A global geometric decomposition of vector fields and applications to topological conjugacy

TL;DR

This paper introduces a global geometric decomposition for differentiable vector fields on Euclidean space, enabling a new criterion for topological conjugacy based on decomposed components.

Contribution

It provides a unique global decomposition of vector fields into gradient-like and orthogonal parts relative to a geometric structure, with applications to topological conjugacy.

Findings

Decomposition is unique and globally defined.

Criterion for topological conjugacy based on decomposed parts.

Applicable to complete vector fields on Euclidean space.

Abstract

We give a global geometric decomposition of continuously differentiable vector fields on $\mathbb{R}^n$. More precisely, given a vector field of class $\mathcal{C}^{1}$ on $\mathbb{R}^{n}$, and a geometric structure on $\mathbb{R}^n$, we provide a unique global decomposition of the vector field as the sum of a left (right) gradient--like vector field (naturally associated to the geometric structure) with potential function vanishing at the origin, and a vector field which is left (right) orthogonal to the identity, with respect to the geometric structure. As application, we provide a criterion to decide topological conjugacy of complete vector fields of class $\mathcal{C}^1$ on $\mathbb{R}^{n}$ based on topological conjugacy of the corresponding parts given by the associated geometric decompositions.