On the geometry of the second fundamental form of the Torelli map

In this paper we give a geometric interpretation of the second fundamental form of the period map of curves and we use it to improve the upper bounds on the dimension of a totally geodesic subvariety Y of A_g generically contained in the Torelli locus obtained in [3], [7]. We get dim Y < 2g if g is even, dim Y < 2g+1 if g is odd. We also study totally geodesic subvarieties Z of A_g generically contained in the hyperelliptic Torelli locus and we show that dim Z < g+2.