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In this paper, we investigate the geometry of the moduli space of curves byusing the curvature properties of direct image sheaves of vector bundles. Weshow that the moduli space $(M_g, \omega_{WP})$ of curves with genus $g>1$ hasdual-Nakano negative and semi-Nakano-negative curvature, and in particular, ithas non-positive Riemannain curvature operator and also non-positive complexsectional curvature. As applications, we prove that any submanifold in $M_g$which is totally geodesic in $A_g$ with finite volume must be a ball quotient.

Curvatures of moduli space of curves and applications