Generalized Gauss-Bonnet theorem
- Generalized Gauss-Bonnet theorem
In mathematics, the generalized-Gauss-Bonnet theorem presents the Euler characteristic of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature. It is a direct generalization of the Gauss-Bonnet theorem to higher dimensions.
Let "M" be a compact 2"n"-dimensional Riemannian manifold without boundary, and let be the curvature form of the Levi-Civita connection. This means that is an -valued 2-form on "M". So can be regarded as a skew-symmetric 2"n" × 2"n" matrix whose entries are 2-forms, so it is a matrix over the commutative ring . One may therefore take the Pfaffian of , , which turns out to be a 2"n"-form.
The generalized-Gauss-Bonnet theorem states that:where denotes the Euler characteristic of "M".
Further generalizations
As with the Gauss-Bonnet theorem, there are generalizations when "M" is a manifold with boundary.
ee also
*Chern-Weil homomorphism
*Pontryagin number
*Pontryagin class
References
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