## 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 | xix |

1 Smooth Manifolds | 1 |

2 Submanifolds | 15 |

3 Fiber Bundles | 26 |

4 Tangent Vectors Bundles and Fields | 37 |

5 Differential Forms | 50 |

Looking for the Forest in the Leaves Foliations | 63 |

1 Integral Curves | 64 |

6 Cartan Space Forms | 215 |

7 Symmetric Spaces | 223 |

Riemannian Geometry | 225 |

1 The Model Euclidean Space | 226 |

2 Euclidean and Riemannian Geometry | 232 |

3 The Equivalence Problem for Riemannian Metries | 235 |

4 Riemannian Space Forms | 239 |

5 Subgeometry of a Riemannian Geometry | 242 |

2 Distributions | 72 |

3 Integrability Conditions | 73 |

4 The Frobenius Theorem | 74 |

5 The Frobenius Theorem in Terms of Differential Forms | 77 |

6 Foliations | 79 |

7 Leaf Holonomy | 81 |

8 Simple Foliations | 88 |

The Fundamental Theorem of Calculus | 93 |

2 Lie Algebras | 99 |

3 Structural Equation | 106 |

4 Adjoint Action | 108 |

5 The Darboux Derivative | 113 |

Local Version | 114 |

Global Version | 116 |

8 Monodromy and Completeness | 126 |

Shapes Fantastic Klein Geometries | 135 |

1 Examples of Planar Klein Geometries | 138 |

Characteri2ation and Reduction | 142 |

3 Klein Geometries | 148 |

4 A Fundamental Property | 158 |

5 The Tangent Bundle of a Klein Geometry | 160 |

6 The Meteor Tracking Problem | 162 |

7 The Gauge View of Klein Geometries | 164 |

Shapes High Fantastical Cartan Geometries | 169 |

1 The Base Definition of Cartan Geometries | 171 |

2 The Principal Bundle Hidden in a Cartan Geometry | 176 |

3 The Bundle Definition of a Cartan Geometry | 182 |

4 Development Geometric Orientation and Holonomy | 200 |

5 Flat Cartan Geometries and Uniformization | 209 |

6 Isoparametric Submanifolds | 257 |

Möbius Geometry | 263 |

1 The Möbius and Weyl Models | 265 |

2 Möbius and Weyl Geometries | 275 |

3 Equivalence Problems for a Conformal Metric | 282 |

4 Submanifolds of Möbius Geometry | 292 |

5 Immersed Curves | 316 |

6 Immersed Surfaces | 321 |

Projective Geometry | 329 |

2 Projective Cartan Geometries | 336 |

3 The Geometry of Geodesics | 341 |

4 The Projective Connection in a Riemannian Geometry | 347 |

5 A Brief Tour of Projective Geometry | 351 |

Ehresmann Connections | 355 |

2 The Reductive Case | 360 |

3 Ehresmann Connections Generalize Cartan Connections | 363 |

4 Covariant Derivative | 369 |

Rolling Without Slipping or Twisting | 373 |

2 The Existence and Uniqueness of Rolling Maps | 376 |

3 Relation to LeviCivita and Normal Connections | 380 |

4 Transitivity of Rolling Without Slipping or Twisting | 386 |

Classification of OneDimensional Effective Klein Pairs | 389 |

Differential Operators Obtained from Symmetry | 395 |

2 Operators on Riemannian Surfaces | 398 |

Characterization of Principal Bundles | 405 |

Bibliography | 409 |

415 | |

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

### Common terms and phrases

action associated assume basis Bianchi identity bundle called canonical Cartan geometry chart choice closed commutative compact complete component condition conformal connected consider constant coordinate Corollary corresponding covering curvature curve defined Definition denote derivative described determined diffeomorphism differential dimension distribution effective Ehresmann element equation equivalent Euclidean example Exercise expression exterior derivative fact fiber flat follows function fundamental gauge given group H hence identity immersion implies induced integral isomorphism Klein geometry Lemma Lie algebra Lie group linear locally manifold metric Möbius geometry normal Note notion oriented pair particular path principal bundle projective Proof properties Proposition reduction regarded respectively restriction result Riemannian satisfying Show smooth smooth manifold space structure subgroup submanifold Suppose symmetric takes tangent Theorem Theory topology trivial unique values vanishes vector field write yields

### Popular passages

Page iv - Lorentz-transformations) is too narrow, ie, that an invariance of the laws must be postulated also relative to non-linear transformations of the co-ordinates in the four-dimensional continuum. This happened in 1908. Why were another seven years required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that co-ordinates must have an immediate metrical meaning.

Page iv - I belong to the latter category — though being eventually able to use it for simple applications, feel insuperable difficulty in mastering more than a rather elementary and superficial knowledge of it.

Page xiv - The authors are indebted to the National Science and Engineering Research Council of Canada for its financial support We also address our thanks to Dr.