New Hsiung-Minkowski identities

TL;DR

This paper derives the most general Hsiung-Minkowski identities relating total mean curvatures of hypersurfaces in Riemannian manifolds, including special cases for position vector fields and Einstein manifolds, with applications to constant curvature hypersurfaces.

Contribution

It introduces the first three most general Hsiung-Minkowski identities for hypersurfaces in Riemannian manifolds, extending classical identities to broader geometric contexts.

Findings

Classical Minkowski identity is natural in all Riemannian manifolds.

A 1st degree Hsiung-Minkowski identity holds for Einstein manifolds.

Results apply to hypersurfaces with constant mean curvatures.

Abstract

We find the first three most general Minkowski or Hsiung-Minkowski identities relating the total mean curvatures $H_i$, of degrees $i=1,2,3$, of a closed hypersurface $N$ immersed in a given orientable Riemannian manifold $M$ endowed with any given vector field $P$. Then we specialise 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 $H_1,H_2$.