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

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.