Smooth torus actions are described by a single vector field

Abstract

Consider a smooth effective action of a torus T n on a connected C ∞ -manifold M of dimension m. Then n ≤ 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 T n × R, where the factor R comes from the flow of X and T n is regarded as a subgroup of Diff(M ).

Smooth actions of connected compact Lie groups with a free point are determined by two vector fields

“…In two previous works [7,8], fitted within the framework of the inverse Galois problem, we proved that any effective action on a manifold of a finite group or a torus is described by a single vector field. But, as Example 6.3 of [8] shows, this result cannot be extended to compact connected Lie groups. Thus the question of determining the actions of these groups by means of a family of vector fields arises.…”

“…The first hypothesis gives rise to a G-principal fibre bundle π : M → B where B is a connected k-manifold. As it was stated at the beginning of Section 3 of [8], there exists a Riemannian metric g on B such that the gradient vector field Y of µ is complete and, moreover, around each p ∈ C there are coordinates (x 1 , . .…”

“…Proof. The first part of the proof of Lemma 2.4 of [8] (see [6] and Proposition 2.1 of [7] as well) allows us to suppose ξ = ξ.…”

“…Consider an action of a Lie group G on the manifold M. A point p ∈ M is called free if its isotropy group (or stabilizer) reduces to the neutral element e of G. Obviously if p is free all the points of its orbit are free too. Observe that the existence of a free point implies that the action is effective, and the converse holds if G is a torus (see Proposition 7.1 of [8]).…”

“…The reader is supposed to be familiarized with our two previous papers [7,8]. Our basic reference sources for differential topology, differential geometry and Lie groups actions are [2], [3] and [4] respectively.…”