View source for Poincaré–Hopf theorem

{{short description|Counts 0s of a vector field on a differentiable manifold using its Euler characteristic}}
In [[Mathematics|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|differential topology]]. It is named after [[Biography:Henri Poincaré|Henri Poincaré]] and [[Biography:Heinz Hopf|Heinz Hopf]].

The '''Poincaré–Hopf''' theorem is often 
illustrated by the special case of the [[Hairy ball theorem|hairy ball theorem]], which simply states that there is no smooth [[Vector field|vector field]] on an even-dimensional [[N-sphere|n-sphere]] having no sources or sinks.

[[File:Poincare-Hopf.svg|thumb|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 [[Physics:Hamiltonian system|Hamiltonian system]])]]

==Formal statement==

Let <math>M</math> be a differentiable manifold, of dimension <math>n</math>, and <math>v</math> a vector field on <math>M</math>. Suppose that <math>x</math> is an isolated zero of <math>v</math>, and fix some [[Local coordinates|local coordinates]] near <math>x</math>. Pick a closed ball <math>D</math> centered at <math>x</math>, so that <math>x</math> is the only zero of <math>v</math> in <math>D</math>. Then the [[Vector field#Index of a vector field|index]] of <math>v</math> at <math>x</math>, <math>\operatorname{index}_x(v)</math>, can be defined as the [[Degree of a continuous mapping#Differential topology|degree]] of the map <math>u : \partial D \to \mathbb S^{n-1}</math> from the [[Manifold#Boundary and interior|boundary]] of <math>D</math> to the <math>(n-1)</math>-sphere given by  <math>u(z)=v(z)/\|v(z)\|</math>.

'''Theorem.''' Let <math>M</math> be a [[Compact space|compact]] [[Differentiable manifold|differentiable manifold]].  Let <math>v</math> be a [[Vector field|vector field]] on <math>M</math> with isolated zeroes. If <math>M</math> has boundary, then we insist that <math>v</math> be pointing in the outward normal direction along the boundary.  Then we have the formula

:<math>\sum_i \operatorname{index}_{x_i}(v) = \chi(M)\,</math>

where the sum of the indices is over all the isolated zeroes of <math>v</math> and <math>\chi(M)</math> is the [[Euler characteristic]] of <math>M</math>. 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 [[Biography:Henri Poincaré|Henri Poincaré]]<ref>Henri Poincaré, On curves defined by differential equations (1881&ndash;1882)</ref> and later generalized to higher dimensions by [[Biography:Heinz Hopf|Heinz Hopf]].<ref>H. Hopf, Vektorfelder in n-dimensionalen Mannigfaltigkeiten, Math. Ann. '''96''' (1926), pp. 209&ndash;221.</ref>

==Significance==
The Euler characteristic of a closed surface is a purely [[Topology|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 [[Integral|integration]], and, in particular, [[Physics:Stokes' theorem|Stokes' theorem]], which states that the integral of the [[Exterior derivative|exterior derivative]] of a [[Differential form|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|integer]] amounts (known as the index) to the total, and they must all sum to 0. This result may be considered{{by whom|date=March 2015}} one of the earliest of a whole series of theorems (e.g. [[Atiyah–Singer index theorem]], [[De Rham's theorem]], [[Grothendieck–Riemann–Roch theorem]]) establishing deep relationships between [[Geometry|geometric]] and analytical or [[Physics:Physics|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, ''N''<sub>ε</sub>. 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''<sub>ε</sub> 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 ''N''<sub>ε</sub> to the {{nowrap|1=(''n''–1)-dimensional}} sphere. Thus, the sum of the indices is independent of the actual vector field, and depends only on the manifold ''M''.{{pb}} 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 {{nowrap|1=(''n''–1)-dimensional}} sphere, that can be extended to the whole n-dimensional manifold, is zero.{{Citation needed|date=December 2020}}
# 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 (topology)|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 {{harv|Brasselet|Seade|Suwa|2009}}.

== See also ==
* [[Eisenbud–Levine–Khimshiashvili signature formula]]
* [[Hopf theorem]]

==References==
{{Reflist}}
*{{Springer|id=p/p110160|title=Poincaré–Hopf theorem}}
*{{cite book|last1=Brasselet|first1=Jean-Paul|last2=Seade|first2=José|last3=Suwa|first3=Tatsuo|title=Vector fields on singular varieties|date=2009|publisher=Springer|location=Heidelberg|isbn=978-3-642-05205-7}}

{{DEFAULTSORT:Poincare-Hopf theorem}}
[[Category:Theorems in differential topology]]
[[Category:Differential topology]]

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