We investigate weak and strong refocusing of light rays in a space-time and related concepts. A strongly causal space-time (X^{n +1}, g) is strongly refocusing at x ∈ X if there is a point y ≠ x such that all null-geodesics through y pass through x. A space-time is strongly refocusing if it is strongly refocusing at some point.

Robert Low introduced three definitions of (weak) refocusing. We prove that these definitions are indeed equivalent. Following a sketch provided by Low, we give a thorough proof of his statement that a strongly causal non-refocusing space-time is homeomorphic to its sky space.

A strongly refocusing space-time is refocusing. The converse is unknown. We construct examples of space-times which are refocusing, but not strongly so, at a particular point. These space-times are strongly refocusing at other points. The geometrization conjecture proved by Perelman implies that a globally hyperbolic refocusing space-time of dimension ≤ 4 admits a strongly refocusing Lorentz metric.

We show that the set of points at which a strongly causal space-time is refocusing is closed. We prove that a Lorentz covering space of a strongly causal refocusing space-time is a strongly causal refocusing space-time. This generalizes the result of Chernov and Rudyak for globally hyperbolic space-times.

We compare refocusing and strong refocusing with their Riemannian analogues, Y˜^x- and Y_l^x-manifolds. A complete connected Riemannian manifold M is called a Y_l^x-manifold if there exist x ∈ M and l ∈ [special characters omitted] such that all unit speed geodesics starting at x at time 0 return to x at time l. In our work with Chernov and Sadykov we introduce Y˜^x-manifolds that generalize Y_l^x-manifolds. There we prove that some conclusions of the Bérard-Bergery Theorem for Y_l^x-manifolds hold in fact for Y˜^x-manifolds. This result is discussed in this thesis.

Following the sketch of Chernov we provide the thorough proof of the statement in their paper with Rudyak that a timelike curve in a globally hyperbolic space-time can be perturbed so that it is transverse to a null-cone and avoids the singular and multiple points of the null-cone. We investigate a possible generalization.