Often, hairy ball theorem is mentioned as an application of degree theory derived from singular homology in algebraic topology or from de Rham cohomology in differential topology; we also mentionned it as an application of Brouwer’s degree in Brouwer’s Topological Degree (III): First Applications. Here we describe an argument due to Milnor, exposed in Gallot, Hullin and Lafontaine’s book Riemannian Geometry, using only integration of differential forms.
For convenience, we begin by exposing some basic definitions and first properties about integration on manifolds.
If is a smooth manifold, let and denote respectively its tangent bundle and its space of differential forms of degree (ie. the vector space of sections of the -th extorior power of the cotangent bundle). We say that is orientable if there exists an atlas , said oriented, such that is orientation-preserving for all .
Property: A smooth manifold is orientable if and only if there exists satisfying for all . Such a differential form is a volume form.
Sketch of proof. Let be a volume form and let be a family of charts covering . Because is a volume form, we can write
where does not vanish. Without loss of generality, we may suppose that (otherwise, switch two coordinates of the chart). Then is an oriented atlas.
Conversely, let be an oriented atlas and be an associated partition of unity. Then
is a volume form.
As an example, we may mention the differential form
Then induces a volume form on viewed as the unit sphere in . [In fact, is the canonical volum form of as a Riemannian manifold. In particular, we define .]
An interesting property of volum forms is that, if is a volume form and , then there exists such that . It is clear locally, and then it is sufficient to extend it globally using a partition of unity.
Definition: Let be an orientable compact -dimensional smooth manifold. There exists one and only one linear map such that for every supported in an open chart ,
[Where the right-hand side is the usual integral of a differential form defined on an open subspace in .]
Sketch of proof. First, suppose that such a linear map exists. Let be a finite family of charts covering (such a family exists by compactness) and be an associated partition of unity. Then necessarily:
hence is uniquely defined by the mentionned properties. Conversely, it is sufficient to show that the above equality does not depend on the family of charts and on the partition of unity chosen. An argument for that is to notice that if is a diffeomorphism between two open subsets , and , then
where if is orientation-preserving and otherwise. We can write for some , and
Using the proof above, we may notice:
Property: Let be two compact orientable -dimensional smooth manifolds and . If is a diffeomorphism then
where if is orientation-preserving and otherwise.
Theorem: (Hairy ball theorem) If is even, any continuous vector field on has a zero.
Proof. Let be a continuous vector field on , that is is a map satisfying for every . By contradiction, suppose that does not vanish.
According to Stone-Weierstrass theorem, it is possible to approximate by a sequence of smooth maps . Then, if denotes the orthogonal projection of on , is a sequence of smooth vector fields on converging uniformely to . Therefore, we may suppose without loss of generality that is smooth.
[Such an argument may be generalized to other manifolds; see Tubular neighborhood.]
Let be the volume form as defined above by the restriction on of
and let . For every and , we may write
where is such that and with or . Therefore, there exist such that
First, let us show how conclude the proof if is a diffeomorphism:
Let for convenience, be a diffeomorphism from to and as mentioned above . Then
so according to , the expression is a polynomial in . Of course, it is only possible when is odd.
To conclude the proof, we shall see that is a diffeomorphism when is sufficiently small. From , if is such that ,
By compactness, the are bounded so the quantity does not vanish when is small enough. Therefore, is a volume form and is an immersion.
Moreover, if were not injective for small enough, there would exist sequences , and such that , and , hence
The first term has norm one, and the last term is bounded according to mean value theorem: a contradiction.
Therefore, for small enough, is an injective immersion, and consequently a local diffeomorphism according to inverse function theorem. In particular, is open but also closed by compactness of ; by connectedness, and is a diffeomorphism.