Moscow Mathematical Journal

Volume 17, Issue 1, January–March 2017 pp. 79–95.

Simplicial Isometric Embeddings of Polyhedra

Authors: Barry Minemyer (1)
Author institution:(1) Department of Mathematics, The Ohio State University, Columbus, Ohio 43210

Summary:

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.

2010 Math. Subj. Class. Primary 51F99, 52B11, 52B70, 57Q35, 57Q65; Secondary 52A38, 53B21, 53B30, 53C50.

Keywords: Differential geometry, Discrete geometry, indefinite metric polyhedra, metric geometry, polyhedral space.

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