The Euler characteristic and signature of four-dimensional closed manifolds and the normalized Ricci flow equation

Abstract

Let M be a closed manifold of dimension four, and let [0, T) be the maximal time interval for the normalized Ricci flow equation. We prove that, if the normalized Ricci flow equation has a solution on the non-negative real line, i.e., T = infinity, then the Euler characteristic chi(M) of M is non-negative. Under suitable assumptions on the solution of the normalized Ricci flow equation on M x [0, T), we prove one more theorem stating that the Hitchin-Thorpe type inequality 2 chi(M) >= 3 vertical bar sigma(M)vertical bar holds between the Euler characteristic chi(M) and the signature sigma(M) of M. To obtain these results, we utilize the Riccati comparison theorem. In this respect, we present a new application of the Riccati comparison theorem.