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) states:

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

<math>\Sigma_i index_v(x_i) = \Chi(M)<math>

where the sum is over all the isolated zeroes of v and <math>\Chi(M)<math> 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.