A Lifting principle of curves under exponential-type maps

Abstract: We introduce a novel framework for lifting smooth paths via the exponential map on semi-Riemannian manifolds, addressing the long-standing difficulties posed by its singularities. We prove that every smooth path -- up to a nondecreasing reparametrization -- can be (partially) lifted to a curve which is inextensible in the domain of definition of the exponential map. Under a natural and purely topological condition -- the so-called path-continuation property for the exponential map -- we also establish the existence of global lifts, leading to a general path-lifting theorem. This lifting approach yields new, alternative proofs of (generalizations of) a number of foundational results in semi-Riemannian geometry: the Hopf--Rinow theorem and Serre's classic theorem about multiplicity of connecting geodesics in the Riemannian case, as well as the Avez--Seifert theorem for globally hyperbolic spacetimes in Lorentzian geometry. More broadly, our results reveal the central role of the continuation property in obtaining geodesic connectivity across a wide range of semi-Riemannian geometries. This offers a unifying geometric principle that is complementary to the more traditional analytic, variational methods used to investigate geodesic connectedness, and provides new insight into the structure of geodesics, both on geodesically complete and non-complete manifolds. We also briefly point out how the lifting theory developed here can extend to more general flow-inducing maps on the tangent bundle other than the geodesic flow, suggesting broader geometric applicability beyond the exponential map.