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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.

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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.

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