Editing Chern–Gauss–Bonnet theorem (section)

===Four-dimensional manifolds===
In dimension <math>2n=4</math>, for a compact oriented manifold, we get

:<math>\chi(M) = \frac{1}{32\pi^2} \int_M \left( |\text{Riem}|^2 - 4 |\text{Ric}|^2 + R^2 \right) \, d\mu  </math>

where <math>\text{Riem}</math> is the full [[Riemann curvature tensor]], <math>\text{Ric}</math> is the [[Ricci curvature|Ricci curvature tensor]], and <math>R</math> is the [[scalar curvature]]. This is particularly important in [[general relativity]], where spacetime is viewed as a 4-dimensional manifold. 

In terms of the orthogonal [[Ricci decomposition]] of the Riemann curvature tensor, this formula can also be written as

:<math>\chi(M) = \frac{1}{8\pi^2} \int_M \left( \frac{1}{4}|W|^2 - \frac{1}{2} |Z|^2 + \frac{1}{24}R^2 \right) \, d\mu  </math>

where <math>W</math> is the [[Weyl tensor]] and <math>Z</math> is the traceless Ricci tensor.