Let $M$ be an algebraic manifold. Let $P:[0,1]\rightarrow M$ be a finite length $C^\infty$ path on $M$. Let $d_M(x,y)$ indicate the shortest length geodesic from $x$ to $y$ on $M$.
Let $s\in M$ be some point on $M$. Consider a function $f_s:[0,1]\rightarrow\mathbb{R}$ such that
$$f_s(t) = d_M(s,P(t))$$
Does $f_s(t)$ have a bounded number of local minima for any given $s$, $P$, and $M$?
Any results that might be useful in proving this or counterexamples would be appreciated.