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.



What is the second fundamental form of moduli space?
The following papers might be useful:


$(1)$ E. Colombo- G. Pirola- A. Tortora


"Hodge-Gaussian maps"


Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), no. 1, 125–146. 


$(2)$ E. Colombo - P. Frediani


"Siegel metric and curvature of the moduli space of curves"


Trans. Amer. Math. Soc. 362 (2010), no. 3, 1231–1246.


