第二基本形式

The author not only presents pseudo sphere, ellipsoid ,conoid,uniported hyperboloid,elliptic biparted hyperboloid,paraboloid and hyperbolic paraboloid of the second elemental form,but also curve of the second elemental form.

不但给以了伪球面、椭圆面、劈锥曲面、单叶双曲面、双叶双曲面、椭圆抛物面及双曲抛物面的第二基本形式，而且给以了一般曲面的第二基本形式。

Chapter 2 introduces some propaedeutics of classic differential geometry, which include the first basic form, the second basic form, normal curvature, main curvature, mean curvature, Gauss curvature, unit normal vector, main vector, etc.

第二章介绍一系列的预备知识，主要是关于古典微分几何中第一，第二基本形式的定义，法曲率，主曲率，平均曲率，Gauss曲率，单位法向量，主方向等的定义。

If we give some restrician to the intrinsic quantities of submanifolds, such as second fundamental form, scalar curvature,Ricci curvature or sectional curvature,then we can get some new property of the submanifolds.The procedure is called pinching problem of submanifolds.

对子流形的第二基本形式模长平方s，数量曲率R，Ricci曲率R_及截面曲率R_等内在量，加以某种限制，从而得到子流形的某些性质，叫做子流形的pinching问题。

Submanifold; Homology group; Topological sphere theorem; L~2 harmonic 1-forms; Ends; Eigenvalue; Mean curvature; Second fundamental form

基础科学，数学，几何、拓扑子流形；同调群；拓扑球面定理； L~2调和1-形式；端；特征值；平均曲率；第二基本形式

This pepediscusses compact submanifoldsoftheunit sphere Study is made on conditions ofsubmanifolds with parallel mean curvature vector,submanifolds with flat normal bundle and pseudo-umbilical submanifolds to be umbilical respectively.

本文讨论球的紧致子流形，研究具平行平均曲率向量的，具平坦法丛的、以及伪脐子流形成为全脐子流形的条件，得到几个关于第二基本形式长度千方S的pinching条件。

The Pinching problems are discussed on the sectional curvature of compact space-like pseudo-umbilical submanifolds M~n with parallel mean curvature vector in De Sitter space S~_p, and the theory of reduction of the codimension is obtained in De Sitter space through evaluating the Laplacian of square of the length.

讨论了De Sitter空间Snp+p中，具有平行平均曲率向量的紧致类空伪脐子流形Mn的截面曲率的拼挤问题，通过估计第二基本形式模长平方的Laplacian，得到了De Sitter空间中的余维数压缩定理。

Then use this moving frame to compute the coefficients of the first fundamental form E, F, G and the coefficients of the second fundamental form L, M, N.

利用这个标架，计算曲面S的第一基本形式的系数E，F，G和第二基本形式的系数L，M，N。

Then, the first and second fundamental forms, Gaussian curvature and mean curvature of the surfaces are directly calculated according to the principles of differential geometry.

首先，根据微分几何中的基本知识，得到了该种度量形式下的平移曲面的第一、第二基本形式以及高斯曲率和平均曲率；然后，主要利用高斯曲率和平均曲率之间的线性关系和平方关系，得到了这类平移曲面的分类定理。

Let x: M→S(superscript n+1)(n≥3) be a hypersurface in the (n+1)-dimensional unit sphere without umbilic points. Denote the Mbius metric, the Mbius form, the Blaschke form and the Mbius second fundamental form of x by G,Ф, A and B, repcectively, which are Mbius invariants in Mbius differential geometry.

设x： M→S(上标 n+1)(n≥3)是n+1-维单位球中的无脐点超曲面，Mbius不变量G，Ф，A和B分别表示x的Mbius度量，Mbius形式，Blaschke形式和Mbius第二基本形式。

In this article,we mainly study the space-like submanifolds with parallel mean curva-ture vector,constant scalar curvature or constant square length of the second fundamental form,respectively.We obtain some pinching theorems and rigidity results by estimating Laplace of the square length of the second fundamental form of the such submanifolds and using yaus maximum principle or Stokes Theorem.

本论文主要研究de Sitter空间中具有平行平均曲率向量、常数量曲率或第二基本形式模长平方是常数的三类类空子流形，并通过分别估计三种情形下子流形的第二基本形式模长平方的Laplace，利用Yau的极大值原理和Stokes定理，获得了这些子流形的一些拼挤定理和刚性定理。

Cynanchum Lingtai apricot production in the average weight 65 grams, the brightly-colored fruit, juicy rich, sweet-sour taste, sweet from the nucleolus, when the late Qing Dynasty famous Shaanxi, Gansu provinces, the Qing imperial court Tongzhi tribute for years.

灵台生产的牛心杏平均单果重65克，果实色泽鲜艳，汁多味浓，甜酸适口，离核仁甜，清末时就驰名陕、甘两省，清同治年间曾为朝廷贡品。

Chenopodium album,Solanum nigrum, and Amaranthus retroflexus were very susceptible to the herbicides. Polygonum persicaria and Abutilon theophrasti were relatively less susceptible to the herbicides, and Lycopersicon esculentum was not susceptible to it. The relationship between reduction rates of weed biomass and PPM values of weed leaves 2,4, and 6 days after treatment was established.

供试的6种杂草对该混剂的敏感性存在显著差异：红心藜Chenopodium album、龙葵Solanum nigrum和反枝苋Amaranthus retroflexus对该混剂最敏感，ED90值分别为47.65、71.67和29.17g/hm2；春蓼Polygonum persicaria和苘麻Abutilon theophrasti敏感，ED90值分别为96.91、114.20g/hm2；而番茄不敏感。