The Schlafli formula for polyhedra and piecewise smooth hypersurfaces

The classical Schlafli formula relates the variations of the dihedral angles of a smooth family of polyhedra in a space form to the variation of the enclosed volume. We extend here this formula to immersed piecewise smooth hypersurfaces in Einstein manifolds. This leads us to introduce a natural notion of total mean curvature of piecewise smooth hypersurfaces and a consequence of our formula is, for instance, in Ricci-flat manifolds, the invariance of the total mean curvature under bendings. We also give a simple and unified proof of the Schlafli formula for polyhedra in Riemannian and pseudoRiemannian space forms. Moreover, we show that the formula makes sense even for polyhedra which are not necessarily embedded.