Spaces of Geodesics: Products, Coverings, Connectedness

TL;DR

This paper explores the structure of geodesic spaces on manifolds, providing conditions for their manifold structure, analyzing coverings, and characterizing geodesic connectedness, with specific results on Hadamard manifolds.

Contribution

It offers new criteria for when the space of geodesics forms a manifold and relates geodesic spaces of coverings to base manifolds, advancing understanding of geodesic topology.

Findings

Product manifolds can have geodesic spaces that are manifolds under certain conditions

The space of geodesics of an n-dimensional Hadamard manifold matches that of Euclidean space

Conditions for geodesic connectedness are linked to the topology of the geodesic space

Abstract

We continue our study of the space of geodesics of a manifold with linear connection. We obtain sufficient conditions for a product to have a space of geodesics which is a manifold. We investigate the relationship of the space of geodesics of a covering manifold to that of the base space. We obtain sufficient conditions for a space to be geodesically connected in terms of the topology of its space of geodesics. We find the space of geodesics of an $n$-dimensional Hadamard manifold is the same as that of $\R^n$.