The Cotton tensor in Riemannian spacetimes

TL;DR

This work systematically develops the Cotton 2-form in Riemannian spacetimes, deriving it from Bianchi identities and performing a full irreducible curvature decomposition to count independent components. In three dimensions, the Weyl tensor vanishes identically, making the Cotton tensor the central gauge-invariant object governing conformal structure, with $C_\alpha$ conserved and expressible as a variational derivative of a Chern–Simons term. The authors classify the 3D Cotton tensor via its eigenvalues (Cotton–York tensor), presenting Euclidean (Class A,B,C) and Lorentzian (Petrov-like I, D, II, N, III, 0) types, and provide explicit examples across these classes, including conformally flat perfect-fluid solutions. They also connect Cotton to matter via energy-momentum forms in the DJT gravity framework and supply extensive appendices on conventions, CS variation, and computational tools. The results illuminate the role of the Cotton tensor in 3D conformal geometry and topologically massive gravity, with practical impact on exact solutions and classification schemes.

Abstract

Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension. We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components for the first time. Subsequently, we exhibit its characteristic properties and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einstein's field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw, and Templeton. For each class examples are given. Finally we investigate the relation between the Cotton tensor and the energy-momentum in Einstein's theory and derive a conformally flat perfect fluid solution of Einstein's field equations in three dimensions.