The mixed Einstein-Hilbert action and extrinsic geometry of foliated manifolds

TL;DR

This paper develops variation formulas for extrinsic geometry on foliated manifolds, deriving Euler-Lagrange equations for the total mixed scalar curvature, an analogue of Einstein-Hilbert action, with applications to solutions like twisted products.

Contribution

It introduces new variation formulas for extrinsic geometry on foliated manifolds and derives Euler-Lagrange equations for the mixed scalar curvature functional.

Findings

Derived the directional derivative of the mixed scalar curvature functional.

Presented Euler-Lagrange equations in extrinsic and intrinsic forms.

Identified classes of solutions such as twisted products.

Abstract

We develop variation formulas for the quantities of extrinsic geometry for adapted variations of metrics on almost-product (e.g. foliated) Riemannian manifolds, and apply them to study the total mixed scalar curvature of a distribution -- analogue of the classical Einstein-Hilbert action. The mixed scalar curvature ${\rm S}_{\,\rm mix}$ is the averaged sectional curvature over all planes that contain vectors from both distributions of an almost-product structure and the variations we consider preserve orthogonality of the distributions. We derive the directional derivative $D J_{\,\rm mix}$ (of the total ${\rm S}_{\,\rm mix}$) for adapted variations of metrics on closed almost-product manifolds and foliations of arbitrary dimension. The obtained Euler-Lagrange equations are presented in two equiva\-lent forms: in terms of extrinsic geometry and intrinsically using the partial Ricci tensor. Certainly, these mixed field equations admit amount of solutions (e.g., twisted products).