  We shall give a definition of the curvature operator for a family of weighted
Bergman spaces $\{\mathcal H_t\}$ associated to a smooth family of smoothly
bounded strongly pseudoconvex domains $\{D_t\}$. In order to study the boundary
term in the curvature operator, we shall introduce the notion of geodesic
curvature for the associated family of boundaries $\{\partial D_t\}$. As an
application, we get a variation formula for the norms of Bergman projections of
currents with compact support. A flatness criterion for $\{\mathcal H_t\}$ and
its applications to triviality of fibrations are also given in this paper.
