Euclidean geometry is the local model of Riemannian geometry. Klein geometry, on the other hand, is a homogeneous space, meaning its symmetry group acts transitively on the entire space. Combining these two perspectives, Cartan proposes a geometry where a manifold is modeled on a Klein space, rather than necessarily on Euclidean space. First, the concept of a fibered space equipped with a "standard" connection will be reviewed. Then, starting from this, a Cartan connection will be constructed to provide a formal definition of Cartan geometry and clarify the information it encodes. This involves selecting a model Klein space and attaching it to each point of our manifold, effectively defining a global section on a larger fiber bundle. Once this definition is established, we will look at examples of how Cartan geometry corresponds to different geometric structures. This generalization offers powerful tools for studying compact pseudo-Riemannian manifolds, particularly in the case of a flat, compact Lorentzian manifold, where the model is the famous Einstein space. Some contemporary results related to this will also be discussed.
