Refocusing of null-geodesics in Lorentz manifolds

We investigate weak and strong refocusing of light rays in a space-time and related concepts. A strongly causal space-time (Xn +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 Ylx-manifolds. A complete connected Riemannian manifold M is called a Ylx-manifold if there exist x ∈ M and l ∈ R+ 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 Ylx-manifolds. There we prove that some conclusions of the Berard-Bergery Theorem for Ylx-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.