 Poincaré–Hopf theorem

For a compact oriented smooth manifold $M$, the Euler class of its tangent bundle satisfiesEquivalently, the sum of the indices of the isolated zeros of a vector field equals the Euler characteristic.

The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the topology of a compact manifold to the behavior of vector fields defined on it. Specifically, it provides a formula for the Euler characteristic of a manifold in terms of the zeros of a smooth vector field on that manifold. Here's a more detailed breakdown of the theorem’s key concepts: 1. **Setting**: Let \( M \) be a compact, oriented \( n \)-dimensional manifold without boundary.