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Cartan geometry

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Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

Cartan geometry is a common generalization of Riemannian geometry, conformal geometry? and Klein geometry, which generalizes the linear tangent spaces of the former to the more general homogeneous spaces of the latter.

Intuitively, Cartan geometry studies the geometry of a manifold by ‘rolling’ another manifold, the ‘model geometry’ on it. The model geometry may be any Klein geometry.

Definition

See Cartan connection.

local modelglobal geometry
Klein geometryCartan geometry
Klein 2-geometryCartan 2-geometry
higher Klein geometryhigher Cartan geometry

References

A standard textbook is

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

See also

Revised on August 31, 2011 17:13:37 by Urs Schreiber (131.211.238.114)
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