Smooth torus actions are described by a single vector field

Abstract

Consider a smooth effective action of a torus ${\mathbb{T}}^n$ on a connected $C^{\infty }$-manifold $M$. Assume that $M$ is not a torus endowed with the natural action. Then we prove that there exists a complete vector field $X$ on $M$ such that the automorphism group of $X$ equals ${\mathbb{T}}^n\times \mathbb{R}$, where the factor $\mathbb{R}$ comes from the flow of $X$ and ${\mathbb{T}}^n$ is regarded as a subgroup of Diff$(M)$. Thus one may reconstruct the whole action of ${\mathbb{T}}^n$ from a single vector field.