不定度量曲面的若干分类问题研究

Surface is one of the most fundamental research objects in differential geometry. From Riemannian geometric point of view, one of the most fundamental problems in the study of surfaces is to classify surfaces in spaces. At present the results of Lorentzian surfaces research are much less, that's mainly because many problems cannot be solved by the classical methods in Riemannian geometry under the indefinite metric. This project intends to examine the classification problems of surfaces with some specific geometric conditions in both pseudo-Riemannian spaces and Lorentzian complex space forms. We will introduce the local coordinates, establish the system of differential equations and determine the parametric equations of surfaces by solving the system of equations. The research includes: the property and classification of spacelike and timelike bi-conservative surfaces in 3-dimensional pseudo-Riemannian space forms; the classification and geometry of constant slope surfaces and generalized constant slope surfaces in Minkowski 3-spaces; the classification problems of ortho-umbilical slant surfaces in Lorentzian complex space forms and the classification of bi-conservative slant surfaces in Lorentzian complex space forms, etc. From these aspects, we hope that these can reveal some interesting geometric phenomenon for surfaces with indefinite metrics.

曲面是微分几何中最基本的研究对象之一。从黎曼几何的角度，研究曲面最基本的问题之一就是给出空间中曲面的分类。目前针对不定度量曲面的研究结果较少，其主要原因是度量的不确定性导致许多问题不能利用黎曼几何中的经典方法来解决。本项目拟考察伪黎曼空间以及Lorentz复空间形式中曲面在特定几何条件下的分类问题。我们将引入局部坐标系，建立方程组，通过求解方程组的方法来确定曲面的参数形式。研究内容包括：三维伪黎曼空间形式中类空和类时的Bi-conservative曲面的性质和分类；Minkowski空间中常Slope曲面以及广义的常Slope曲面的分类和何；Lorentz复空间形式具有正交脐性Slant曲面的分类问题和Bi-conservative Slant曲面的分类问题等等。在本项目中，希望从这些方面揭示不定度量曲面所对应的有趣几何现象。

曲面是微分几何中最基本的研究对象之一。从黎曼几何的角度，研究曲面最基本的问题之一就是给出空间中曲面的分类。目前针对Lorentz曲面的研究结果较少，其主原因是度量的不确定性导致许多问题不能利用黎曼几何中的经典方法来解决。本项目在申请人前期工作基础上，研究了伪黎曼空间中曲面在特定几何条件下的分类问题，包括：给出了Lorentz形式中类空和Lorent biconservative曲面的完全显式分类 ；研究了欧氏和Minkowski空间中广义的常Slope曲面的几何性质；研究了Lorentz空间形式中的黄金型超曲面的分类。除了上述计划内的研究内容外，我们还研究了欧氏空间中具有三个不同主曲率的null 2-type超曲面的几何，特别是证明了这类超曲面一定具有常平均曲率。