## Differential Geometry: Cartan's Generalization of Klein's Erlangen ProgramCartan geometries were the first examples of connections on a principal bundle. They seem to be almost unknown these days, in spite of the great beauty and conceptual power they confer on geometry. The aim of the present book is to fill the gap in the literature on differential geometry by the missing notion of Cartan connections. Although the author had in mind a book accessible to graduate students, potential readers would also include working differential geometers who would like to know more about what Cartan did, which was to give a notion of "espaces généralisés" (= Cartan geometries) generalizing homogeneous spaces (= Klein geometries) in the same way that Riemannian geometry generalizes Euclidean geometry. In addition, physicists will be interested to see the fully satisfying way in which their gauge theory can be truly regarded as geometry. |

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### Contents

In the Ashes of the Ether Differential Topology | 1 |

1 Smooth Manifolds | 3 |

2 Submanifolds | 17 |

3 Fiber Bundles | 28 |

4 Tangent Vectors Bundles and Fields | 39 |

5 Differential Forms | 52 |

Looking for the Forest in the Leaves Foliations | 65 |

1 Integral Curves | 66 |

7 Symmetric Spaces | 225 |

Riemannian Geometry | 227 |

1 The Model Euclidean Space | 228 |

2 Euclidean and Riemannian Geometry | 234 |

3 The Equivalence Problem for Riemannian Metries | 237 |

4 Riemannian Space Forms | 241 |

5 Subgeometry of a Riemannian Geometry | 244 |

6 Isoparametric Submanifolds | 259 |

2 Distributions | 74 |

3 Integrability Conditions | 75 |

4 The Frobenius Theorem | 76 |

5 The Frobenius Theorem in Terms of Differential Forms | 79 |

6 Foliations | 81 |

7 Leaf Holonomy | 83 |

8 Simple Foliations | 90 |

The Fundamental Theorem of Calculus | 95 |

1 The MaurerCartan Form | 96 |

2 Lie Algebras | 101 |

3 Structural Equation | 108 |

4 Adjoint Action | 110 |

5 The Darboux Derivative | 115 |

Local Version | 116 |

Global Version | 118 |

8 Monodromy and Completeness | 128 |

Shapes Fantastic Klein Geometries | 137 |

1 Examples of Planar Klein Geometries | 140 |

Characteri2ation and Reduction | 144 |

3 Klein Geometries | 150 |

4 A Fundamental Property | 160 |

5 The Tangent Bundle of a Klein Geometry | 162 |

6 The Meteor Tracking Problem | 164 |

7 The Gauge View of Klein Geometries | 166 |

Shapes High Fantastical Cartan Geometries | 171 |

1 The Base Definition of Cartan Geometries | 173 |

2 The Principal Bundle Hidden in a Cartan Geometry | 178 |

3 The Bundle Definition of a Cartan Geometry | 184 |

4 Development Geometric Orientation and Holonomy | 202 |

5 Flat Cartan Geometries and Uniformization | 211 |

6 Cartan Space Forms | 217 |

Möbius Geometry | 265 |

1 The Möbius and Weyl Models | 267 |

2 Möbius and Weyl Geometries | 277 |

3 Equivalence Problems for a Conformal Metric | 284 |

4 Submanifolds of Möbius Geometry | 294 |

5 Immersed Curves | 318 |

6 Immersed Surfaces | 323 |

Projective Geometry | 331 |

1 The Projective Model | 332 |

2 Projective Cartan Geometries | 338 |

3 The Geometry of Geodesics | 343 |

4 The Projective Connection in a Riemannian Geometry | 349 |

5 A Brief Tour of Projective Geometry | 353 |

Ehresmann Connections | 357 |

1 The Geometric Origin of Ehresmann Connections | 358 |

2 The Reductive Case | 362 |

3 Ehresmann Connections Generalize Cartan Connections | 365 |

4 Covariant Derivative | 371 |

Rolling Without Slipping or Twisting | 375 |

1 Rolling Maps | 376 |

2 The Existence and Uniqueness of Rolling Maps | 378 |

3 Relation to LeviCivita and Normal Connections | 382 |

4 Transitivity of Rolling Without Slipping or Twisting | 388 |

Classification of OneDimensional Effective Klein Pairs | 391 |

Differential Operators Obtained from Symmetry | 397 |

2 Operators on Riemannian Surfaces | 400 |

Characterization of Principal Bundles | 407 |

411 | |

417 | |

### Other editions - View all

Differential Geometry: Cartan's Generalization of Klein's Erlangen Program R.W. Sharpe No preview available - 2000 |

### Common terms and phrases

action algebra g atlas canonical Cartan connection Cartan geometry chart commutative compact component condition conformal constant coordinate system Corollary corresponding covariant derivative curvature function Darboux derivative defined Definition denote determined diagram diffeomorphism differential dimension Ehresmann connection equivalent Euclidean space example fiber flat foliation follows G bundle gauge geodesies geometrically oriented given group H hence holonomy homomorphism immersion induced integral curve isomorphism Klein geometry Klein pair leaf Lemma Let G Lie algebra Lie group linear locally ambient geometry loop matrix Maurer-Cartan form metric Mobius geometry neighborhood normal bundle notion On(R open set Or(R path plaque Pnor principal bundle projective Proof Proposition Ptan reduction Ricci Riemannian geometry satisfying simply connected smooth manifold smooth map structural equation subgroup submanifold submodule subspace symmetric takes values tangent bundle Theory topology torsion free trivial Tx(M unique vanishes vector bundle vector field vector space Weyl