Chern-Simons classes and the Ricci flow on 3-manifolds (Q2839317)

For a 3-manifold \(M\) with a Riemannan metric, the Chern-Simons form \({TP}(\omega)\), as the transgression for the first Pontryagin form, defines a cohomology class in \(H^3(M, \mathbb{R}/\mathbb{Z})\). A priori, the Chern-Simons class depends on the Riemannian metric and its associated Riemannian connection \(\omega\). The cohomology class is shown to be invariant under a conformal change of the metric by \textit{S. Chern} and \textit{J. Simons} [Ann. Math. (2) 99, 48--69 (1974; Zbl 0283.53036)].NEWLINENEWLINEThis paper studies the behavior of the Chern-Simons class under the Ricci flow of the metric. It is shown that in general it is not invariant under the Ricci flow. The example used is the Berger sphere.