Is there a Morse function that does not arise from a minimal-dimensional height function?

Question

Let $M$ be a smooth, finite dimensional manifold of dimension $n$, and let $m$ be the minimal dimension for which $M$ admits a smooth embedding into $\mathbb R^m$.

Question: Does every Morse function arise from a height function in dimension $m$? Formally, is every Morse function $f: M \to \mathbb{R}$ of the form: $$f: M \hookrightarrow \mathbb{R}^m \stackrel{\pi}{\to} \mathbb{R}?$$

Answer 1

Take $M:=S^1$ and $f:=\sin \varphi$. Its critical points are min->max->min->max. Any immersion of $S^1\to \mathbb R^2$ which realizes these maxima and minima with respect to a height function must have a self-intersection.