They play an important role in the modern study of both fields. Sketch of proof Embed M in some high-dimensional Euclidean space. (Use the Whitney embedding theorem.) Take a small neighborhood of M in that Euclidean space, Nε. Extend the vector field to this neighborhood so that it still has the same zeroes and the zeroes have the same indices. In addition, make sure that the extended vector field at the boundary of Nε is directed outwards. The sum of indices of the zeroes of the old (and new) vector field is equal to the degree of the Gauss map from the boundary of Nε to the sphere. Thus, the sum of the indices is independent of the actual vector field, and depends only on the manifold M. Technique: cut away all zeroes of the vector field with small neighborhoods. Then use the fact that the degree of a map from the boundary of an n-dimensional manifold to an sphere, that can be extended to the whole n-dimensional manifold, is zero. Finally, identify this sum of indices as the Euler characteristic of M. To do that, construct a very specific vector field on M using a triangulation of M for which it is clear that the sum of indices is equal to the Euler characteristic. Generalization It is still possible to define the index for a vector field with nonisolated zeroes. A construction of this index and the extension of Poincaré–Hopf theorem for vector fields with nonisolated zeroes is outlined in Section 1.1.2 of . See also Eisenbud–Levine–Khimshiashvili signature formula Hopf theorem References Theorems in differential topology Differential topology