Manifolds with a unique embedding

Abstract

Let X be a smooth, compact manifold of dimension k: We show that any two smooth embeddings X !R n are tamely equivalent provided n ‚ 2k + 2: Moreover, if additionally X is a real analytic manifold, then we show that any two real analytic embeddings X !R n are real analytically equivalent. We extend this result to some interesting sub-categories of the category of real smooth manifold. In particular we will prove that if X;Y are Nash kidimensional submanifolds of R n (where n ‚ 2k + 2) and ` : X ! Y is a difieomorphism (Nash isomorphism, real analytic isomorphism), thencan be extended to a tame difieomorphism (tame Nash isomorphism, tame real analytic isomorphism) ' :R n !R n : We prove also that if X;Y are kidimensional smooth algebraic subvarieties of C n (where n ‚ 2k + 2), and ` : X ! Y is a biholomorphism (or polynomial isomorphism), thencan be extended to a global tame biholomorphism (polynomial automorphism) ' :C n !C n : Finally we prove: if X;Y are kidimensional closed semi-algebraic subsets of R n (where n ‚ 2k + 2), and ` : X ! Y is a semi-algebraic homeomorphism, thatcan be extended to a global tame semi-algebraic homeomorphism ' :R n !R n : This last result is a semi-algebraic counterpart of a classical result of Herman Gluck (6), on extension of homeomorphisms of compact polyhedrons.