- Research Article
- Open access
- Published:
A Set of Axioms for the Degree of a Tangent Vector Field on Differentiable Manifolds
Fixed Point Theory and Applications volume 2010, Article number: 845631 (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
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
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,
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:
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
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
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
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:
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
Let 
be identity-like on an open subset 
of 
. Then,
Additivity
Given an admissible pair 
, if 
and 
are two disjoint open subsets of 
such that 
, then
Homotopy Invariance
If 
is an admissible homotopy on 
, then
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
If 
is admissible, then 
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
As a consequence, we have the following property.
Solution
If 
, then 
.
To obtain it, observe that if 
, taking 
, we get
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
Proof.
It is well-known (see, e.g., [11]) that 
has exactly two connected components. Equivalently, the following two subsets of 
are connected:
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
It remains to prove that 
when 
. For this purpose consider the vector field 
determined by
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
of the equation 
and observe that 
, 
.
By Additivity (and taking into account the Localization property), we get
Now, observe that 
in 
coincides with the vector field 
determined by
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
which, by (4.6), implies that
Notice that 
in 
coincides with the vector field 
defined by
which is admissibly homotopic (in 
) to the linear vector field 
defined by 
with
Thus, by Homotopy Invariance, Excision, Localization, and formula (4.9)
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
where 
is a continuous function such that 
. Consider the vector field 
determined by 
, and let 
be the homotopy on 
, joining 
with 
, defined by
For all 
in 
we have
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,
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
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
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
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
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
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
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
To show this, denote by 
the set of admissible pairs 
with 
and consider the map 
defined by
Our claim means that the restriction 
of 
to 
coincides with 
. Observe that 
is invertible and
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
and, by the above claim, we get
By Lemma 4.2 and Remark 2.1
for 
. Thus
As in the case 
, the uniqueness of the degree of a tangent vector field is now a consequence of Lemma 5.1.
References
Krasnosel'skiĭ MA: The Operator of Translation along the Trajectories of Differential Equations, Translations of Mathematical Monographs, vol. 19. American Mathematical Society, Providence, RI, USA; 1968:vi+294.
Krasnosel'skiĭ MA, Zabreĭko PP: Geometrical Methods of Nonlinear Analysis, Grundlehren der Mathematischen Wissenschaften. Volume 263. Springer, Berlin, Germany; 1984:xix+409.
Guillemin V, Pollack A: Differential Topology. Prentice-Hall, Englewood Cliffs, NJ, USA; 1974:xvi+222.
Hirsch MW: Differential Topology. Springer, New York, NY, USA; 1976:x+221. Graduate Texts in Mathematics, no. 33
Milnor JW: Topology from the Differentiable Viewpoint. The University Press of Virginia, Charlottesville, Va, USA; 1965:ix+65.
Tromba AJ: The Euler characteristic of vector fields on Banach manifolds and a globalization of Leray-Schauder degree. Advances in Mathematics 1978,28(2):148–173. 10.1016/0001-8708(78)90061-0
Amann H, Weiss SA: On the uniqueness of the topological degree. Mathematische Zeitschrift 1973, 130: 39–54. 10.1007/BF01178975
Führer L: Ein elementarer analytischer beweis zur eindeutigkeit des abbildungsgrades im
. Mathematische Nachrichten 1972, 54: 259–267. 10.1002/mana.19720540117Staecker PC: On the uniqueness of the coincidence index on orientable differentiable manifolds. Topology and Its Applications 2007,154(9):1961–1970. 10.1016/j.topol.2007.02.003
Furi M, Pera MP, Spadini M: On the uniqueness of the fixed point index on differentiable manifolds. Fixed Point Theory and Applications 2004,2004(4):251–259. 10.1155/S168718200440713X
Warner FW: Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics. Volume 94. Springer, New York, NY, USA; 1983:ix+272.
. Mathematische Nachrichten 1972, 54: 259–267. 10.1002/mana.19720540117Acknowledgment
The author is dedicated to Professor William Art Kirk for his outstanding contributions in the theory fixed points
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/845631