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