Simplicial Isometric Embeddings of Polyhedra

Abstract

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.

The isometric embedding problem for length metric spaces

“…There are various results concerning continuous and piecewise-linear isometries and isometric embeddings of Euclidean polyhedra into Euclidean space, as well as piecewise-linear and simplicial isometric embeddings of indefinite metric polyhedra into Minkowski space. These results show a surprising similarity to the theorems about differentiable manifolds mentioned above, and are due to Zalgaller [Zal58], Burago and Zalgaller [BZ96], Krat [Kra04], Akopyan [Ako07], the author [Min15], [Mi16], and Galashin and Zolotov [GZ15]. Of course though, these results are not really generalizations of the Nash isometric embedding theorems since clearly not all Riemannian metrics will be piecewise linear on some simplicial triangulation of the manifold.…”

“…is the signed squared function. Let , g denote the symmetric bilinear form associated to G. A simple calculation, worked out in [Mi16] and [Min16], shows that…”

Extensions of isometric embeddings of pseudo-Euclidean metric polyhedra

“…The question about isometrical simplicial embeddings of indefinite metric polyhedra into a Minkowski space was recently considered in [6]. We show that the results from [6] hold for non-compact indefinite metric polyhedra as well, and give an explicit construction.…”

“…The question about isometrical simplicial embeddings of indefinite metric polyhedra into a Minkowski space was recently considered in [6]. We show that the results from [6] hold for non-compact indefinite metric polyhedra as well, and give an explicit construction. We also show that every partial simplicial isometric embedding of indefinite metric polyhedra into a Minkowski space such that the images of the vertices are in d-general position extends to a simplicial isometric embedding of the whole space.…”

“…In particular, a Euclidean polyhedron (of bounded vertex degree) admits a simplicial isometric embedding into a (non-Euclidean) Minkowski space of an appropriate dimension. Now we are ready to cite a theorem from [6]:…”