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

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