Topological consequences of null-geodesic refocusing and applications to Z manifolds

Let $(M,h)$ be a connected, complete Riemannian manifold, x∈ M, and $l>0$. Then M is called a Zˣ manifold if all geodesics starting at x return to x, and it is called a Yˣₗ manifold if every unit-speed geodesic starting at x returns to x at time l. It is unknown whether there are Zˣ manifolds that are not Yˣₗ manifolds for any $l>0$. By the Bérard-Bergery theorem, any Yˣₗ manifold of dimension at least $2$ is compact with finite fundamental group. We prove the same result for Zˣ manifolds M for which all unit-speed geodesics starting at x return to x in uniformly bounded time. We also prove that any Zˣ manifold $(M,h)$ with h analytic is a Yˣₗ manifold for some $l>0$. We start by defining a class of globally hyperbolic spacetimes (called observer-refocusing) such that any Zˣ manifold is the Cauchy surface of some observer-refocusing spacetime. We then prove that under suitable conditions the Cauchy surfaces of observer-refocusing spacetimes are compact with finite fundamental group, and we show that analytic observer-refocusing spacetimes of dimension at least $3$ are strongly refocusing. We end by stating a contact-theoretic conjecture analogous to our results in Riemannian and Lorentzian geometry.