Away from the hyperelliptic locus, the moduli of curves immerses in the moduli of principally polarized abelian varieties. The ambient space has a riemannian metric, so one can ask about the second fundamental form, the first-order deviation of the submanifold from being totally geodesic. What is this second fundamental form? Is anything known about it?
I think one could translate this into the language of algebraic geometry by using the Serre-Tate formal coordinates, which exist at each point of $A_g$. With respect to these coordinates, $M_g$ is not linear; what is its quadratic approximation? One could interpret this as a version of the Schottky problem, which suggests that existing solutions to it might be applicable.