Let $(M,g)$ be a smooth Riemannian manifold and let $f: M \to \mathbb{R}$ be a Morse function (i.e. critical points are nondegenerate). The Morse lemma says that around every critical point $p$ of $f$ there is a chart $(x_1, \dots, x_n)$ in which we have $$f(x) = f(p) - x_1^2 - x_2^2 - \dots - x_{\lambda}^2 + x_{\lambda+1}^2 + \dots + x_n^2$$
One is interested in \emph{gradient-like} vector fields on $M$, i.e. a vector field $X$ such that
- The directional derivative $X \cdot f$ is strictly positive away from critical points,
- Around every critical point there is a Morse chart $(x_1, x_2, \dots, x_n)$ such that $$X = -2x_1 \frac{\partial}{\partial x_1} - \dots - 2x_{\lambda}\frac{\partial}{\partial x_{\lambda}} + 2x_{\lambda+1} \frac{\partial}{\partial x_{\lambda+1}} \dots + 2x_n \frac{\partial}{\partial x_n}$$
Here's the question: given a prescribed Riemannian metric $g$ on $M$ and a critical point $p$ of $f$, can one pick a coordinate chart $(x_1, \cdot, x_n)$ around $p$ so that
- $f$ is in Morse form
- The gradient $\operatorname{grad}_{g}f$ agrees with (2)?
Given the name "gradient-like", one would expect the answer to be yes, but I couldn't find a construction of this.