The Cotton tensor and Chern-Simons invariants in dimension $3$: an introduction

TL;DR

This survey introduces Chern-Simons invariants in 3-dimensional Riemannian geometry, explores their relation to the Cotton tensor, and discusses their connections to eta invariants and Selberg zeta functions.

Contribution

It provides a comprehensive overview of Chern-Simons invariants and the Cotton tensor, emphasizing their properties and interrelations in 3D geometry.

Findings

Cotton tensor is the first variation of Chern-Simons invariant.

Cotton tensor characterizes local conformal flatness in 3D.

Links between Chern-Simons invariants, eta invariant, and Selberg zeta function are discussed.

Abstract

Chern-Simons invariants of closed oriented Riemannian $3$-manifolds are introduced and studied from the basics. Their first-order variation is the Cotton tensor. The properties of the Cotton tensor: symmetry, conformal covariance, trace- and divergence-freedom, are recovered as corollaries of the Chern-Simons invariant. We prove that the Cotton tensor is the obstruction to local conformal flatness in dimension $3$. Finally we discuss the link between Chern-Simons invariants, the eta invariant, and the central value of the Selberg zeta function. This is a survey paper without original results, based on lecture notes for a doctoral course at the University of Bucharest, March 2014.