On submanifolds whose tubular hypersurfaces have constant higher order mean curvatures |
| |
Authors: | Tian Shou Jin Jian Quan Ge |
| |
Affiliation: | 1. Information and Control Engineering Institute, ZheJiang Guangsha College of Applied Construction Technology, Dongyang 322100, P. R. China;2. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, P. R. China |
| |
Abstract: | Motivated by the theory of isoparametric hypersurfaces, we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures. Here a k-th order mean curvature Q k ν (k ≥ 1) of a submanifold M n is defined as the k-th power sum of the principal curvatures, or equivalently, of the shape operator with respect to the unit normal vector ν. We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures, then the submanifold M itself has some constant higher order mean curvatures Q k ν independent of the choice of ν. Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained. In particular, we generalize several classical results in isoparametric theory given by E. Cartan, K. Nomizu, H. F. M¨unzner, Q. M. Wang, et al. As an application, we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold. |
| |
Keywords: | Isoparametric hypersurface constant mean curvature austere submanifold |
本文献已被 CNKI SpringerLink 等数据库收录! |
| 点击此处可从《数学学报(英文版)》浏览原始摘要信息 |
|
点击此处可从《数学学报(英文版)》下载免费的PDF全文 |
|