  In this paper we study totally geodesic subvarieties $Y \subset \mathsf{A}_g$
of the moduli space of principally polarized abelian varieties with respect to
the Siegel metric, for $g\geq 4$. We prove that if $Y$ is generically contained
in the Torelli locus, then $\dim Y \leq (7g -2)/3$.
