1/27/2024 0 Comments Hair swirlSome more work would be needed to show that this implies there must actually be a zero of the vector field. Hence they have fixed points (since the Lefschetz number is nonzero). Therefore, they all have Lefschetz number 2, also. By integrating a vector field we get (at least a small part of) a one-parameter group of diffeomorphisms on the sphere and all of the mappings in it are homotopic to the identity. the Lefschetz number (total trace on homology) of the identity mapping is 2. Since the Betti numbers of a 2-sphere are 1, 0, 1, 0, 0. There is a closely related argument from algebraic topology, using the Lefschetz fixed-point theorem. However, the hairy ball theorem says there exists no continuous function that can do this for every point on the sphere (equivalently, for every given vector). To see this, consider the given vector as the radius of a sphere and note that finding a non-zero vector orthogonal to the given one is equivalent to finding a non-zero vector that is tangent to the surface of that sphere where it touches the radius. This is a corollary of the hairy ball theorem. There is no single continuous function that can do this for all non-zero vector inputs. In this regard, it follows that for any compact regular 2-dimensional manifold with non-zero Euler characteristic, any continuous tangent vector field has at least one zero.Īpplication to computer graphics Ī common problem in computer graphics is to generate a non-zero vector in R 3 that is orthogonal to a given non-zero vector. In the case of the torus, the Euler characteristic is 0 and it is possible to "comb a hairy doughnut flat". This is a consequence of the Poincaré–Hopf theorem. Therefore, there must be at least one zero. Counting zeros Įvery zero of a vector field has a (non-zero) " index", and it can be shown that the sum of all of the indices at all of the zeros must be two, because the Euler characteristic of the 2-sphere is two. The theorem has been expressed colloquially as "you can't comb a hairy ball flat without creating a cowlick" or "you can't comb the hair on a coconut". The theorem was first proved by Henri Poincaré for the 2-sphere in 1885, and extended to higher even dimensions in 1912 by Luitzen Egbertus Jan Brouwer. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R 3 to every point p on a sphere such that f( p) is always tangent to the sphere at p, then there is at least one pole, a point where the field vanishes (a p such that f( p) = 0). The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres.
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