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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?

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up vote 0 down vote accepted

Here is a trivial counterexample. Let $M=\mathbb{R}^n$ and let $f(x_1,\dots,x_n)=\sum_{i=1}^nx_i^2$.

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