 Poincaré–Hopf theorem

For a closed, oriented smooth manifold $M$ and a smooth vector field $X$ with isolated zeros, the Poincaré-Hopf theorem statesThe index of a vector field at an isolated zero is the mapping degree of $X/\Vert X\Vert$ on a small surrounding sphere, computed in an oriented coordinate chart. Thus local winding at the zeros adds up to the global Euler characteristic.^[ref]

For the gradient of height on a two-dimensional sphere, the north and south zeros each have index one; their sum is $\chi (S^2)=2$. For a compact manifold with boundary, the same sum requires an outward-pointing boundary condition. A constant nonzero vector field on a closed disk shows why unrestricted boundary behavior cannot be allowed.

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.