I am trying to prove or give a counterexample for the following assertion.
Given a smooth connected manifold $M$ with a Riemannian metric $g$ on it, suppose that $f:M\to[a,b]$ is a smooth function such that $|\nabla f|=\mathfrak{b}(f)$, where $\nabla f$ is the gradient of $f$ and $\mathfrak{b}:\mathbb{R}\to\mathbb{R}$ is a smooth function. We know that each of the critical points of function $f$ is isolated. By the way, $x\in M$ is a critical point if $f'(x)=0$. Now my question is as follows:
One can prove that $f$ has at most two critical points?
In fact, it is not difficult to prove:
given some critical point $o$, each level set of $f$ is a sphere of center $o$. According to this fact, I think one can say that $f$ having more than two critical points contradicts the fact that each level set is a sphere of the center of each of its critical points.
I appreciate any comments or answers.