Ralph Cohen (professor at Stanford) is teaching a class on algebraic topology and moduli spaces this quarter, beginning by reviewing his perspective of Morse theory. He defined "nice" metrics, proved that they are dense in the $L^2$ space of metrics on $\mathbb R^n$, proved one result using that, and doesn't need them anymore. But given the previous assumptions, I and some classmates want to know: are not all metrics "nice"?
The setup: Assume a real-valued $C^\infty$ function $f$ on a closed smooth manifold $M$ of dimension $n$ is Morse: critical points are nondegenerate in the sense of having full-rank Hessians. Given some critical point $p$ of index $k$, find a small neighborhood $U$ and a diffeomorphism $U\cong\mathbb R^n$ such that $f$ becomes a function $\sum_{i=k+1}^nx_i^2-\sum_{i=1}^kx_i^2$.