A Morse function without critical points

Question

Let $M \subset \mathbb{R}^n$ be a smooth and connected manifold and let $f:M \rightarrow \mathbb{R}$ be a Morse function.

I wonder if $f$ has only as critical points maxima and minima points on $M$, then Is $M$ topologically equivalent to a hyper sphere ?

Thanks in advance

Answer 1

This is not true if one does not assume compactness. A simple example is

$$ M = \left\{ \left(x, \frac{x}{1+x^2}\right) : x\in \mathbb R\right\}$$

in $\mathbb R$ with the morse function $f(x, y) = y$. It has exactly one maximum and minimum and is not $\mathbb S^1$.

Or more generally, one can take any morse function $f$ on any submanifold $N$ of $\mathbb R^n$ and consider $M = N\setminus S$, where $S$ is the critical points which are not maxima nor minima.

If $M$ is compact, then the answer is yes and is answered in this post.