The Einstein{Hilbert Type Action on Pseudo-Riemannian Almost-Product Manifolds

摘要

We develop variation formulas for the quantities of extrinsic geometryof almost-product pseudo-Riemannian manifolds, and we consider the variationsof metric preserving orthogonality of the distributions. These formulasare applied to study the Einstein–Hilbert type actions for the mixed scalarcurvature and the extrinsic scalar curvature of a distribution. The Euler–Lagrange equations for these variations are derived in full generality and inseveral particular cases (foliations that are integrable plane fields, conformalsubmersions, etc.). The obtained Euler–Lagrange equations generalize theresults for codimension-one foliations to the case of arbitrary codimension,and admit a number of solutions, e.g., twisted products and isoparametricfoliations.