Editing Chern–Gauss–Bonnet theorem (section)

== Special cases ==

===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.

===Even-dimensional hypersurfaces===
For a compact, even-dimensional [[hypersurface]] <math> M </math> in <math> \mathbb{R}^{n+1} </math> we get<ref>{{cite book|last1=Guillemin |first1=V. |last2=Pollack |first2=A.| title=Differential topology | location=New York, NY |publisher=Prentice-Hall |year=1974 |page=196| isbn=978-0-13-212605-2}}</ref>

:<math>\int_M K\,dV = \frac{1}{2}\gamma_n\,\chi(M) </math>
where <math> dV </math> is the [[volume element]] of the hypersurface, <math>K</math> is the [[Jacobian matrix and determinant|Jacobian determinant]] of the [[Gauss map#Generalizations|Gauss map]], and <math>\gamma_n</math> is the [[N-sphere#Volume_and_surface_area|surface area of the unit n-sphere]].

=== Gauss–Bonnet theorem ===
{{Main|Gauss–Bonnet theorem}}
The [[Gauss–Bonnet theorem]] is a special case when <math> M </math> is a 2-dimensional manifold. It arises as the special case where the topological index is defined in terms of [[Betti number]]s and the analytical index is defined in terms of the Gauss–Bonnet integrand.

As with the two-dimensional Gauss–Bonnet theorem, there are generalizations when <math> M </math> is a [[manifold|manifold with boundary]].