Variations of the total mixed scalar curvature of a distribution (Q1656640)

This paper addresses the general question, ``What are the best pseudo-Riemannian metrics on a smooth manifold endowed with a smooth distribution?'' If the pseudo-Riemannian metric \(g\) is nondegenerate on the distribution \({\mathcal D}\), then it defines an almost product structure, and hence a mixed scalar curvature function on the manifold. By integrating this curvature function over the manifold, one obtains the total mixed scalar curvature as a functional on the space of metrics. The authors find the variation formulas for total mixed scalar curvature under arbitrary variations of the metric as well as for variations which preserve volume or which preserve the metric on \({\mathcal D}\) and then derive the Euler-Lagrange equations that define the critical metrics for the total mixed scalar curvature functional. They conclude the paper with examples of critical metrics drawn from contact geometry, Riemannian geometry, foliations, etc.