Let $f \colon M \to \mathbb{R}$ be a smooth function on a manifold $M$. If $f$ is Morse then all the critical points of $f$ are non-degenerate; that is, if $p$ is a critical point of $f$, then $\det \text{Hess}_f(p) \neq 0$.
If $f$ is Morse and $p$ is a critical point of $f$, are the eigenvalues of $\text{Hess}_f(p)$ simple?
If so, are there any references to this?