Conformal flatness of compact three-dimensional Cotton-parallel manifolds

TL;DR

This paper proves that all compact three-dimensional Cotton-parallel pseudo-Riemannian manifolds are conformally flat, establishing a key geometric property and classifying such manifolds.

Contribution

It demonstrates that compact three-dimensional Cotton-parallel manifolds must be conformally flat, resolving a geometric classification question.

Findings

Compact three-dimensional Cotton-parallel manifolds are conformally flat.

Noncompactness is necessary for non-flat Cotton-parallel manifolds.

Provides a classification result for three-dimensional pseudo-Riemannian manifolds.

Abstract

A three-dimensional pseudo-Riemannian manifold is called essentially conformally symmetric (ECS) if its Cotton tensor is parallel but nowhere-vanishing. In this note we prove that three-dimensional ECS manifolds must be noncompact or, equivalently, that every compact three-dimensional Cotton-parallel pseudo-Riemannian manifold must be conformally flat.