Smooth actions of connected compact Lie groups with a free point are determined by two vector fields

Abstract

Consider a smooth action $\mathbf{G}\times M\to M$ of a compact connected Lie group $\mathbf{G}$ on a connected manifold $M$. Assume the existence of a point of $M$ whose isotropy group has a single element (free point). Then we prove that there exist two complete vector field $X,X_1$ such that their group of automorphisms equals $\mathbf{G}$ regarded as a group of diffeomorphisms of $M$ (the existence of a free point implies that the action of $\mathbf{G}$ is effective). Moreover, some examples of effective actions with no free point where this result fails are exhibited.

arXiv comment30 pages, no figures. V2: Major revision. Exposition improved and added a new section presenting some open questions