Poincaré–Hopf theorem

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.

, and fix some local coordinates near

Pick a closed ball

, can be defined as the degree of the map

be a compact differentiable manifold.

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

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] 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.

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 geometric and analytical or physical concepts.

They play an important role in the modern study of both fields.

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 (Brasselet, Seade & Suwa 2009).

Another generalization that use only compact triangulable space and continuous mappings with finitely many fixed points is Lefschetz-Hopf theorem.

Since every vector field induce flow on manifold and fixed points of small flows corresponds to zeroes of vector field (and indices of zeroes equals indices of fixed points), then Poincare-Hopf theorem follows immediately from it.