Gauss-Bonnet Theorem / Invariant

In General > s.a. euler class and euler number.
* Idea: (a.k.a. Gauss-Bonnet-Chern theorem) An important result in differential geometry relating the geometry of a closed surface to its topology.
* Conditions: It applies only for Riemannian (positive-definite) metrics.
* Relationships: It can be considered as a special case of the Atiyah-Singer index theorem.
$ 2D version: If M is a compact two-dimensional Riemannian manifold with Gaussian curvature K, and K_g is the the geodesic curvature of the boundary ∂M

2π χ(M) = ∑_i (π − α_i) + ∫_∂M K_g ds + ∫_M K dA ,

where α_i are the internal angles of ∂M (i = 1, ..., n) (∫ K dA can be a surface integral; In 2D, the Einstein tensor is identically zero).
$ 4D version: If R_abcd is the Riemann curvature tensor of a compact oriented 4D manifold,

χ(M) = ∫_M d⁴v (R_abcd R^abcd − 4 R_ab R^ab + R²)

(it vanishes for M homeomorphic to \(\mathbb R\)⁴).
* Even-dimensional, orientable manifold: If e(F) is the Euler class, ∫_M e(F) = χ(M).
* Complex n-dimensional manifold: (They are all even-dimensional and orientable) ∫_M c_n(F) = χ(M).

References > s.a. Gauss-Bonnet Gravity.
@ General: Labbi Sigma(07)-a0709-proc [Gauss-Bonnet invariants in arbitrary dimensions and applications]; Szczęsny et al IJGMP(09)-a0810 [new elementary proof]; Li a1111 [overview].
@ Lorentzian: in Hartle & Sorkin GRG(81).
@ Generalizations: Alty JMP(95) [with boundary and arbitrary signature]; Bao & Chern AM(96) [for Finsler spaces]; Arnlind et al a1001 [for matrix analogues of embedded surfaces]; Zhao JGP(15)-a1408 [for a general connection].
> Online resources: see MathWorld page; Wikipedia page.
> In higher dimensions: see Wikipedia page.

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