Poincaré–Hopf theorem

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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 in differential topology. It is named after Henri Poincaré and Heinz Hopf.

Theorem. Let M be a compact differentiable manifold. Let v be a vector field on M with isolated zeroes. If M has boundary, then we insist that v be pointing in the outward normal direction along the boundary. Then we have the formula

where the sum is over all the isolated zeroes of v and χ(M) is the Euler characteristic of M.

The theorem was proven for two dimensions by Henri Poincaré and later generalized to higher dimensions by Heinz Hopf.

The Poincaré–Hopf theorem is often illustrated by the special case of the Hairy ball theorem.