关于两曲面交线的不变量

本文研究了R3中相交子流形的不变量.利用活动标架法,得出了一个类似的欧拉公式. 即两曲面交线的挠率可以用两曲面的测地挠率、法曲率及两曲面的夹角表示.     

常平均曲率曲面的谱变换与Weierstrass表示

利用可积系统的方法, 研究了三维双曲空间常平均曲率曲面的形变, 以及三维双曲空间常平均曲率大于1的单连通曲面的Weierstrass表示公式.     

R~(n+1)中常高阶平均曲率超曲面

本文给出了Rn+1中超曲面的一些积分公式,并利用这些积分公式得到了以球面为边界的常高阶平均曲率超曲面的一些唯一性结果.     

静电场理论中微分几何方法的一个应用

引用微分几何的方法导出了静电场中曲面导体表面附近场强的法向分量的法向变化率与曲面两个主曲率的关系式,并作为特例,导出了静电场理论的一个众所周知的经典公式.     

αK bH=c曲面的Bcklund变换

本文首先对Ｇａｕｓｓ曲率Ｋ和平均曲率Ｈ满足线性关系ａＫ＋ｂＨ＝ｃ的曲面，证明了其ＧａｕｓｓＷｅｉｎｇａｒｔｅｎ公式不仅可看作曲面的Ｇａｕｓｓ－Ｃｏｄａｚｚｉ方程的Ｌａｘ对，而且给出了Ｂ¨ａｃｋｌｕｎｄ变换．然后，将著名的关于负常曲率的Ｂ¨ａｃｋｌｕｎｄ定理推广到主曲率ｋ１和ｋ２满足关系（ｋ１－ｍ）（ｋ２－ｍ）＝－ｌ２的曲面     

中曲率大于零的非凸曲面

本文研究了曲面的曲率问题.利用积分几何方法和旋转曲面性质,构造出一个欧氏空间R3中紧致光滑的中曲率H大于零的非凸曲面,并得到了关于紧致光滑曲面曲率的几个不等式.     

3维双曲空间中给定平均曲率曲面的Weierstrass表示

首先叙述双曲空间中曲面的Gauss映照的定义;导出Gauss映照所满足的Beltrami方程;给出了给定平均曲率曲面的Weierstrass表示公式;并讨论了这种表示的完全可积条件.     

曲面上的曲率新讲

阐述一种讲解曲面上的各种曲率形式的新教学模式.我们利用平面向量分析,矩阵,求导运算等工具推导出所有的结论和公式.这种新观点和方法避免了微分几何传统讲法中使用Weingarten映射及其特征值的思路,而较之更加直观明了.     

Dirac-Witten算子特征值的下界估计

在某些条件下,我们获得了Lorentzian流形中紧致spin类空超曲面（无边或带边）的Dirac-Witten算子特征值的一个优化下界估计.该估计依赖于超曲面的数量曲率、平均曲率以及旋量诱导的能量动量张量.在极限情形下,我们发现类空超曲面或者是极大的且具有正数量曲率的Einstein流形,或者是具有非零常平均曲率的Ricci平坦流形.     

双三次Bezier曲面片的光滑拼接

本文根据曲面的总曲率和平均曲率推导出双三次Bezier曲面片拼接的C^2连续条件。     

三维欧氏空间中的一类Weingarten螺旋面

通过求解相关的非线性常微分方程,构造了三维欧氏空间中主曲率之差为常数的螺旋面,并证明这类曲面的广泛存在性.     

Euler characteristic and Lipschitz–Killing curvatures of closed semi-algebraic sets

We prove a formula that relates the Euler?CPoincaré characteristic of a closed semi-algebraic set to its Lipschitz?CKilling curvatures.     

On mean curvatures in submanifolds geometry

By using moving frame theory,first we introduce 2p-th mean curvatures and(2p 1)-th mean curvature vector fields for a submanifold.We then give an integral expression of them that characterizes them as mean values of symmetric functions of principle curvatures.Next we apply it to derive directly the celebrated Weyl-Gray tube formula in terms of integrals of the 2p-th mean curvatures and some Minkowski-type integral formulas.     

Expression of curvature tensors and some applications

We get an explicit expression of curvature operators in terms of at most eight terms of sectional curvatures. Some applications of this result are also given, particularly we improve a result of Chen-Tian related to the first Chern class of admissible surfaces in pinched manifolds. We also characterize in a simple way all functionsk(x, y) which can be sectional curvatures of some curvature operatorR.Supported by CNPq, Brazil and NNSFC.     

Polynomial recurrence with large intersection over countable fields

Applying a local Gauss–Bonnet formula for closed subanalytic sets to the complex analytic case, we obtain characterizations of the Euler obstruction of a complex analytic germ in terms of the Lipschitz–Killing curvatures and the Chern forms of its regular part. We also prove analogous results for the global Euler obstruction. As a corollary, we give a positive answer to a question of Fu on the Euler obstruction and the Gauss–Bonnet measure.     

Normal Euler classes of knotted surfaces and triple points on projections

We present a new formula relating the normal Euler numbers of embedded surfaces in -space and the number of triple points on their projections into -space. This formula generalizes Banchoff's formula between normal Euler numbers and branch points on the projections.

     

Local Euler obstructions of toric varieties

We use Matsui and Takeuchi's formula for toric A-discriminants to give algorithms for computing local Euler obstructions and dual degrees of toric surfaces and 3-folds. In particular, we consider weighted projective spaces. As an application we give counterexamples to a conjecture by Matsui and Takeuchi. As another application we recover the well-known fact that the only defective normal toric surfaces are cones.