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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 Ag generically contained in the Torelli locus obtained by Elisabetta Colombo, Paola Frediani, and Alessandro Ghigi [Internat. J. Math. 26 (2015), no. 1, 1550005] and A. Ghigi, P. Pirola, and S. Torelli (to appear on Communications in Contemporary Mathematics, https:// doi.org/10.1142/S0219199720500200). We get dim Y ≤ 2g − 1 if g is even, dim Y ≤ 2g if g is odd. We also study totally geodesic subvarieties Z of Ag generically contained in the hyperelliptic Torelli locus and we show that dim Z ≤ g + 1. [ABSTRACT FROM AUTHOR]

The article presents a study on intrinsic geometry of disk complex D(M,S). The disk complex has a natural tendency to form a curve complex C(S) which is not a quasi-isometric embedding since some disks may be close to the curve complex but are far in the disk complex. Subsurfaces holes may be formed and these paths in the disk complex must traverse into and out of these holes. The paths formed in the curve complex may skip over a hole by using a vertex to show the boundary of the subsurface.