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} d*s* + ∫_{M} *K* d*A* ,

where *α*_{i} are
the internal angles of ∂*M* (*i* =
1, ..., *n*)
(∫ *K* d*A* 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} d^{4}*v* (*R*_{abcd} *R*^{abcd}
– 4 *R*_{ab} *R*^{ab} + *R*^{2})

(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) ∫

**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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send feedback and suggestions to bombelli at olemiss.edu – modified 26
sep 2016