On The Existence of Globally Solvable Vector Fields in Smooth Manifolds

TL;DR

This paper characterizes when Lie subalgebras of vector fields on smooth manifolds have orbits that are diffeomorphic to open convex sets in Euclidean space, providing necessary and sufficient conditions.

Contribution

It establishes a complete characterization of Lie subalgebras with orbits diffeomorphic to convex open sets, advancing understanding of global solvability of vector fields.

Findings

Derived necessary and sufficient conditions for orbit convexity.

Connected the structure of Lie subalgebras to geometric properties of orbits.

Provided criteria for the existence of globally solvable vector fields.

Abstract

Let $(\mathrm{M}, \omega_{0})$ be a connected paracompact smooth oriented manifold. We establish a necessary and sufficient conditions on Lie subalgebra $\mathfrak{a}$ of $\mathrm{T M}$ such that its orbits becomes diffeomorphic to an open convex set of $\mathbb{R}^{n}$, where $n$ is the orbit dimension.