Smooth toric actions are described by a single vector field

TL;DR

This paper demonstrates that for a smooth, effective torus action on a connected manifold with dimension greater than the torus, there exists a single complete vector field whose automorphism group precisely captures the torus action.

Contribution

It establishes that smooth effective torus actions of lower dimension can be uniquely represented by a single vector field with a specific automorphism group.

Findings

Existence of a complete vector field representing the torus action

Automorphism group of the vector field equals the torus times real numbers

Applicable to manifolds with dimension greater than the torus

Abstract

Consider a smooth effective action of a torus $\mathbb{T}^n$ on a connected $C^{\infty}$-manifold $M$ of dimension $m$. Then $n\leq m$. In this work we show that if $n<m$, then there exist a complete vector field $X$ on $M$ such that the automorphism group of $X$ equals $\mathbb T^n \otimes \mathbb{R}$, where the factor $\mathbb{R}$ comes from the flow of $X$ and $\mathbb{T}^n$ is regarded as a subgroup of the full group of diffeomorphisms of ${\operatorname{Diff}}(M)$.