Differential geometry Connections and curvature Concept
Curvature form
Summary
The curvature form of a connection on a principal bundle is a -form on the total space. Its values lie in the Lie algebra of the structure group. It measures the failure of the horizontal distribution of the connection to be integrable, and it vanishes exactly when the connection is flat. For connections on vector bundles it becomes a -form with values in endomorphisms, and for the Levi-Civita connection it recovers the Riemann curvature tensor. Its invariant polynomials give the characteristic forms of Chern–Weil theory.
Where this leads
15 pages build on this one.
- Index of the de Rham complex
- Gauge group
- Space of connections modulo gauge
- Uhlenbeck compactness theorem
- Atiyah–Bott theorem on Yang–Mills over Riemann surfaces
- Uhlenbeck's removable singularity theorem
- Yang–Mills functional
- Chern–Weil homomorphism
- Instanton
- Chern–Gauss–Bonnet theorem
- Yang–Mills equations
- Ambrose–Singer theorem
- Bianchi identities
- Moduli space of instantons
- Chern–Simons form
Further reading
About this page
| Tag | ZAJFW1S078 |
|---|---|
| URI | m2://ZAJFW1S078 |
| Kind | concept |
| Writing | W0 stub |
| MSC | 53C05 |
| Same as |
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Used by
- Gauge group
- Uhlenbeck compactness theorem
- Atiyah–Bott theorem on Yang–Mills over Riemann surfaces
- Uhlenbeck's removable singularity theorem
- Yang–Mills functional
- Chern–Weil homomorphism
- Instanton
- Chern–Gauss–Bonnet theorem
- Yang–Mills equations
- Ambrose–Singer theorem
- Bianchi identities
- Chern–Simons form