Critical points and manifolds

Question

This might be trivial but I was wandering if given any manifold $M$ and point $a\in M$ why is it that we can't just define a function that has $a$ as a non-degenerate critical point? I.e., what property of the manifold prevents certain points from being critical for any function?

Answer 1

I'm unsure of what exactly you're asking. If $M$ has dimension $n$ then I can take any function $f$ on $\mathbb R^n$ with a "nondegnerate" critical point at $0$ and which vanishes outside a ball, and then take a smooth chart on $M$ about my point $a\in M$ and identify it with $\mathbb R^n$. I then extend $f$ to be zero outside of this chart. In this way every point on $M$ can be made into the critical point of some function.