On Conjugate Points and Geodesic Loops in a Complete Riemannian Manifold

摘要

A well-known lemma in Riemannian geometry by Klingenberg says that if x(0) is a minimum point of the distance function d(p, .) 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 Fq (.) = d(p, .) + d(q, .) 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, .) 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 injectivity radius estimate would hold under weaker conditions.