Every point in a Riemmanian manifold is critical

Abstract

We show that for any point $p$ in a closed Riemannian manifold $M$, there exists at least one point $q\in M$ such that $p$ is critical for the distance function from $q$. We also show that such a point $q$ cannot always be reached with geodesic loops based at $q$ with midpoint $p$.

7 pages, 2 figures. Minor changes. Accepted for publication in Calculus of Variations and Partial Differential Equations