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Möbius space
Contents
Contents
Idea
The local model space for conformal geometry regarded as parabolic Cartan geometry.
geometric context | gauge group | stabilizer subgroup | local model space | local geometry | global geometry | differential cohomology | first order formulation of gravity |
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differential geometry | Lie group/algebraic group | subgroup (monomorphism) | quotient (“coset space”) | Klein geometry | Cartan geometry | Cartan connection | |
examples | Euclidean group | rotation group | Cartesian space | Euclidean geometry | Riemannian geometry | affine connection | Euclidean gravity |
| Poincaré group | Lorentz group | Minkowski spacetime | Lorentzian geometry | pseudo-Riemannian geometry | spin connection | Einstein gravity |
| anti de Sitter group | | anti de Sitter spacetime | | | | AdS gravity |
| de Sitter group | | de Sitter spacetime | | | | deSitter gravity |
| linear algebraic group | parabolic subgroup/Borel subgroup | flag variety | parabolic geometry | | | |
| conformal group | conformal parabolic subgroup | Möbius space | | conformal geometry | conformal connection | conformal gravity |
supergeometry | super Lie group | subgroup (monomorphism) | quotient (“coset space”) | super Klein geometry | super Cartan geometry | Cartan superconnection | |
examples | super Poincaré group | spin group | super Minkowski spacetime | Lorentzian supergeometry | supergeometry | superconnection | supergravity |
| super anti de Sitter group | | super anti de Sitter spacetime | | | | |
higher differential geometry | smooth 2-group | 2-monomorphism | homotopy quotient | Klein 2-geometry | Cartan 2-geometry | | |
| cohesive ∞-group | ∞-monomorphism (i.e. any homomorphism) | homotopy quotient of ∞-action | higher Klein geometry | higher Cartan geometry | higher Cartan connection | |
examples | | | extended super Minkowski spacetime | | extended supergeometry | | higher supergravity: type II, heterotic, 11d |
References
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R. Sulanke, Differential geometry of the Möbius space I (pdf)
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Felipe Leitner, part 1, section 6 of Applications of Cartan and Tractor Calculus to Conformal and CR-Geometry, 2007 (pdf)
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Pierre Anglès, section 2.2.1.2 of Conformal Groups in Geometry and Spin Structures, Progress in Mathematical Physics 2008
Last revised on March 25, 2015 at 17:28:13.
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