A Set of Axioms for the Degree of a Tangent Vector Field on Differentiable Manifolds
© Massimo Furi et al. 2010
Received: 28 September 2009
Accepted: 7 February 2010
Published: 10 May 2010
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 of , a tangent vector field on can be identified with a map , and its degree, when defined, coincides with the Brouwer degree with respect to zero of the corresponding map . As is well known, the Brouwer degree in 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.
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]). Here, we depart from the established tradition by choosing the name "degree" because of the following consideration: in the case that the ambient manifold is an open subset of , there is a natural identification of a vector field on with a map , and the degree of on , when defined, is just the Brouwer degree of on with respect to zero. Thus the degree of a vector field can be seen as a generalization to the context of differentiable manifolds of the notion of Brouwer degree in . As is well-known, this extension of does not require the orientability of the underlying manifold, and is therefore different from the classical extension of for maps acting between oriented differentiable manifolds.
A result of Amann and Weiss  (see also ) asserts that the Brouwer degree in is uniquely determined by three axioms: Normalization, Additivity, and Homotopy Invariance. A similar statement is true (e.g., as a consequence of a result of Staecker ) for the degree of maps between oriented differentiable manifolds of the same dimension. In this paper, which is closely related in both spirit and demonstrative techniques to , 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].
Incidentally, we point out that sets and local maps (with the obvious composition) constitute a category. Although the notation would be acceptable in the context of category theory, it will be reserved for the case when .
Whenever it makes sense (e.g., when source and target spaces are differentiable manifolds), local maps are tacitly assumed to be continuous.
By a local tangent vector field on we mean a local map having as source and as target, with the property that the composition is the identity on . Therefore, given a local tangent vector field on , for all there exists such that .
Let be an open subset of and suppose that is a local tangent vector field on with . We say that is identity-like on if there exists a diffeomorphism of onto such that and the identity in correspond under . Notice that any diffeomorphism from an open subset of onto induces an identity-like vector field on .
Using the fact that is a zero of , it is not difficult to prove that does not depend on the choice of . This endomorphism of is called the linearization of at . Observe that, when , the linearization of a tangent vector field at a zero is just the Fréchet derivative at of the map associated to .
The following fact will play an important rôle in the proof of our main result.
3. Degree of a Tangent Vector Field
We will show that there exists at most one function that, to any admissible pair , assigns a real number called the degree (or characteristic or rotation) of the tangent vector field on , which satisfies the following three properties that will be regarded as axioms. Moreover, this function (if it exists) must be integer valued.
As a consequence of the Additivity Property and Remark 3.1, one easily gets the following (often neglected) property, which shows that the degree of an admissible pair does not depend on the behavior of outside . To prove it, take and in the Additivity Property.
A further important property of the degree of a tangent vector field is the following.
As a consequence, we have the following property.
4. The Degree for Linear Vector Fields
By we will mean the normed space of linear endomorphisms of , and by we will denote the group of invertible ones. In this section we will consider linear vector fields on , namely, vector fields with the property that . Notice that , with a linear vector field, is an admissible pair if and only if .
5. The Uniqueness Result
Given a local tangent vector field on , a zero of is called nondegenerate if is smooth in a neighborhood of and its linearization at is an automorphism of . It is known that this is equivalent to the assumption that is transversal at to the zero section of (for the theory of transversality see, e.g., [3, 4]). We recall that a nondegenerate zero is, in particular, an isolated zero.
Let be a local tangent vector field on . A pair will be called nondegenerate if is a relatively compact open subset of , is smooth on a neighborhood of the closure of , being nonzero on , and all its zeros in are nondegenerate. Note that, in this case, is an admissible pair and is a discrete set and therefore finite because it is closed in the compact set .
The following result, which is an easy consequence of transversality theory, shows that the computation of the degree of any admissible pair can be reduced to that of a nondegenerate pair.
Let be a local tangent vector field on and let be admissible. Let be a relatively compact open subset of containing and such that . Then, there exists a local tangent vector field on which is admissibly homotopic to in and such that is a nondegenerate pair. Consequently, .
Theorem 5.2 below provides a formula for the computation of the degree of a tangent vector field that is valid for any nondegenerate pair. This implies the existence of at most one real function on the family of admissible pairs that satisfies the axioms for the degree of a tangent vector field. We recall that the property of Localization as well as Lemmas 5.1 and 4.2, which are needed in the proof of our result, are consequences of the properties of Normalization, Additivity and Homotopy Invariance.
Theorem 5.2 (uniqueness of the degree).
Consequently, there exists at most one function on the family of admissible pairs satisfying the axioms for the degree of a tangent vector field, and this function, if it exists, must be integer valued.
Moreover if two pairs and correspond under , then the sets and correspond under . It is also evident that the function satisfies the axioms. Thus, by the first part of the proof, it coincides with the restriction , and this implies our claim.
The author is dedicated to Professor William Art Kirk for his outstanding contributions in the theory fixed points
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