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On a smooth manifold with distributions ${\cal D}_1$ and ${\cal D}_2$ havingtrivial intersection, we consider the integral of their mutual curvature, as afunctional of Riemannian metrics that make the distributions orthogonal. Themutual curvature is defined as the sum of sectional curvatures of planesspanned by all pairs of vectors from an orthonormal basis, such that one vectorof the pair belongs to ${\cal D}_1$ and the second vector belongs to ${\calD}_2$. As such, it interpolates between the sectional curvature of a planefield (if both distributions are one-dimensional), and the mixed scalarcurvature of a Riemannian almost product structure (if both distributionstogether span the tangent bundle). We derive Euler-Lagrange equations for thefunctional, formulated in terms of extrinsic geometry of distributions, i.e.,their second fundamental forms and integrability tensors. We give examples ofcritical metrics for distributions defined on domains of Riemanniansubmersions, twisted products and $f$-$K$-contact manifolds.

Variations of the mutual curvature of two orthogonal non-complementary distributions