  In this paper, we investigate the geometry of the moduli space of curves by
using the curvature properties of direct image sheaves of vector bundles. We
show that the moduli space $(M_g, \omega_{WP})$ of curves with genus $g>1$ has
dual-Nakano negative and semi-Nakano-negative curvature, and in particular, it
has non-positive Riemannain curvature operator and also non-positive complex
sectional 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.
