Analytic and Nash equivalence relations of Nash maps

Abstract

Let M and N be Nash manifolds, and f and g be 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∞ 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 (R2, 0) to (R4, 0) which are C∞ R-L equivalent but not analytically R-L equivalent.