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.

Abstract. In this paper we consider piecewise linear (pl) isometric embeddings of Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a metric space P which admits a triangulation T such that each n-dimensional simplex of T is affinely isometric to a simplex in E n . We prove that any 1-Lipschitz map from an n-dimensional Euclidean polyhedron P into E 3n is ǫ-close to a pl isometric embedding for any ǫ > 0. If we remove the condition that the map be pl then any 1-Lipschitz map into E 2n+1 can be approximated by a (continuous) isometric embedding. These results are extended to isometric embedding theorems of spherical and hyperbolic polyhedra into Euclidean space by the use of the Nash-Kuiper C 1 isometric embedding theorem ([14] and [11]). Finally, we discuss how these results extend to various other types of polyhedra.

Abstract:In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into E n if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a "nice" inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E n+ . Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C version of) the famous Nash isometric embedding theorem from [10].

We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the 1-skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices in the triangulation gives another integer called the combinatorial volume. We show that a class of smooth manifolds satisfies a systolic inequality for all Riemannian metrics if and only if it satisfies a corresponding combinatorial systolic inequality for all smooth triangulations. Along the way, we show that any closed Riemannian manifold has a smooth triangulation which "remembers" the geometry of the Riemannian metric, and conversely, that every smooth triangulation gives rise to Riemannian metrics which encode the combinatorics of the triangulation. We give a few applications of these results.

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