Variations of the Mutual Curvature of Two Orthogonal Non-complementary Distributions

On a smooth manifold with distributions \(\mathcal {D}_1\) and \(\mathcal {D}_2\) having trivial intersection, we consider the integral of their mutual curvature, as a functional of Riemannian metrics that make the distributions orthogonal. The mutual curvature is defined as the sum of sectional curvatures of planes spanned by all pairs of vectors from an orthonormal basis, such that one vector of the pair belongs to \(\mathcal {D}_1\) and the second vector belongs to \(\mathcal {D}_2\). As such, it interpolates between the sectional curvature of a plane field (if both distributions are one-dimensional), and the mixed scalar curvature of a Riemannian almost product structure (if both distributions together span the tangent bundle). We derive Euler–Lagrange equations for the functional, formulated in terms of extrinsic geometry of distributions, i.e., their second fundamental forms and integrability tensors. We give examples of critical metrics for distributions defined on domains of Riemannian submersions, twisted products and f–K-contact manifolds.