Euler Characteristic

Question

Can someone provide the exact formula for the Euler Characteristic in 4D? I know it goes as $\epsilon^{\mu\nu\rho\lambda}R_{\mu\nu}^{\alpha\beta}R_{\rho\lambda}^{\gamma\delta}\epsilon_{\alpha\beta\gamma\delta}$, but I am not sure about the exact numerical factor that tags along. Can you please also cite your source, I have been unable to find one.

Answer 1

You are asking abou the Chern-Gauss-Bonnet Theorem. The Euler characteristic in 4D is $$ \xi(M)=\frac{1}{32\pi^2}\int_M (|\text{Riem}|^2-4|\text{Ric}|^2+R^2)\,d\mu $$ where Riem is the full Riemann curvature tensor, Ric is the Ricci tensor and $R$ is the scalar curvature.