Reparametrizations of vector fields and their shift maps

Abstract

LetM be a smooth manifold, F be a smooth vector field on M, and (Ft) be the local flow of F. Denote by Sh(F) the subset of C^∞(M,M) consisting of maps h : M → M of the following form: h(x) = Fα(x)(x), where _ runs over all smooth functions M → R which can be substituted into F instead of t. This space often contains the identity component of the group of diffeomorphisms preserving orbits of F. In this note it is shown that Sh(F) is not changed under reparametrizations of F, that is for any smooth strictly positive function μ : M → (0,+∞) we have that Sh(F) = Sh(μF). As an application it is proved that F can be reparametrized to induce a circle action on M if and only if there exists a smooth function μ : M → (0,+∞) such that F(x, μ(x)) ≡ x.