Hedgehog combing theorem

Vector field on a sphere with a single singular point ( dipole of index 2).

The hedgehog combing theorem states that on a sphere it is impossible to choose the tangent direction at each point, which is defined at all points of the sphere and continuously depends on the point. Informally speaking, it is impossible to comb a hedgehog curled up in a ball so that not a single needle sticks out - hence the mention of the hedgehog in the title of the theorem.

The theorem is a consequence of the fixed-point theorem proved in 1912 by Brauer .

Wording

There is no continuous tangent vector field on a sphere that does not vanish anywhere.

In other words, if $f$ ${\ displaystyle f}$ $f$ Is a continuous function defining a vector tangent to the sphere at each of its points, then there exists at least one point $p$ ${\ displaystyle p}$ $p$ such that $f(p)=0$ ${\ displaystyle f (p) = 0}$ $f(p)=0$ .

Another version of the "hedgehog theorem" looks like this:

Let be $\mathbf {f}$ ${\ displaystyle \ mathbf {f}}$ ${\mathbf {f}}$ Is a nonzero continuous vector field on the sphere. Then there is a point at which the field is perpendicular to the sphere.

Consequences and Applications

Any continuous mapping of a sphere onto itself either has a fixed point or maps some point to its diametrically opposite one.: This becomes clear if we transform the map into a continuous vector field as follows. Let be $s$ ${\ displaystyle s}$ $s$ - mapping the sphere onto itself, and $v$ ${\ displaystyle v}$ $v$ - the desired vector field. For any point $p$ ${\ displaystyle p}$ $p$ construct a stereographic projection of a point $s(p)$ ${\ displaystyle s (p)}$ $s(p)$ to the tangent plane at $p$ ${\ displaystyle p}$ $p$ . Then $v(p)$ ${\ displaystyle v (p)}$ $v(p)$ Is the displacement vector of the projection relative to $p$ ${\ displaystyle p}$ $p$ . By the hedgehog combing theorem, there is such a point $p$ ${\ displaystyle p}$ $p$ , what $v(p)=0$ ${\ displaystyle v (p) = 0}$ $v(p)=0$ , so that $s(p)=p$ ${\ displaystyle s (p) = p}$ $s(p)=p$ .

The proof does not pass only if for some point

p

{\ displaystyle p}

s(p)

{\ displaystyle s (p)}

the opposite

p

{\ displaystyle p}

, since in this case it is impossible to construct its stereographic projection onto the tangent plane at a point

p

{\ displaystyle p}

.

There must be a cyclone on Earth.: An interesting meteorological application of this theorem is obtained if we consider the wind as a continuous vector field on the surface of the planet. Let us consider the idealized case in which the field component normal to the surface is negligible. The hedgehog combing theorem states that there will always be a point on the planet’s surface where there will be no wind (zero tangent vector field). Such a point will be the center of the cyclone or anticyclone: the wind will spin around this point (it cannot be directed to or from this point). Thus, according to the theorem on combing the hedgehog, if at least some wind blows on the Earth, then somewhere there must necessarily be a cyclone .

There is no unambiguously defined continuous “top” vector for a virtual camera.: There is no continuous function in $\mathbb {R} ^{3}$ ${\ displaystyle \ mathbb {R} ^ {3}}$ which for each vector generates perpendicular. In computer graphics, the traditional position of the camera , which looks from point A to object B, is as follows: a certain direction (“top”) is selected, and the desired vector (“top of the frame”) is the orthogonal component of the direction-top to vector AB. Of course, when the camera should look straight up or down, this vector is zero. The theorem says that even in space, where there is no “top” and “bottom”, it is impossible to make such a mapping so that it is simultaneously unambiguous and without such special directions.

Variations and generalizations

From a more general point of view, it can be shown that a certain sum of zeros of the tangent vector field must be equal to 2, the Euler characteristic of the two-dimensional sphere, therefore, at least one zero must exist. This is a consequence of Poincare’s vector field theorem . For the two-dimensional torus, the Euler characteristic is 0, so it can be “combed”. In general, any continuous tangent vector field on a compact regular two-dimensional manifold with nonzero Euler characteristic has at least one zero.

Relationship with the Euler Characteristic $\chi$ ${\ displaystyle \ chi}$ suggests the correct generalization: to $2n$ ${\ displaystyle 2n}$ -dimensional sphere does not exist anywhere nonzero continuous vector field ( $n\geqslant 1$ ${\ displaystyle n \ geqslant 1}$ ) The difference between even and odd dimensions is that $k$ ${\ displaystyle k}$ -dimensional Betty numbers $m$ ${\ displaystyle m}$ -dimensional spheres equal 0 for all $k$ ${\ displaystyle k}$ , Besides $k=0$ ${\ displaystyle k = 0}$ and $k=m$ ${\ displaystyle k = m}$ , therefore their alternating sum $\chi$ ${\ displaystyle \ chi}$ equal to 2 for even $m$ ${\ displaystyle m}$ and 0 for odd ones.

Lefschetz Theorem

There is a very close statement from algebraic topology based on the Lefschetz fixed point theorem . Since the Betti numbers of the two-dimensional sphere are 1, 0, 1, 0, 0, ..., the Lefschetz number (complete trace on homology ) of the identity map is 2 . Integrating the vector field, we obtain (at least in a small neighborhood of 0) a one-parameter group of diffeomorphisms on the sphere, all of which maps are homotopic to the identity. Therefore, all of them also have a Lefschetz number 2, therefore, they have fixed points (since their Lefschetz number is nonzero). It can be proved that these points will indeed be zeros of the vector field. This suggests the formulation of a more general Poincaré vector field theorem .

Literature

Murray Eisenberg, Robert Guy. A Proof of the Hairy Ball Theorem . - The American Mathematical Monthly. - Vol. 86. - No. 7 (Aug. - Sep., 1979). - pp. 571-574.