Gauss-Bonnet
Theorem| Gauss-Bonnet
Theorem |
In General
* Conditions: Applies only for Riemannian (positive-definite) metrics.
* Relationships: Can
be considered as a special case of the Atiyah-Singer index
theorem.
$ 2D manifold with boundary:
2
(M)
= 
i ( –
i)
+ 
bdry
of M Kg ds + M K dA ,
where i are
the internal angles of 
M (i =
1, ..., n), Kg is
the geodesic curvature of M,
and K is the Gaussian
curvature of M.
(
K dA
can be a surface integral; In 2D, the Einstein tensor is identically zero.)
$ 4D manifold:
(M)
= M d4v (Rabcd Rabcd
– 4 Rab Rab + R2)
(vanishes for M homeomorphic to R4).
* Even-dimensional, orientable
manifold:
If e(F) is the Euler class, M e(F)
= (M).
* Complex n-dimensional
manifold:
(They are all even-dimensional and orientable) M cn(F)
= (M).
References > s.a. Gauss-Bonnet
Gravity.
@ General: Labbi a0709-in [Gauss-Bonnet invariants in arbitrary dimensions
and applications].
@ Lorentzian: in Hartle & Sorkin GRG(81).
@ Generalizations: Alty JMP(95)
[with boundary and arbitrary signature]; Bao & Chern AM(96)
[for Finsler spaces].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
27 jan 2008