Theorem of the hairy ball

The theorem of the hairy ball , as called lemma of Milnor affirms as all Champ of continuous vector on a Sphère of even Dimension is cancelled in at least a point.

Intuitive approach

Intuitively, any hairstyle on the hairy ball of even size comprises at least an ear. In dimension 2, that means that, at every moment, there is at least a point of the Ground where the Vent is cancelled, to suppose that the direction of the wind is at least continuous.

Statement

The sphere of dimension N >0 is the topological subspace of

\ R^ {n+1}

of the vectors of Euclidian norm 1, that is to say thus:

S_ {N} = \ {(x_0, \ dowries, x_n) \ in \ mathbb {R} ^ {n+1}, x_0^2 + \ cdots + x_n^2 = 1 \}

. This space is under related Variété and Compact E of dimension N . Intuitively,

S_n

can in the vicinity of v approach by the hyperplane closely connected

\ R^ {n+1}

passing by v and orthogonal with v ; dimension refers to the dimension of this space closely connected. A field of vectors on

S_n

can be defined as an application X

S_n \ rightarrow \ R^ {n+1}

such as for all v in S _N, X ( v ) that is to say orthogonal with

v

. The field is known as continuous if the application is continuous.

Theorem of the hairy ball: If N is even, any continuous field of vectors X on $S_n$ admits with the moint a point of cancellation or not criticizes, in other words: there exists v (dependant on X ) such as: $X (v) =0$ .

In odd dimension, there exist fields of continuous vectors and better still, differentiable, which are not cancelled in any point. For example, it is possible to identify $\ R^ {2n}$ with $\ mathbb {C} ^ {N}$ . For any vector v , the vectors v and I v are orthogonal for the metric Euclidean one on R ^N.

Demonstrations

August 1st

Consequences

August 1st

If has is a antisymmetric Opérateur on an Euclidean space V of dimension odd, then has is noninjective. In particular, there do not exist antisymmetric bilinear forms not degenerated into odd dimension.

Indeed, for any vector v , one a:

Av.v=-v.Av=-Av.v

and thus a fortiori

Av.v=0

. In particular,

X (v) =Av

defines by restricition a continuous field of vectors on the sphere unit of V , which is a sphere of even size. This field admits a point of cancellation v . This unit vector v thus belongs to the core of has .

See too

Théorème of the point fixes of Brouwer

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