子流形几何研究中一个基本的问题是寻找最精简的不变量,在相差外围空间变换群的一个变换下,完全决定子流形在外围空间中的形状.在Riemann空间形式中,大多数超曲面完全由它内蕴的度量决定其形状,这种超曲面称为具有刚性.其余不能由其内蕴度量决定其形状的超曲面称为可形变超曲面,这类可形变超曲面被完全确定并分类．Riemann空间形式中子流形的刚性及形变的研究也比较丰富,而且对它的认识也比较深入.但对Lorentz空间形式中的子流形的刚性及形变的研究比较少,还处于一个前期发展阶段.本项目研究Lorentz空间形式中类空子流形和类时子流形关于它的度量,度量共形类,Moebius度量的刚性和形变问题；同时也研究Lorentz空间形式中的一些具有特殊等距和共形不变量的超曲面.研究的目的是想弄清楚Lorentz空间形式中子流形的内蕴几何在多大程度决定其外蕴几何,以及子流形的等距不变量和共形不变量之间的关系.

In submanifold theory a fundamental problem is to investigae which data are sufficient to determine a submanifold under a transformation of some transformation group of ambient space. In Riemannian space form, general hypersurfaces are determined by the intrinsic metric, which are called of rigidity. The other hypersurfaces, which can not be determined by the intrinsic metric, is called deformable hypersurface. Deformable hypersurfaces were classified completely. The studies of rigidity and deformation of submanifolds in Riemannian space form are much plentiful and profound. But for submanifolds in Lorentzian space form, the study is in early developing stage. This project is to study the rigidity and deformation of spacelike and timelike submanifolds in Lorentzian space form with respect to the metric under the isometry group, conformal class of the metric and Moebius metric under the conformal transformation group. In addition we study some hypersurfaces with special isometry and conformal invariants.The purpose of the project is to make clear what extent the intrinsic geometry determines the extrinsic geometry, and relations between isometry invariants and conformal invariants of submanifolds in Lorentzian space form.