Every point in a Riemmanian manifold is critical
  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 $p$. 
Fernando Galaz-Garcia, Luis Guijarro
math.DG math.MG