On submanifolds whose tubular hypersurfaces have constant higher order mean curvatures

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 Qkν (k ≥ 1) of a submanifold Mn 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 Qkν 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.