For a complete non-compact Riemannian manifold with sectional curvature positive, it is diffeomorphic to $\mathbb{R}^n$ by known result. Choose a point $p$ on the manifold, is it possible that the distance function $d(p, \cdot)$ has a sequence of critical points going to the infinity of the manifold?



complete non-compact Riemannian manifolds and critical points
Yes, it is a classical result.


Let $q\_1,\dots,q\_n$ be a sequence of critical points such that $$|q\_{n+1}-p|\ge 2\cdot |q\_n-p|.$$
By Toponogov's comparison, 
$$\measuredangle [p\,^{q\_i}\_{q\_j}]\ge \tfrac\pi3.$$
Hence we get a bound on $n$.


