The Einstein-Hilbert type action on almost $k$-product manifolds

TL;DR

This paper extends the Einstein-Hilbert action to Riemannian manifolds with multiple orthogonal distributions, deriving the associated Euler-Lagrange equations and Einstein-like conditions for these complex structures.

Contribution

It introduces a new variational framework for mixed scalar curvature on almost $k$-product manifolds with $k>2$, deriving the corresponding Einstein equations.

Findings

Derived Euler-Lagrange equations for the action.

Presented Einstein equations for almost $k$-product structures.

Applicable to multiply warped products and foliations.

Abstract

A Riemannian manifold endowed with $k>2$ orthogonal complementary distributions (called here a Riemannian almost $k$-product structure) appears in such topics as multiply warped products, the webs composed of several foliations, and proper Dupin hypersurfaces of real space-forms. In the paper, we consider the mixed scalar curvature of such structure for $k>2$, derive Euler-Lagrange equations for the Einstein-Hilbert type action with respect to adapted variations of metric, and present them in a nice form of Einstein equation.