- Part I. General theory of Lorentzian G-manifold.
- A.The orbit space M/G of a Lorentzian manifold M with an isometry group G
- B. Structure of a Lorentzian G-manifold in a neighborhood of a closed orbit P
- C. Lorentzian G-manifolds with semisimple group G.
- D. Compact Lorentzian G-manifolds.

- Part II. Lorentzian manifolds with large isometry group.
- A. Homogeneous Lorentzian manifolds.
- B. Lorentzian G-manifolds with isometry group G of big dimension.
- C. Cohomogeneity 1 Lorentzian manifolds.

- Part III. Lorentzian spin manifolds with many Killing spinors.

- Lorentzian manifolds with large isometry groups
- Homogeneity in Lorentzian geometry
- Curvature homogeneous Lorentzian metrics in low dimensions

In this brief course of Lectures on Lorentzian Geometry we are going to review
the main ideas involved in the formulation and proof of the so called "Rieman-
nian and Lorentzian Splitting Theorems". We will put special emphasis on the
similarities and differences in the arguments when passing from the Riemannian
to the Lorentzian case, and the way in which these obstacles are overcome in
the proofs.

We will begin by establishing the Riemannian Splitting Theorem by Cheeger
and Gromoll, including some comments about the initial motivation and its
precedents. Next, we will give some basic notions and previous results useful
for the proof of the theorem. Then, we will outline the main ideas of the
original proof due to Cheeger and Gromoll. We will also study an alternative
proof provided by Eschenburg and Heintze, which minimizes the use of elliptic
theory, and thus, becomes very useful in the Lorentzian version of the theorem.

Next, we will consider the Lorentizian case. After recalling some basic no-
tions and results of this geometry, we will establish the Lorentzian Splitting
Theorem, including some comments about the main hits in the history of its
proof. For the proof of this result, which will be described with certain detail,
we will essentially follow the approach by Galloway in [J. Diff. Geom. 29 (1989),
373-387]. Finally, we will conclude the course by analyzing some open problems
related to these theorems.

In General Relativity, spacetime is modeled as a Lorentzian manifold and light rays are modeled as lightlike geodesics. If there are two or more future-oriented light rays from a timelike curve (worldline of a light source) to a point (observation event), one speaks of multiple imaging by the gravitational lens effect. This means that an observer at the observation event sees two or more images of the light source. In these lectures I demonstrate how a variational principle (Fermat's principle) can be used for investigating the gravitational lens effect. Among other things, I will discuss the notion of conjugate points and the notion of cut points and I will outline their relevance in view of gravitational lensing.