Simplicial isometric embeddings of polyhedra
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
In this paper, isometric embedding results of Greene, Gromov and Rokhlin are extended to what are called “indefinite metric polyhedra”. An indefinite metric polyhedron is a locally finite simplicial complex where each simplex is endowed with a quadratic form (which, in general, is not necessarily positive-definite, or even non-degenerate). It is shown that every indefinite metric polyhedron (with the maximal degree of every vertex bounded above) admits a simplicial isometric embedding into Minkowski space of an appropriate signature. A simple example is given to show that the dimension bounds in the compact case are sharp, and that the assumption on the upper bound of the degrees of vertices cannot be removed. These conditions can be removed though if one allows for isometric embeddings which are merely piecewise linear instead of simplicial.
Key words and phrases:
differential geometry, discrete geometry, indefinite metric polyhedra, metric geometry, polyhedral space.
|National Science Foundation
|This research was supported in part by the NSF grant of Tom Farrell and Pedro Ontaneda, DMS-1103335.
MSC: Primary 51F99, 52B11, 52B70, 57Q35, 57Q65; Secondary 52A38, 53B21, 53B30, 53C50
Received: September 30, 2015; in revised form January 10, 2017
Barry Minemyer, “Simplicial isometric embeddings of polyhedra”, Mosc. Math. J., 17:1 (2017), 79–95
Citation in format AMSBIB
\paper Simplicial isometric embeddings of polyhedra
\jour Mosc. Math.~J.
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