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

## 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., [16]). 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 [15].

## 2. Preliminaries

Given two sets and , by a 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

(2.1)

with domain .

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.

Throughout the paper all of the differentiable manifolds will be assumed to be finite dimensional, smooth, real, Hausdorff, and second countable. Thus, they can be embedded in some . Moreover, and will always denote arbitrary differentiable manifolds. Given any , will denote the tangent space of at . Furthermore will be the tangent bundle of ; that is,

(2.2)

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 .

Let be a local tangent vector field on and let be a 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:

(2.3)

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.

Let , , , and be as above. Then, the commutativity of diagram (2.3) implies

(2.4)

## 3. Degree of a Tangent Vector Field

Given an open subset of and a local tangent vector field on , the pair is said to be admissible on if and the set

(3.1)

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 .

Given an open subset of and a (continuous) local map with source and target , we say that is a homotopy of tangent vector fields on if , and if is a local tangent vector field for all . If, in addition, the set

(3.2)

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

(3.3)

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 fieldon, which satisfies the following three properties that will be regarded as axioms. Moreover, this function (if it exists) must be integer valued.

Normalization

Let be identity-like on an open subset of . Then,

(3.4)

Given an admissible pair , if and are two disjoint open subsets of such that , then

(3.5)

Homotopy Invariance

If is an admissible homotopy on , then

(3.6)

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 .

To prove this property, observe that by Additivity, Remark 3.1, and Localization one gets

(3.7)

As a consequence, we have the following property.

Solution

If , then .

To obtain it, observe that if , taking , we get

(3.8)

## 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.

Let be a nonsingular linear operator in . Then

(4.1)

Proof.

It is well-known (see, e.g., [11]) that has exactly two connected components. Equivalently, the following two subsets of are connected:

(4.2)

Since the connected sets and are open in , they are actually path connected. Consequently, given a linear tangent vector field on with , Homotopy Invariance implies that depends only on the component of containing . Therefore, if , one has , where is the identity on . Thus, by Normalization, we get

(4.3)

It remains to prove that when . For this purpose consider the vector field determined by

(4.4)

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 .

Let and denote, respectively, the open half-spaces of the points in with negative and positive last coordinate. Consider the two solutions

(4.5)

of the equation and observe that , .

By Additivity (and taking into account the Localization property), we get

(4.6)

Now, observe that in coincides with the vector field determined by

(4.7)

which is admissibly homotopic (in ) to the tangent vector field , given by . Therefore, because of the properties of Localization, Excision, Homotopy Invariance, and Normalization, one has

(4.8)

which, by (4.6), implies that

(4.9)

Notice that in coincides with the vector field defined by

(4.10)

which is admissibly homotopic (in ) to the linear vector field defined by with

(4.11)

Thus, by Homotopy Invariance, Excision, Localization, and formula (4.9)

(4.12)

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.

Since is Fréchet differentiable at and , we have

(4.13)

where is a continuous function such that . Consider the vector field determined by , and let be the homotopy on , joining with , defined by

(4.14)

For all in we have

(4.15)

where is positive because is invertible. This shows that there exists a neighborhood of such that coincides with the compact set . Thus, by Excision and Homotopy Invariance,

(4.16)

Let be the linear tangent vector field given by . Clearly, is admissibly homotopic to in . By Excision, Homotopy Invariance, and Lemma 4.1, we get

(4.17)

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.

Without loss of generality we can assume that . Let

(5.1)

From the Transversality theorem (see, e.g., [3, 4]) it follows that one can find a smooth tangent vector field that is transversal to the zero section of and such that

(5.2)

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.

The conclusion follows by observing that the homotopy on of tangent vector fields given by

(5.3)

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).

Let be a real function on the family of admissible pairs satisfying the properties of Normalization, Additivity, and Homotopy Invariance. If is a nondegenerate pair, then

(5.4)

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.

Consider first the case . Let be a nondegenerate pair in and, for any , let be an isolating neighborhood of . We may assume that the neighborhoods are pairwise disjoint. Additivity and Localization together with Lemma 4.2 yield

(5.5)

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

Let us now consider the general case and denote by the dimension of . Let be any open subset of which is diffeomorphic to and let be any diffeomorphism onto . Denote by the set of all pairs which are admissible and such that . We claim that for any one necessarily has

(5.6)

To show this, denote by the set of admissible pairs with and consider the map defined by

(5.7)

Our claim means that the restriction of to coincides with . Observe that is invertible and

(5.8)

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.

Now let be a given nondegenerate pair in . Let and let be pairwise disjoint open subsets of such that , for . Since any point of has a fundamental system of neighborhoods which are diffeomorphic to , we may assume that each is diffeomorphic to by a diffeomorphism . Additivity and Localization yield

(5.9)

and, by the above claim, we get

(5.10)

By Lemma 4.2 and Remark 2.1

(5.11)

for . Thus

(5.12)

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

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## Acknowledgment

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

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Correspondence to Massimo Furi.

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Furi, M., Pera, M. & Spadini, M. A Set of Axioms for the Degree of a Tangent Vector Field on Differentiable Manifolds. Fixed Point Theory Appl 2010, 845631 (2010). https://doi.org/10.1155/2010/845631