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

- Massimo Furi
^{1}Email author, - MariaPatrizia Pera
^{1}and - Marco Spadini
^{1}

**2010**:845631

**DOI: **10.1155/2010/845631

© Massimo Furi et al. 2010

**Received: **28 September 2009

**Accepted: **7 February 2010

**Published: **10 May 2010

## 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
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.

## 1. Introduction

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 [7] (see also [8]) 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 [9]) 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 [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].

## 2. Preliminaries

*local map*with

*source*and

*target*we mean a triple , where , the

*graph*of , is a subset of such that for any there exists at most one with . The domain of is the set of all for which there exists such that ; that is, , where denotes the projection of onto the first factor. The

*restriction*of a local map to a subset of is the triple

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.

The map
given by
will be the *bundle projection* of
. It will also be convenient, given any
, to denote by
the zero element of
.

Given a smooth map , by we will mean the map that to each associates . Here denotes the differential of at . Notice that if is a diffeomorphism, then so is and one has .

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
and
be differentiable manifolds and let
be a diffeomorphism. Recall that two tangent vector fields
and
*correspond under*
if the following diagram commutes:

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
.

*zero*of ; that is, . Consider a diffeomorphism of a neighborhood of onto and let be the tangent vector field on that corresponds to under . Since , then the map associated to sends into itself. Assuming that is smooth in a neighborhood of , the function is Fréchet differentiable at . Denote by its Fréchet derivative and let be the endomorphism of which makes the following diagram commutative:

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.

Remark 2.1.

## 3. Degree of a Tangent Vector Field

*admissible on*if and the set

of the zeros of in is compact. In particular, is admissible if the closure of is a compact subset of and is nonzero on the boundary of .

*homotopy of tangent vector fields on*if , and if is a local tangent vector field for all . If, in addition, the set

*admissible*. Thus, if is compact and , a sufficient condition for to be admissible on is the following:

which, by abuse of terminology, will be referred to as " is nonzero on ".

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.

Normalization

Additivity

Homotopy Invariance

From now on we will assume the existence of a function defined on the family of all admissible pairs and satisfying the above three properties that we will regard as axioms.

Remark 3.1.

The pair is admissible. This includes the case when is the empty set ( is coherent with the notion of local tangent vector field). A simple application of the Additivity Property shows that and .

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.

Localization

A further important property of the degree of a tangent vector field is the following.

Excision

Given an admissible pair and an open subset of containing , one has .

As a consequence, we have the following property.

Solution

## 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
.

The following consequence of the axioms asserts that the degree of an admissible pair , with , is either or .

Lemma 4.1.

Proof.

Notice that is well defined because is compact. Observe also that is zero, because is admissibly homotopic in to the never-vanishing vector field given by .

of the equation and observe that , .

Hence, being path connected, we finally get for all linear tangent vector fields on such that , and the proof is complete.

We conclude this section with a consequence as well as an extension of Lemma 4.1. The Euclidean norm of an element will be denoted by .

Lemma 4.2.

Let be a local vector field on and let be open and such that the equation has a unique solution . If is smooth in a neighborhood of and the linearization of at is invertible, then .

Proof.

The assertion now follows from (4.16), (4.17), and the fact that coincides with .

## 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.

Lemma 5.1.

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, .

Proof.

Since is closed in , the set is a compact subset of . Thus, this inequality shows that is admissible. Moreover, at any zero the endomorphism is invertible. This implies that is nondegenerate.

is nonzero on and therefore it is admissible on . The last assertion follows from Excision, and Homotopy Invariance.

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.

Proof.

Now the uniqueness of the degree of a tangent vector field on follows immediately from Lemma 5.1.

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.

As in the case , the uniqueness of the degree of a tangent vector field is now a consequence of Lemma 5.1.

## Declarations

### Acknowledgment

The author is dedicated to Professor William Art Kirk for his outstanding contributions in the theory fixed points

## Authors’ Affiliations

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