- 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ1_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ2_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equa_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equb_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ5_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ6_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ7_HTML.gif)
Additivity
Given an admissible pair , if
and
are two disjoint open subsets of
such that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ8_HTML.gif)
Homotopy Invariance
If is an admissible homotopy on
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ10_HTML.gif)
As a consequence, we have the following property.
Solution
If , then
.
To obtain it, observe that if , taking
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ12_HTML.gif)
Proof.
It is well-known (see, e.g., [11]) that has exactly two connected components. Equivalently, the following two subsets of
are connected:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ13_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ14_HTML.gif)
It remains to prove that when
. For this purpose consider the vector field
determined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ16_HTML.gif)
of the equation and observe that
,
.
By Additivity (and taking into account the Localization property), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ17_HTML.gif)
Now, observe that in
coincides with the vector field
determined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ19_HTML.gif)
which, by (4.6), implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ20_HTML.gif)
Notice that in
coincides with the vector field
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ21_HTML.gif)
which is admissibly homotopic (in ) to the linear vector field
defined by
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ22_HTML.gif)
Thus, by Homotopy Invariance, Excision, Localization, and formula (4.9)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ24_HTML.gif)
where is a continuous function such that
. Consider the vector field
determined by
, and let
be the homotopy on
, joining
with
, defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ25_HTML.gif)
For all in
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ26_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ27_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ28_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ29_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ30_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ32_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ33_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ34_HTML.gif)
To show this, denote by the set of admissible pairs
with
and consider the map
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ35_HTML.gif)
Our claim means that the restriction of
to
coincides with
. Observe that
is invertible and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ36_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ37_HTML.gif)
and, by the above claim, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ38_HTML.gif)
By Lemma 4.2 and Remark 2.1
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ39_HTML.gif)
for . Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F845631/MediaObjects/13663_2009_Article_1353_Equ40_HTML.gif)
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.19720540117
Staecker 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.
Acknowledgment
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