Reducible Riemannian manifolds with conformal product structures

TL;DR

This paper classifies reducible Riemannian manifolds with conformal product structures, showing they are either conformally flat or locally isometric to a specific type of triple product manifold under certain conditions.

Contribution

It provides a classification of reducible Riemannian manifolds with conformal product structures, identifying conditions under which they are conformally flat or triple products.

Findings

Manifolds are conformally flat or triple products under technical assumptions.

Triple product manifolds are of the form $(M,g)$ with $g=e^{2f}g_1+g_2+g_3$.

The conformal factor $f$ depends on $M_1 imes M_2$.

Abstract

We study conformal product structures on compact reducible Riemannian manifolds, and show that under a suitable technical assumption, the underlying Riemannian mani\-folds are either conformally flat, or triple products, \emph{i.e.} locally isometric to Riemannian manifolds of the form $(M,g)$ with $M=M_1\times M_2\times M_3$ and $g=e^{2f}g_1+g_2+g_3$, where $g_i$ is a Riemannian metric on $M_i$, for $i\in\{1,2,3\}$, and $f\in C^\infty(M_1\times M_2)$.