Note on vector fields in manifolds

Abstract:

We give a direct geometric proof of Hopf’s theorem on the sum of indices at the zeros of a vector field in a manifold M, or rather of that part of the theorem that says that the sum is the same for any two vector fields. The main idea is to connect the two fields by a one-parameter family of fields and to make everything transversal (to $M \times I$). The resulting system of curves permits one to read off the theorem.