nLab
Cartan geometry

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

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          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Riemannian geometry

          Contents

          Idea

          Cartan geometry is geometry of spaces that are locally (infinitesimally, tangentially) like coset spaces G/HG/H, i.e. like Klein geometries. Intuitively, Cartan geometry studies the geometry of a manifold by ‘rolling without sliding’ the ‘model geometry’ G/HG/H along it. Hence Cartan geometry may be thought of as the globalization of the program of Klein geometry initiated in the Erlangen program.

          Cartan geometry subsumes many types of geometry, such as notably Riemannian geometry, conformal geometry, parabolic geometry and many more. As a Cartan geometry is defined by principal connection data (hence by cocycles in nonabelian differential cohomology) this means that it serves to express all these kinds of geometries in connection data.

          This is used notably in the first order formulation of gravity, which was the motivating example in the original text (Cartan 22). The physics literature tends to use the term “Cartan moving frame method” instead of “Cartan geometry”.

          Definition

          A Cartan geometry is a space equipped with a Cartan connection. See there for more.

          Examples

          geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
          differential geometryLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
          examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
          Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
          anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
          de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter 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 GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
          examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
          super anti de Sitter groupsuper anti de Sitter spacetime
          higher differential geometrysmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
          cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
          examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

          References

          The original articles are

          • Élie Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces á torsion, C. R. Acad. Sci. 174

            , 593-595 (1922).

          • Élie Cartan, Comptes rendus hebdomadaires des séances de l’Académie des sciences, 174, 437-439, 593-595, 734-737, 857-860, 1104-1107 (January 1922).

          • Élie Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie). Annales scientifiques de l’École Normale Supérieure, Sér. 3, 40 (1923), p. 325-412 (NUMDAM)

          Textbook accounts are in

          • R. Sharpe, Differential Geometry – Cartan’s Generalization of Klein’s Erlangen program Springer (1997)

          • Andreas Cap, Jan Slovák, chapter 1 of Parabolic Geometries I – Background and General Theory, AMS 2009 (publisher)

          • ps

          For more see at Cartan connection – References.

          Discussion in modal homotopy type theory is in

          See also

          Last revised on August 2, 2018 at 11:41:39. See the history of this page for a list of all contributions to it.