Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
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 .
For smooth manifolds
There is a generalization for an orbifold due to (Satake).
The Chern-Gauss-Bonnet theorem goes back to
- Shiing-shen Chern, A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds , Annals, 45 (1944), 747-752.
A classical textbook reference is chapter X of volume II of
Discussion is for instance in
- Denis Bell, The Gauss-Bonnet theorem for vector bundles (pdf)
The generalization to orbifolds is considered in
- Ichiro Satake, The Gauss-Bonnet Theorem for V-manifolds J. Math. Soc. Japan Volume 9, Number 4 (1957), 464-492. (ProjectEclid)
For an approach via 0|2-dimensional supersymmetric Euclidean field theory?, see
- Daniel Berwick-Evans, The Chern-Gauss-Bonnet Theorem via supersymmetric Euclidean field theories, Communications in Mathematical Physics, Volume 335 (2015), (arXiv:1310.5383).
Revised on October 20, 2016 08:46:29
by David Corfield