Smooth actions of connected compact Lie groups with a free point are determined by two vector fields (Q6542419)

Let \(M\) be a connected manifold and \(G\) a compact connected Lie group acting smoothly on \(M\). Assume that the action has a free point (i.e., with trivial isotropy). The authors use tools from differential geometry, especially from vector fields on manifolds and from Lie groups, to prove the existence of two complete vector fields \(X,X_1\) on \(M\) such that \(G=\Aut(X,X_1)\). Several examples illustrating their results are described. Moreover, the authors give an example showing that the hypothesis of having a free point is sufficient but not necessary for this result. In the end of the paper, the authors discuss some interesting open problems related to the result.