The Schläfli formula for polyhedra and piecewise smooth hypersurfaces.
type: Item
instance of: scholarly article
title: The Schläfli formula for polyhedra and piecewise smooth hypersurfaces.
zbMATH Open document ID: 1065.52015
DOI: 10.1016/S0926-2245(03)00054-8
author: Rabah Souam
published in: Differential Geometry and its Applications
publication date: 2004-02-03T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 52C25
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 52B70
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 2037275
zbMATH Keywords: polyhedra
zbMATH Keywords: Schläfli
zbMATH Keywords: (pseudo)-Riemannian space forms
zbMATH Keywords: Einstein manifold
zbMATH Keywords: mean curvature
zbMATH Keywords: volume
zbMATH Keywords: dihedral angle
Wikidata QID: Q115337993
reviewed by: Charles Leytem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2037275