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

TL;DR

This paper shows that for smooth actions of connected compact Lie groups with a free point, the entire group action can be characterized by two specific vector fields, highlighting a unique link between group actions and vector fields.

Contribution

It proves that such group actions are uniquely determined by two complete vector fields, extending understanding of the structure of smooth group actions with free points.

Findings

Existence of two vector fields characterizes the group action.

Effective actions with no free point may not be determined by two vector fields.

Examples demonstrate the necessity of the free point condition.

Abstract

Consider a smooth action $\mathbf G\times M \rightarrow M$ 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.