Approximating continuous maps by isometries

TL;DR

This paper extends the Nash-Kuiper Theorem to Minkowski space, showing that isometric embeddings are dense among all continuous maps without the 1-Lipschitz restriction, broadening the scope of isometric embedding approximations.

Contribution

The paper generalizes existing results by removing the 1-Lipschitz condition in Minkowski space, enabling density of isometric embeddings among all continuous maps.

Findings

Isometric embeddings are dense among all continuous maps in Minkowski space.

Removal of the 1-Lipschitz condition broadens the applicability of the Nash-Kuiper Theorem.

Extensions to Euclidean polyhedra and indefinite metrics are discussed.

Abstract

The Nash-Kuiper Theorem states that the collection of $C^1$-isometric embeddings from a Riemannian manifold $M^n$ into $\mathbb{E}^N$ is $C^0$-dense within the collection of all smooth 1-Lipschitz embeddings provided that $n < N$. This result is now known to be a consequence of Gromov's more general $h$-principle. There have been some recent extensions of the Nash-Kuiper Theorem to Euclidean polyhedra, which in some sense provide a very specialized discretization of the $h$-principle. In this paper we will discuss these recent results and provide generalizations to the setting of isometric embeddings of spaces endowed with indefinite metrics into Minkowski space. The new observation is that, when dealing with Minkowski space, the assumption "1-Lipschitz" can be removed. Thus, we obtain results about isometric embeddings that are $C^0$-dense within the collection of {\it all} continuous maps.