Geometric, algebraic, and analytic descendants of nash isometric embedding theorems

Abstract

Is there anything interesting left in isometric embeddings after the problem had been solved by John Nash? We do not venture a definite answer, but we outline the boundary of our knowledge and indicate conjectural directions one may pursue further. Our presentation is by no means comprehensive. The terrain of isometric embeddings and the fields surrounding this terrain are vast and craggy with valleys separated by ridges of unreachable mountains; people cultivating their personal gardens in these "valleys" only vaguely aware of what happens away from their domains and the authors of general accounts on isometric embeddings have a limited acquaintance with the original papers. Even the highly cited articles by Nash have been carefully read only by a handful of mathematicians. In order not to mislead the reader, we try be open about what we do and what we do not know firsthand and to provide references to what is missing from the present paper.