noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
The Chern-Gauss-Bonnet theorem gives a formula that computes the Euler characteristic of an even-dimensional smooth manifold as the integration of a curvature characteristic form of the Levi-Civita connection on its tangent bundle.
For surfaces the theorem simplifies and in this simpler version is the older Gauss-Bonnet theorem .
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Let $X$ be a compact smooth manifold of even dimension $dim X = 2k$. Write $\chi(X)$ for its Euler characteristic.
For $\nabla$ any Levi-Civita connection on its tangent bundle, write $F_\nabla$ for its curvature 2-form, valued in the orthogonal Lie algebra $\mathfrak{so}(2k)$ and $Pf(F_\nabla)$ for its Pfaffian $2k$-form.
Then
There is a generalization for $X$ an orbifold due to (Satake).
The Chern-Gauss-Bonnet theorem goes back to
A classical textbook reference is chapter X of volume II of
Discussion is for instance in
Expositions include
Liviu I. Nicolaescu, The many faces of the Gauss-Bonnet theorem (pdf)
Denis Bell, The Gauss-Bonnet theorem (pdf)
Chenchang Zhu, The Gauss-Bonnet theorem and its applications (pdf)
The generalization to orbifolds is considered in
For an approach via 0|2-dimensional supersymmetric Euclidean field theory?, see