Abstract
AbstractGiven 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 "Equation missing" of "Equation missing", a tangent vector field "Equation missing" on "Equation missing" can be identified with a map "Equation missing", and its degree, when defined, coincides with the Brouwer degree with respect to zero of the corresponding map "Equation missing". As is well known, the Brouwer degree in "Equation missing" 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.
Highlights
The degree of a tangent vector field on a differentiable manifold is a very well-known tool of nonlinear analysis used, in particular, in the theory of ordinary differential equations and dynamical systems
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
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
Summary
The degree of a tangent vector field on a differentiable manifold is a very well-known tool of nonlinear analysis used, in particular, in the theory of ordinary differential equations and dynamical systems. This notion is more often known by the names of rotation or of (Euler) characteristic of a vector field see, e.g., 1–6. In this paper, which is closely related in both spirit and demonstrative techniques to 10 , we will prove that suitably adapted versions of the above axioms are sufficient to uniquely determine the degree of a tangent vector field on a not necessarily orientable differentiable manifold. We will not deal with the problem of existence of such a degree, for which we refer to 1–5
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