <InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"14"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"15"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"16"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"17"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"18"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"19"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"20"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"21"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>M^n$ be a Riemannian $n$-manifold with $n\ge 4$. Consider the Riemannian invariant $\sigma(2)$ defined by <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \sigma(2)=\tau-\frac{(n-1)\min \Ric}{n^2-3n+4}, $$ where $\tau$ is the scalar curvature of $M^n$ and $(\min \Ric)(p)$ is the minimum of the Ricci curvature of $M^n$ at $p$. In an earlier article, B. Y. Chen established the following sharp general inequality: $$ \sigma(2)\le \frac{n^2{(n-2)}^2}{2(n^2-3n+4)}H^2 $$ for arbitrary $n$-dimensional conformally flat submanifolds in a Euclidean space, where $H^2$ denotes the squared mean curvature. The main purpose of this paper is to completely classify the extremal class of conformally flat submanifolds which satisfy the equality case of the above inequality. Our main result states that except open portions of totally geodesic $n$-planes, open portions of spherical hypercylinders and open portion of round hypercones, conformally flat submanifolds satifying the equality case of the inequality are obtained from some loci of $(n-2)$-spheres around some special coordinate-minimal surfaces.     

Curvature flow with a general forcing term in Euclidean spaces

We consider closed convex hypersurfaces moving in Euclidean spaces with normal velocity equal to h−F, where h=h(t) is a nonnegative continuous function of t and F is evaluated at the principal curvatures and satisfies the standard conditions. We study long time existence and convergence of the evolving hypersurfaces in three different cases, which include Andrews' contractive case, McCoy's mixed volume preserving case and the additional expanding case.