Some variational properties of the weighted {\sigma}_{r}-curvature for submanifolds in Riemannian manifolds

TL;DR

This paper investigates how the r-th weighted curvature functional varies on hypersurfaces within Riemannian manifolds, providing insights and applications to Euclidean space and spheres.

Contribution

It introduces new variational properties of the weighted urvature for hypersurfaces, extending understanding of curvature functionals in Riemannian geometry.

Findings

Derived variational formulas for the weighted urvature functional.

Applied results to hypersurfaces in Euclidean space.

Analyzed curvature properties on the unit sphere.

Abstract

The objet of this paper is the study of the variations of a functional whose integrant is the r-th weighted curvature on the hypersurface of a closed Riemannian manifold. Some applications to hypersurfaces of the Euclidean space and the unit round sph\`ere are given.