Extensions of isometric embeddings of pseudo-Euclidean metric polyhedra

Abstract

We extend the results of B. Minemyer by showing that any indefinite metric polyhedron (either compact or not) with the vertex degree bounded from above admits an isometric simplicial embedding into a Minkowski space of the lowest possible dimension. We provide a simple algorithm of constructing such embeddings. We also show that every partial simplicial isometric embedding of such space in general position extends to a simplicial isometric embedding of the whole space.2010 Mathematics Subject Classification. 51F99.

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.…”

Simplicial Isometric Embeddings of Polyhedra

“…D. Thesis [Min13]. In January of 2015, a paper was posted on arXiv by Pavel Galashin and Vladimir Zolotov [GZ15] which extends some of these results. Specifically, they show that the dimension requirements of Theorem 3 can be reduced to the same as those of Theorem 1.…”