  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.
