Prescribing the mixed scalar curvature of a foliated Riemann-Cartan manifold

TL;DR

This paper investigates how to prescribe the mixed scalar curvature in foliated Riemann-Cartan manifolds through conformal changes, reducing the problem to solving elliptic equations with multiple solutions under specific conditions.

Contribution

It introduces a method to prescribe mixed scalar curvature via conformal changes in foliated Riemann-Cartan manifolds, reducing the problem to elliptic equations with multiple solutions.

Findings

The problem reduces to solving a leafwise elliptic equation.

Under certain conditions, three stable solutions exist.

Only one solution corresponds to the Riemannian case.

Abstract

The mixed scalar curvature is one of the simplest curvature invariants of a foliated Riemannian manifold. We explore the problem of prescribing the mixed scalar curvature of a foliated Riemann-Cartan manifold by conformal change of the structure in tangent and normal to the leaves directions. Under certain geometrical assumptions and in two special cases: along a compact leaf and for a closed fibred manifold, we reduce the problem to solution of a leafwise elliptic equation, which has three stable solutions -- only one of them corresponds to the case of a foliated Riemannian manifold.