We find the first three most general Minkowski or Hsiung-Minkowski identities relating the total mean curvatures Hi, 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… Expand