  We examine the total mixed scalar curvature of a fixed distribution as a
functional of a pseudo-Riemannian metric. We develop variational formulas for
quantities of extrinsic geometry of the distribution to find the critical
points of this action. Together with the arbitrary variations of the metric, we
consider also variations that preserve the volume of the manifold or partially
preserve the metric (e.g. on the distribution). For each of those cases we
obtain the Euler-Lagrange equation and its several solutions. The examples of
critical metrics that we find are related to various fields of geometry such as
contact and 3-Sasakian manifolds, geodesic Riemannian flows and codimension-one
foliations of interesting geometric properties (e.g. totally umbilical,
minimal).
