# mobius space

Contents

## Idea

The local model space for conformal geometry regarded as parabolic Cartan geometry.

geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group $GG$subgroup (monomorphism) $H↪GH \hookrightarrow G$quotient (“coset space”) $G/HG/H$Klein geometryCartan geometryCartan connection
examplesEuclidean group $Iso(d)Iso(d)$rotation group $O(d)O(d)$Cartesian space $ℝ d\mathbb{R}^d$Euclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group $Iso(d−1,1)Iso(d-1,1)$Lorentz group $O(d−1,1)O(d-1,1)$Minkowski spacetime $ℝ d−1,1\mathbb{R}^{d-1,1}$Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group $O(d−1,2)O(d-1,2)$$O(d-1,1)$anti de Sitter spacetime $AdS dAdS^d$AdS gravity
de Sitter group $O(d,1)O(d,1)$$O(d-1,1)$de Sitter spacetime $dS ddS^d$deSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group $O(d,t+1)O(d,t+1)$conformal parabolic subgroupMöbius space $S d,tS^{d,t}$conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group $GG$subgroup (monomorphism) $H↪GH \hookrightarrow G$quotient (“coset space”) $G/HG/H$super Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime $ℝ d−1,1|N\mathbb{R}^{d-1,1\vert N}$Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group $GG$2-monomorphism $H→GH \to G$homotopy quotient $G//HG//H$Klein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) $H→GH \to G$homotopy quotient $G//HG//H$ of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

## References

• 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 tướng 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

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