On the mixed scalar curvature of almost multi-product manifolds

TL;DR

This paper investigates the mixed scalar curvature of Riemannian almost multi-product manifolds, deriving integral and variation formulas with applications to manifold splitting and Einstein-Hilbert action.

Contribution

It introduces new integral and variation formulas for mixed scalar curvature on these manifolds, extending results to pseudo-Riemannian cases and applications in geometric analysis.

Findings

Derived integral formulas for mixed scalar curvature

Established variation formulas related to Einstein-Hilbert action

Extended results to pseudo-Riemannian almost product manifolds

Abstract

A pseudo-Riemannian manifold endowed with $k>2$ orthogonal complementary distributions (called a Riemannian almost multi-product structure) appears in such topics as multiply warped products, the webs composed of several foliations, Dupin hypersurfaces and in stu\-dies of the curvature and Einstein equations. In this article, we consider the following two problems on the mixed scalar curvature of a Riemannian almost multi-product manifold with a linear connection: a) integral formulas and applications to splitting of manifolds, b) variation formulas and applications to the mixed Einstein-Hilbert action, and we generalize certain results on the mixed scalar curvature of pseudo-Riemannian almost product manifolds.