Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 845631, 11 pages
Research Article

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

Dipartimento di Matematica Applicata “Giovanni Sansone”, Università degli Studi di Firenze, Via San Marta 3, 50139 Firenze, Italy

Received 28 September 2009; Accepted 7 February 2010

Academic Editor: Mohamed Amine Khamsi

Copyright © 2010 Massimo Furi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 m, a tangent vector field v on U can be identified with a map v:Um, 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 m 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.