On submanifolds whose tubular hypersurfaces have constant mean curvatures

TL;DR

This paper investigates submanifolds with tubular hypersurfaces exhibiting constant higher order mean curvatures, extending classical isoparametric hypersurface results and providing geometric insights into focal varieties.

Contribution

It generalizes classical isoparametric hypersurface theory by analyzing higher order mean curvatures and establishes new restrictions involving principal curvatures and Jacobi operators.

Findings

Derived necessary restrictions on principal curvatures and higher order mean curvatures.

Generalized classical results in isoparametric hypersurface theory.

Provided a geometric filtration for focal varieties of isoparametric functions.

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 $Q_k$ ($k\geq1$) of a hypersurface $M^n$ is defined as the $k$-th power sum of the principal curvatures, or equivalently, of the shape operator. Many necessary restrictions involving principal curvatures, higher order mean curvatures and Jacobi operators on such submanifolds are obtained, which, among other things, generalize some classical results in the theory of isoparametric hypersurfaces given by E. Cartan, K. Nomizu, H. F. M{\"u}nzner, Q. M. Wang, \emph{etc.}. As an application, we finally get a geometrical filtration for the focal varieties of isoparametric functions on a complete Riemannian manifold.