Let $M$ be a Riemannian manifold with a Morse function $f: M \to \mathbb{R}$. The zeroes of the gradient vector field of $f$ are the critical points of $f$.
How do you show that a critical point of $f$ with Morse index $k$ is a zero of the gradient vector field with index $(-1)^k$?
Update, 8/29: If somebody has something they want to add as an answer, there are 50 points are on the table that will otherwise go to waste.
Notes:
- The latter index is the degree of the Gauss map $\partial B^n \to S^{n-1}$ defined by the vector field on the boundary of a small ball $B^n$ containing the zero, as in Poincaré-Hopf.
- I'm looking for an argument that is intrinsic or at least preserves the Riemannian structure, maybe using the Hessian. (Here's a sketch of my unsatisfying argument: Any two Riemannian structures on $M$ define gradient vector fields for $f$ with the same critical points, and you can define a time-dependent flow connecting any two such vector fields to see that the index of every zero is preserved. Thus it suffices to invoke the Morse lemma and use a Morse chart to pull back to $\mathbb{R}^n$ with the standard Riemannian metric. From there, the claim is a simple, explicit calculation.)