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

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.