New Hsiung-Minkowski identities

Abstract

We find the first three most general Minkowski or Hsiung–Minkowski identities relating the total mean curvatures 𝐻𝑖, of degrees 𝑖=0,1,2,3, of a closed hypersurface N immersed in a given orientable Riemannian manifold M endowed with any given vector field P. Then we specialize the three identities to the case when P is a position vector field. We further obtain that the classical Minkowski identity is natural to all Riemannian manifolds and, moreover, that a corresponding 1st degree Hsiung–Minkowski identity holds true for all Einstein manifolds. We apply the result to hypersurfaces with constant 𝐻1,𝐻2.