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., $T=\infty$, then the Euler characteristic $\chi (M)$ of M is non-negative. Under suitable assumptions on the solution of the normalized Ricci flow equation on $M\times [0,T)$, we prove one more theorem stating that the Hitchin-Thorpe type inequality $2\chi (M)\ge 3|\sigma (M)|$ 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.