Reparametrizations of vector fields and their shift maps

Original · EN

Let M 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 space of smooth maps h:M→ M of the following form: h(x) = Ff₍ₓ₎(x), where f:M runs over all smooth functions on M which can be substituted into the flow Fₜ instead of time. This space often coincides with 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 and pushforwards of F. As an application it is proved that a vector field F without non-closed orbits can be reparametrized to induce a circle action on M if and only if there exists a smooth function f:M→ (0,+∞) such that for each non-singular point x of M, the value f(x) is an integer multiple of the period of x with respect to F.