Every point in a Riemmanian manifold is critical

TL;DR

This paper proves that in any closed Riemannian manifold, each point is critical for the distance function from some other point, but such points cannot always be connected by specific geodesic loops.

Contribution

It establishes the existence of critical points for the distance function in closed Riemannian manifolds and explores limitations of geodesic loops.

Findings

Every point in a closed Riemannian manifold is critical for some distance function.

Such critical points cannot always be connected by geodesic loops with midpoint at the point.

The result highlights geometric constraints on geodesic loops in Riemannian manifolds.

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 $p$.