From the side of differential geometry we are interested in connections, curvature and holonomy and their interaction with submanifolds. Submanifolds enters in the picture in several ways as the holonomy subbundles of a reduction of the holonomy group, as the orbits of the action of the holonomy group, etc. Since Elie Cartan’s work the relation between symmetric spaces and holonomy is well-known. An important role are played by both the holonomy group associated to the Levi-Civita connection and that of the normal connection of a submanifold.
Other holonomy groups are also the object of study e.g. holonomy of the Chern connection, conformal holonomy, etc. We apply the above concepts to the study of special kind of submanifolds, for example the so called helix submanifolds. Kahler submanifolds are also studied in the above framework enriched by Calabi’s Diastasis function. To study complex submanifolds of Hermitian symmetric spaces we take advantage of the two approaches to handle them: the classical Elie Cartan’s approach from Lie Theory and the Max Koecher’s Jordan algebra viewpoint. We are also interested in the uniformization problem for compact or algebraic varieties since under natural assumptions the universal covering of such varieties are Hermitian symmetric spaces.