Homogeneous Structures of Linear Type

Abstract

Among the different applications and possibilities of homogeneous structures, the translation of the algebraic classifications of the tensors (like those provided in the previous chapter) into geometric terms might be the most important one. For example, as we mentioned at the end of the previous chapter, naturally reductive pseudo-Riemannian manifolds are completely characterized by the existence of a homogeneous structure belonging to class$$\mathcal {S}_3$$. The full translation of the different classes is a major research project, of which there is much work to do. From that point of view, it is not surprising that the main bulk of the results in this line of investigation only involve the smallest classes of the classifications, mainly for two reasons.