  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.
