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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 Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...
. The Poincaré–Hopf theorem is often illustrated by the special case of the
hairy ball theorem 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. For the ordinary sphere, or 2‑sphere, if ...
, which simply states that there is no smooth vector field on an even-dimensional n-sphere having no sources or sinks.


Formal statement

Let M be a differentiable manifold, of dimension n, and v a vector field on M. Suppose that x is an isolated zero of v, and fix some
local coordinates Local coordinates are the ones used in a ''local coordinate system'' or a ''local coordinate space''. Simple examples: * Houses. In order to work in a house construction, the measurements are referred to a control arbitrary point that will allow ...
near x. Pick a closed ball D centered at x, so that x is the only zero of v in D. Then the index of v at x, \operatorname_x(v), can be defined as the degree of the map u : \partial D \to \mathbb S^ from the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
of D to the (n-1)-sphere given by u(z)=v(z)/\, v(z)\, . Theorem. Let M be a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
differentiable manifold. Let v be a vector field on M with isolated zeroes. If M has
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
, then we insist that v be pointing in the outward normal direction along the boundary. Then we have the formula :\sum_i \operatorname_(v) = \chi(M)\, where the sum of the indices is over all the isolated zeroes of v and \chi(M) is the Euler characteristic of 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 and later generalized to higher dimensions by
Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...
.H. Hopf, Vektorfelder in n-dimensionalen Mannigfaltigkeiten, Math. Ann. 96 (1926), pp. 209–221.


Significance

The Euler characteristic of a closed surface is a purely
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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

# Embed ''M'' in some high-dimensional Euclidean space. (Use the
Whitney embedding theorem In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: *The strong Whitney embedding theorem states that any smooth real -dimensional manifold (required also to be Hausdorff ...
.) # Take a small neighborhood of ''M'' in that Euclidean space, ''N''ε. 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''ε 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 In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that ' ...
from the boundary of ''N''ε 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. # 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 .


See also

*
Eisenbud–Levine–Khimshiashvili signature formula In mathematics, and especially differential topology and singularity theory, the Eisenbud–Levine–Khimshiashvili signature formula gives a way of computing the Poincaré–Hopf index of a real, analytic vector field at an algebraically isolated ...
*
Hopf theorem The Hopf theorem (named after Heinz Hopf) is a statement in differential topology, saying that the topological degree is the only homotopy invariant of continuous maps to spheres. Formal statement Let ''M'' be an ''n''-dimensional compact connec ...


References

* * {{DEFAULTSORT:Poincare-Hopf theorem Theorems in differential topology Differential topology