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

We review, with complete proofs, the theory of Chern–Simons invariants for oriented Riemannian $3$-manifolds. The Cotton tensor is the first-order variation of the Chern–Simons invariant. We deduce that it is conformally invariant, and trace- and divergence-free, from the corresponding properties of the Chern–Simons invariant. Moreover, the Cotton tensor vanishes if and only if the metric is locally conformally flat. In the last part of the paper we survey the link of Chern–Simons invariants with the eta invariant and with the central value of the Selberg zeta function of odd type.