The Schläfli formula for polyhedra and piecewise smooth hypersurfaces. (Q1422026)

The classical Schläfli formula \[ n K \frac{dV_i}{dt}=\sum_i \operatorname{Vol} (G_{i,t})\frac{d\theta_{i,t}}{dt} \] relates the variations of the dihedral angles of a smooth family of polyhedra in a space form to the variation of the enclosed volume. This paper generalizes the Schläfli formula in various ways. First it establishes a formula for polyhedra in simply connected Riemannian and pseudo-Riemannian space forms. This formula applies to oriented polyhedra which are not necessarily embedded and may have self-intersections. A consequence is the invariance under flex of the total mean curvature of polyhedra in (pseudo-)Euclidean spaces, and the invariance of a linear combination of the volume and the total mean curvature in (pseudo-)Riemannian space forms of non-zero curvature. The second main result gives a Schläfli-type formula for piecewise smooth hypersurfaces in Einstein manifolds, leading to a natural concept of total mean curvature for piecewise smooth hypersurfaces. As a corollary the author establishes the invariance under isometric deformations of a linear combination of the volume bounded by the hypersurface and the total mean curvature.