# Smooth torus actions are described by a single vector field

### Francisco Javier Turiel

Universidad de Málaga, Spain### Antonio Viruel

Universidad de Málaga, Spain

## 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.

## Cite this article

Francisco Javier Turiel, Antonio Viruel, Smooth torus actions are described by a single vector field. Rev. Mat. Iberoam. 34 (2018), no. 2, pp. 839–852

DOI 10.4171/RMI/1005