The classical Schläfli 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 Schläfli formula for polyhedra in Riemannian and pseudo-Riemannian space forms. Moreover, we show that the formula makes sense even for polyhedra which are not necessarily embedded. © 2003 Elsevier B.V. All rights reserved.
Souam, R. (2004). The Schläfli formula for polyhedra and piecewise smooth hypersurfaces. Differential Geometry and Its Application, 20(1), 31–45. https://doi.org/10.1016/S0926-2245(03)00054-8