On conjugate points and geodesic loops in a complete Riemannian manifold

Abstract

A well-known Lemma in Riemannian geometry by Klingenberg says that if $x_0$ is a minimum point of the distance function $d(p,\cdot)$ to $p$ in the cut locus $C_p$ of $p$, then either there is a minimal geodesic from $p$ to $x_0$ along which they are conjugate, or there is a geodesic loop at $p$ that smoothly goes through $x_0$. In this paper, we prove that: for any point $q$ and any local minimum point $x_0$ of $F_q(\cdot)=d(p,\cdot)+d(q,\cdot)$ in $C_p$, either $x_0$ is conjugate to $p$ along each minimal geodesic connecting them, or there is a geodesic from $p$ to $q$ passing through $x_0$. In particular, for any local minimum point $x_0$ of $d(p,\cdot)$ in $C_p$, either $p$ and $x_0$ are conjugate along every minimal geodesic from $p$ to $x_0$, or there is a geodesic loop at $p$ that smoothly goes through $x_0$. Earlier results based on injective radius estimate would hold under weaker conditions.