Let B be a smooth compact orientable surface without boundary and with χ(B)<0. We examine two types of fiber bundles M over B with fiber F. The first is a principle fiber bundle with a two-torus fiber and the second is an S\sp2 fiber bundle with an SO(3) group action. In each case, the tangent space of the bundle can be decomposed into a vertical space, those vectors tangent to fibers, and a horizontal space complementary to the vertical space and invariant under the group action. The bundle can be given a metric that is the direct sum of metrics on the vertical and horizontal spaces. Additionally, with this metric, M, is locally isometric to a product space B×F with metric g\sbb+g\sbf. Here g\sbb is any fixed metric on the base, g\sbf is a constant curvature metric on the fiber invariant under the action of the group. We can obtain a new metric on M by scaling the horizontal component of the original by e\sp2u and the vertical component by f\sp2, where u and f are smooth functions on the base. We put certain constraints on u and f and consider the family of all such variations. In this thesis, we show, using nonlinear elliptic estimates, that among these metrics there is one for which the integral of the norm of the Ricci curvature tensor squared, ∫\sbM|Ric|\sp2dV, is minimized.