Every point in a Riemmanian manifold is critical
Fernando Galaz‐García, Luis Guijarro · arXiv (Cornell University) · 2014
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 $p$.