A Set of Axioms for the Degree of a Tangent Vector Field on Differentiable Manifolds

Abstract

Given a tangent vector field on a finite-dimensional real smooth manifold, its degree (also known as characteristic or rotation) is, in some sense, an algebraic count of its zeros and gives useful information for its associated ordinary differential equation. When, in particular, the ambient manifold is an open subset U of ℝm, a tangent vector field v on U can be identified with a map v→:U→ℝm, and its degree, when defined, coincides with the Brouwer degree with respect to zero of the corresponding map v→. As is well known, the Brouwer degree in ℝm is uniquely determined by three axioms called Normalization, Additivity, and Homotopy Invariance. Here we shall provide a simple proof that in the context of differentiable manifolds the degree of a tangent vector field is uniquely determined by suitably adapted versions of the above three axioms.