New York Journal of Mathematics
Volume 22 (2016) 741-753


Barry Minemyer

Approximating continuous maps by isometries

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Published: July 29, 2016
Keywords: metric geometry, isometric embedding, polyhedral space, Euclidean polyhedra, indefinite metric polyhedra, h-principle, Minkowski space
Subject: Primary 51F99, 52B11, 53B21, 53B30, 57Q35; Secondary 52A38, 52B70, 53C50, 57Q65

The Nash-Kuiper Theorem states that the collection of C1-isometric embeddings from a Riemannian manifold Mn into EN is C0-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 C0-dense within the collection of all continuous maps.

Author information

Department of Mathematics, The Ohio State University, Columbus, Ohio 43210