# Poincaré-Hopf theorem

(Redirected from Poincaré–Hopf theorem)
Let $M$ be a smooth compact manifold with boundary $W=\partial M$, and let $X$ be a vector field on $M$ with isolated zeros such that $X$ points outwards at all points in the boundary $W$.
Then the sum of the indices of the zeros of $V$ (see Singular point, index of a) is equal to the Euler characteristic of $M$.