×

Convex polyhedra in Lorentzian space-forms. (English) Zbl 1020.53046

A. D. Alexandrov characterized metrics induced on (the boundaries of) convex polytopes in the Euclidean 3-space, see, for example, the book [A. D. Aleksandrov, Convex polyhedra (in Russian). Moscow-Leningrad (1950; Zbl 0041.50901)]. That result reads as follows: a Riemannian metric \(g\) on \(S^2\) with conical singularities is induced by a convex polyhedral embedding in the Euclidean 3-space if and only if \(g\) is flat except at a finite number of singular points \(x_1,\dots,x_n\) where the singular curvature is positive; moreover, the embedding is then unique modulo global isometries. Similar characterizations are known for convex polyhedra in the Lobachevskij and spherical 3-spaces.
In the paper under review, the author gives similar results for compact polyhedra in two Lorentzian space-forms, the de-Sitter and the Minkowski spaces \(S^3_1\) and \(E^3_1\). He also describes the induced metrics on complete polyhedra in the de Sitter space \(S^3_1\). The author shows that, in the spaces under consideration, it is necessary to consider induced metrics together with some additional combinatorial data depending on the way the polyhedron is imbedded, namely, on the way the faces ‘degenerate’.

MSC:

53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
52B10 Three-dimensional polytopes

Citations:

Zbl 0041.50901
PDFBibTeX XMLCite
Full Text: DOI