On Submanifolds Whose Tubular Hypersurfaces Have Constant Higher Order Mean Curvatures  

作　　者：Tian Shou JIN Jian Quan GE 

机构地区：[1]Information and Control Engineering Institute,ZheJiang Guangsha College of Applied Construction Technology [2]School of Mathematical Sciences,Laboratory of Mathematics and Complex Systems,Beijing Normal University

出　　处：《Acta Mathematica Sinica,English Series》2016年第4期474-498,共25页数学学报（英文版）

基　　金：partially supported by NSFC(Grant No.11331002);the Fundamental Research Funds for the Central Universities

摘　　要：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^v（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 v.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^v independent of the choice of v.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.Miinzner,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.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^v（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 v.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^v independent of the choice of v.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.Miinzner,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.

关 键 词：Isoparametric hypersurface constant mean curvature austere submanifold 

分 类 号：O186.12[理学—数学]