Differential Geometry: Cartan's Generalization of Klein's Erlangen Program

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Springer Science & Business Media, Nov 21, 2000 - Mathematics - 426 pages
Cartan 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
Index
415
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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.

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