The Einstein-Hilbert type action on foliated pseudo-Riemannian manifolds

TL;DR

This paper develops variation formulas for Einstein-Hilbert type actions on foliated pseudo-Riemannian manifolds, leading to new Euler-Lagrange equations with diverse solutions like twisted products and conformal submersions.

Contribution

It generalizes existing results on codimension-one foliations to arbitrary (co)dimension, providing new formulas and solutions for variational problems on foliated manifolds.

Findings

Derived variation formulas for mixed and extrinsic scalar curvatures.

Obtained Euler-Lagrange equations with multiple solutions.

Extended previous results to arbitrary (co)dimension foliations.

Abstract

We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a distribution. The obtained Euler-Lagrange equations admit a number of solutions, e.g., twisted products, conformal submersions and isoparametric foliations. The paper generalizes recent results about the actions on codimension-one foliations for the case of arbitrary (co)dimension.