Given a tangent vector field on a finite dimensional real smooth manifold, its degree (also known as \emph{characteristic} or \emph{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 $\R\sp{m}$, a tangent vector field $v$ on $U$ can be identified with a map $\vec{v}\colon U \to \R\sp{m}$, and its degree, when defined, coincides with the Brouwer degree with respect to zero of the corresponding map $\vec v$. As is well known, the Brouwer degree in $\R\sp{m}$ is uniquely determined by three axioms, called \emph{Normalization}, \emph{Additivity} and \emph{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.
A set of axioms for the degree of tangent vector fields on differentiable manifolds / M. Furi; M.P. Pera; M. Spadini. - In: FIXED POINT THEORY AND APPLICATIONS. - ISSN 1687-1820. - ELETTRONICO. - 2010:(2010), pp. 1-11. [10.1155/2010/845631]
A set of axioms for the degree of tangent vector fields on differentiable manifolds
FURI, MASSIMO;PERA, MARIA PATRIZIA;SPADINI, MARCO
2010
Abstract
Given a tangent vector field on a finite dimensional real smooth manifold, its degree (also known as \emph{characteristic} or \emph{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 $\R\sp{m}$, a tangent vector field $v$ on $U$ can be identified with a map $\vec{v}\colon U \to \R\sp{m}$, and its degree, when defined, coincides with the Brouwer degree with respect to zero of the corresponding map $\vec v$. As is well known, the Brouwer degree in $\R\sp{m}$ is uniquely determined by three axioms, called \emph{Normalization}, \emph{Additivity} and \emph{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.File | Dimensione | Formato | |
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