Smooth torus actions are described by a single vector field (Q1645297)

Let \({\mathbb{T}}^n\) be a torus acting effectively on a connected smooth manifold \(M\), where \(n\leq \dim M -1\). The main result of this paper states that there is a complete vector field \(X\) on \(M\) such that the flow \(\Phi_t\) of \(X\) commutes with the action of \({\mathbb{T}}^n\), and the group homomorphism \({\mathbb{T}}^n\times {\mathbb{R}}\to \text{Aut}(X)\), \((g,t)\mapsto g\circ \Phi_t\), is an isomorphism. Therefore, it is possible to reconstruct the action of \({\mathbb{T}}^n\) from a single vector field. The proof is divided into three steps: The authors first prove the result for free actions, and then extend it to effective ones, always assuming \(\dim M -n \geq 2\). Finally, they prove the case where \(\dim M - n=1\). They also give examples that show the result cannot be extended for effective actions of general compact Lie groups. In [\textit{F. J. Turiel} and \textit{A. Viruel}, Rev. Mat. Iberoam. 30, No. 1, 317--330 (2014; Zbl 1302.57066)] the authors obtained analogous results for finite group actions.