1. They contain the usual coordinate singularity on the hypersurface but, for this class, this is not a curvature singularity.

2. In this paper, a rigidity theorem of hypersurface in real space form will be given. In addition, we obtain rigidity theorems of submanifold in sphere which improve the result of Hou and Xu.

3. Thus the shape of the whole energy hypersurface is considered, and the convergence on the ground state of system is ensured.

4. At last the complete hypersurface with constant mean curvature in the quasi constant curvature space is investigated, some characterization of totally umbilical hypersurfaces are obtained.

5. Four reasonable geometries on the potential energy hypersurface of glycine and water system are considered with the global minimum being a cyclic double-hydrogen bonded structure.

6. This clearly demonstrates the existence of a scalar curvature singularity on this hypersurface.

7. It shows that there are similarities between the role of the link of a hypersurface singularity and the space of directions of a singularity in an Alexandrov space.

8. In this paper,( ) an intrinsic condition for a Compact Space-like hypersurface with constant scalar curvature in a de Sitter space to be totally umbilical is obtained.

9. Using the principal curvature formula, we prove an existence theorem of Weingarten hypersurface.

10. B-spline inference rule was established and B-spline inference system was constructed, and then the final result of the system was calculated as a B-spline hypersurface.

11. Discussing the VC dimension of neural network with the hypersurface arrangement.

12. The contributions in the first part are as follows:1) The existence of separating hypersurface and the geometric construction of separating hypersurface is studied.

13. Finally, it gives an example that generating regular 3D spatial dataset based on the interpolation of hypersurface spline, it testifies the feasibility and veracity of this method.

14. It mainly deals with the properties for the principal curvatures of the hypersurface and gets two sufficient conditions under which S has a second gap.

15. It is essential to study the principal curvature of hypersurface in the hypersurface geometry.

16. Let S be the second fundamental form of a closed hypersurface which is minimally immersed in a unit sphere. The question whether the value of S is discrete has been attractive for a long time.

17. Our result extends the case when M is a minimal hypersurface in the same condition.

18. The general event horizon formula of black hole is given from null hypersurface equation.

19. It mainly deals with the properties for the principal curvatures of the hypersurface and gets two sufficient conditions under which S has a second gap. Also estimates for the gaps length are obtained.