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 $\sum_{i=k+1}^nx_i^2-\sum_{i=1}^kx_i^2$.

A "nice" (smooth Riemannian) metric on $M$ is defined to be one that, when restricted to $U\cong\mathbb R^n$, gives $f$ a gradient field that after some diffeomorphism of $\mathbb R^n$ can be written as $\sum c_ix_i\partial_i$ for nonzero constants $c_i$. (Edited to add the missing $x_i$s.)

Can anyone make an illuminating example of a non-nice metric on $\mathbb R^n$? In fact (because $U$ is bounded) I might prefer one in the setting where $U$ is mapped an open ball rather than all of $\mathbb R^n$.