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. |
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 algebra g canonical Cartan connection Cartan geometry chart commutative compact component conformal constant coordinate system Corollary corresponding Darboux derivative defined Definition denote determined diagram diffeomorphism differential Ehresmann connection equivalent Euclidean space example Exercise fiber flat foliation follows G bundle gauge geodesics geometrically oriented given Gl(g group H H₁ hence holonomy homomorphism identity immersion induced integral curve isomorphism Klein geometry Klein pair Lemma Let G Let P,w Lie algebra Lie group linear locally ambient geometry M₁ Maurer-Cartan form metric Mn(R Möbius geometry normal bundle notion open set P₁ path plaque Pnor principal bundle projective Proof Proposition Ptan reduction Ricci Riemannian geometry satisfying Show smooth manifold smooth map structural equation subgroup submanifold submodule subspace symmetric takes values tangent bundle Theorem Theory topology torsion free trivial unique V₁ vanishes vector bundle vector field vector space w₁ Weyl
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.