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

Front Cover
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.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

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
Bibliography
411
Index
417
Copyright

Other editions - View all

Common terms and phrases

Bibliographic information