Path-lifting properties of the exponential map with applications to geodesics

TL;DR

This paper explores the path-lifting properties of exponential maps to establish new existence and multiplicity results for geodesics in affine and Lorentzian manifolds, generalizing classical theorems under weaker conditions.

Contribution

It extends the Hadamard-Cartan theorem to affine manifolds and introduces a Lorentzian version with relaxed assumptions, using path-lifting theory for broader geodesic results.

Findings

Generalized Hadamard-Cartan theorem for affine manifolds

New Lorentzian Hadamard-Cartan theorem with weaker assumptions

Described pseudoconvexity and disprisonment in terms of exponential maps

Abstract

We revisit certain path-lifting and path-continuation properties of abstract maps as described in the work of F. Browder and R. Rheindboldt in 1950-1960s, and apply their elegant theory to exponential maps. We obtain thereby a number of novel results of existence and multiplicity of geodesics joining any two points of a connected affine manifold, as well as ${\it causal}$ geodesics connecting any two causally related points on a Lorentzian manifold. These results include a generalization of the well-known Hadamard-Cartan theorem of Riemannian geometry to the affine manifold context, as well as a new version of the so-called Lorentzian Hadamard-Cartan theorem using weaker assumptions than global hyperbolicity and timelike 1-connectedness required in the extant version. We also include a general discription of ${\it pseudoconvexity}$ and ${\it disprisonment}$ of broad classes of geodesics in terms of suitable restrictions of the exponential map. The latter description sheds further light on the the relation between pseudoconvexity and disprisonment of a given such class on the one hand, and geodesic connectedness by members of that class on the other.