| We will describe
recent results on totally geodesic submanifolds and Shimura
subvarieties of A_g contained in the Torelli locus T_g. We will
use the second fundamental form of the Torelli map to give an upper
bound on the dimension of totally geodesic submanifolds contained in
T_g, which depends on the gonality of the curve. We will then discuss some geometric properties of the second fundamental form of the Torelli map. We will finally describe some new examples of Shimura subvarieties in T_g obtained as non-abelian Galois coverings of P^1. These are results in collaboration with E. Colombo, A. Ghigi and M. Penegini. |