Abstract
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.
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