# Revista Matemática Iberoamericana

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**Volume 34, Issue 2, 2018, pp. 839–852**

**DOI: 10.4171/RMI/1005**

Published online: 2018-05-28

Smooth torus actions are described by a single vector field

Francisco Javier Turiel^{[1]}and Antonio Viruel

^{[2]}(1) Universidad de Málaga, Spain

(2) Universidad de Málaga, Spain

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.

*Keywords: *Torus, group action, vector field

Turiel Francisco Javier, Viruel Antonio: Smooth torus actions are described by a single vector field. *Rev. Mat. Iberoam.* 34 (2018), 839-852. doi: 10.4171/RMI/1005