Poincaré–Hopf theorem
In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincaré and Heinz Hopf. The Poincaré–Hopf theorem is often illustrated by the special case of the hairy ball theorem, which simply states that there is no smooth vector field on an even-dimensional n-sphere having no sources or sinks. Formal statement. Let formula_1 be a differentiable manifold, of dimension formula_2, and formula_3 a vector field on formula_1. Suppose  that formula_5 is an isolated zero of formula_3, and fix some local coordinates near formula_5. Pick a closed ball formula_8 centered at formula_5, so that formula_5 is the only zero of formula_3 in formula_8. Then the index of formula_3 at formula_5, formula_15, can be defined as the degree of the map formula_16 from the boundary of formula_8 to the formula_18-sphere given by formula_19. Theorem. Let formula_1 be a compact differentiable manifold. Let formula_3 be a vector field on formula_1 with isolated zeroes. If formula_1 has boundary, then we insist that formula_3 be pointing in the outward normal direction along  the boundary. Then we have the formula where the sum of the indices is over all the isolated zeroes of formula_3 and formula_27 is the Euler characteristic of formula_1. A particularly useful corollary is when there is a non-vanishing vector field implying Euler characteristic 0. The theorem was proven for two dimensions by Henri Poincaré and later generalized to higher dimensions by Heinz Hopf. Significance. The Euler characteristic of a closed surface is a purely topological concept, whereas the index of a vector field is purely analytic. Thus, this theorem establishes a deep link between two seemingly unrelated areas  of mathematics. It is perhaps as interesting that the proof of this theorem relies heavily on integration, and, in particular, Stokes' theorem, which states that the integral of the exterior derivative of a differential form is equal to the integral of that form over the boundary. In the special case of a manifold without boundary, this amounts to saying that the integral is 0. But by examining vector fields in a sufficiently small neighborhood of a source or sink, we see that sources and sinks contribute integer amounts (known as the index) to the total, and they must all sum  to 0. This result may be considered one of the earliest of a whole series of theorems establishing deep relationships between geometric and analytical or physical concepts. They play an important role in the modern study of both fields. Sketch of proof. 1. Embed "M" in some high-dimensional Euclidean space. (Use the Whitney embedding theorem.) 2. Take a small neighborhood of "M" in that Euclidean space, "N". Extend the vector field to this neighborhood so that it still has the same zeroes and the zeroes have the same indices. In addition, make sure that the extended vector field at  the boundary of "N" is directed outwards. 3. The sum of indices of the zeroes of the old (and new) vector field is equal to the degree of the Gauss map from the boundary of "N" to the sphere. Thus, the sum of the indices is independent of the actual vector field, and depends only on the manifold "M". Technique: cut away all zeroes of the vector field with small neighborhoods. Then use the fact that the degree of a map from the boundary of an n-dimensional manifold to an sphere, that can be extended to the whole n -dimensional manifold, is zero. 4. Finally, identify this sum of indices as the Euler characteristic of "M". To do that, construct a very specific vector field on "M" using a triangulation of "M" for which it is clear that the sum of indices is equal to the Euler characteristic. Generalization. It is still possible to define the index for a vector field with nonisolated zeroes. A construction of this index and the extension of Poincaré–Hopf theorem for vector fields with nonisolated zeroes is outlined in Section 1.1.2 of .