Poincaré–Hopf theorem

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.

According to the Poincare-Hopf theorem, closed trajectories can encircle two centres and one saddle or one centre, but never just the saddle. (Here for in case of a Hamiltonian system) Conteúdo 1 Declaração formal 2 Significado 3 Esboço de prova 4 Generalização 5 Veja também 6 References Formal statement Let {estilo de exibição M} be a differentiable manifold, of dimension {estilo de exibição m} , e {estilo de exibição v} a vector field on {estilo de exibição M} . Suponha que {estilo de exibição x} is an isolated zero of {estilo de exibição v} , and fix some local coordinates near {estilo de exibição x} . Pick a closed ball {estilo de exibição D} centered at {estilo de exibição x} , de modo a {estilo de exibição x} is the only zero of {estilo de exibição v} dentro {estilo de exibição D} . Then the index of {estilo de exibição v} no {estilo de exibição x} , {nome do operador de estilo de exibição {index} _{x}(v)} , can be defined as the degree of the map {estilo de exibição você:partial Dto mathbb {S} ^{n-1}} from the boundary of {estilo de exibição D} to the {estilo de exibição (n-1)} -sphere given by {estilo de exibição você(z)=v(z)/|v(z)|} .

Teorema. Deixar {estilo de exibição M} be a compact differentiable manifold. Deixar {estilo de exibição v} be a vector field on {estilo de exibição M} with isolated zeroes. Se {estilo de exibição M} has boundary, then we insist that {estilo de exibição v} be pointing in the outward normal direction along the boundary. Then we have the formula {soma de estilo de exibição _{eu}nome do operador {index} _{x_{eu}}(v)=chi (M),} where the sum of the indices is over all the isolated zeroes of {estilo de exibição v} e {chi de estilo de exibição (M)} is the Euler characteristic of {estilo de exibição M} . 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[1] and later generalized to higher dimensions by Heinz Hopf.[2] Significance The Euler characteristic of a closed surface is a purely topological concept, whereas the index of a vector field is purely analytic. Desta forma, 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, e, em 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[by whom?] one of the earliest of a whole series of theorems[que?] 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 Embed M in some high-dimensional Euclidean space. (Use the Whitney embedding theorem.) Take a small neighborhood of M in that Euclidean space, . Extend the vector field to this neighborhood so that it still has the same zeroes and the zeroes have the same indices. Além disso, make sure that the extended vector field at the boundary of Nε is directed outwards. 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 Nε to the (n–1)-dimensional sphere. Desta forma, 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 (n–1)-dimensional sphere, that can be extended to the whole n-dimensional manifold, is zero.[citação necessária] Finalmente, 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 do (Brasselet, Seade & Suwa 2009).

See also Eisenbud–Levine–Khimshiashvili signature formula Hopf theorem References ^ Henri Poincaré, On curves defined by differential equations (1881–1882) ^ H. Hopf, Vektorfelder in n-dimensionalen Mannigfaltigkeiten, Matemática. Ana. 96 (1926), pp. 209–221. "Poincaré–Hopf theorem", Enciclopédia de Matemática, Imprensa EMS, 2001 [1994] Brasselet, Jean-Paul; Seade, José; Suwa, Tatsuo (2009). Vector fields on singular varieties. Heidelberg: Springer. ISBN 978-3-642-05205-7. Categorias: Theorems in differential topologyDifferential topology

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