Poincaré-Hopf index theorem

Name

This result goes by the name of Poincaré-Hopf theorem, Hopf index theorem.

Statement

For compact manifolds

Let $M$ be a compact differential manifold. Then, the following are true:

1. There exists a vector field on $M$ with isolated zeros. An isolated zero is a point where the vector field vanishes, such that there is an open set containing the point, and not containing any other point where the vector field vanishes. Since $M$ is compact, this is equivalent to demanding that the set of zeros be finite.
2. For any vector field with isolated zeros, the sum of the indices of all isolated zeros equals the Euler characteristic of the manifold.