Chern-Simons classes and the Ricci flow on 3-manifolds

TL;DR

This paper explores how Chern-Simons classes, which depend on geometry, behave under Ricci flow on 3-manifolds, revealing invariance in some cases and non-invariance in others, thus linking geometric structures to flow dynamics.

Contribution

It demonstrates the behavior of Chern-Simons classes under Ricci flow on specific 3-manifolds, highlighting cases of invariance and non-invariance, and deepening understanding of geometric invariants.

Findings

Chern-Simons class for the first Pontryagin class is invariant on warped products S^2×_f S^1 and S^1×_f S^2.

The same class is not invariant on a generalized Berger sphere.

The study links geometric invariants to Ricci flow behavior.

Abstract

In 1974, S.-S. Chern and J. Simons published a paper where they defined a new type of characteristic class - one that depends not just on the topology of a manifold but also on the geometry. The goal of this paper is to investigate what kinds of geometric information is contained in these classes by studying their behavior under the Ricci flow. In particular, it is shown that the Chern- Simons class corresponding to the first Pontryagin class is invariant under the Ricci flow on the warped products $S^2\times_f S^1$ and $S^1 \times_f S^2$ but that this class is not invariant under the Ricci flow on a generalized Berger sphere.