Analytic and Nash equivalence relations of Nash maps

TL;DR

This paper explores the relationship between analytic and Nash equivalence relations of Nash maps, establishing conditions under which they coincide and highlighting differences with analytic map germs.

Contribution

It proves that for compact Nash manifolds, analytic R-L equivalence implies Nash R-L equivalence, and shows local $C^\infty$ R-L equivalence does as well, revealing differences with analytic germs.

Findings

Analytic R-L equivalence implies Nash R-L equivalence for compact Nash manifolds.

Local $C^\infty$ R-L equivalence of Nash map germs implies Nash R-L equivalence.

Existence of analytic map germs that are $C^\infty$ R-L equivalent but not analytically R-L equivalent.

Abstract

Let $M$ and $N$ be Nash manifolds, and $f$ and $g$ Nash maps from $M$ to $N$. If $M$ and $N$ are compact and if $f$ and $g$ are analytically R-L equivalent, then they are Nash R-L equivalent. In the local case, $C^infty$ R-L equivalence of two Nash map germs implies Nash R-L equivalence. This shows a difference of Nash map germs and analytic map germs. Indeed, there are two analytic map germs from $(R^2,0)$ to $(R^4,0)$ which are $C^infty$ R-L equivalent but not analytically R-L equivalent.