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

$\backslash Sigma\_i\; index\_v(x\_i)\; =\; \backslash chi(M)\backslash ,$

where the sum is over all the isolated zeroes of v and $\backslash chi(M)$ 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.

Differential topology Mathematical theorems | 庞加莱-霍夫定理 | Poincaré-Hopf-Theorem