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arXiv · 1711.11268

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

Razvan M. Tudoran
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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.

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Razvan M. Tudoran. 2017-11-30. A global geometric decomposition of vector fields and applications to topological conjugacy. https://doi.org/10.1007/s10440-019-00258-0

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