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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24 of 24 people found the following review helpful:
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Everything via principal bundles.,
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This review is from: Differential Geometry: Cartan's Generalization of Klein's Erlangen Program (Graduate Texts in Mathematics) (Hardcover)
Sharpe's book is a detailed argument supporting the assertion that most of differential geometry can be considered the study of principal bundles and connections on them, disguised as an introductory differential geometry textbook.Some standard introductory material (e.g. Stokes' theorem) is omitted, as Sharpe confesses in his preface, but otherwise this is a truly wonderful place to read about the central role of Lie groups, principal bundles, and connections in differential geometry. The theme is that what one can do for Lie groups, one can do fiberwise for principal bundles, to yield information about the base. The informal style (just look at the table of contents) and wealth of classical examples make this book a pleasure to read. While its somewhat nonstandard approach and preference for classical terminology might confuse those who have never been introduced to the concepts, this is a perfect *second* place to read and marvel about differential geometry.
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Great Book,
This review is from: Differential Geometry: Cartan's Generalization of Klein's Erlangen Program (Graduate Texts in Mathematics) (Hardcover)
I was fortunate enough to have Sharpe as my supervisor at University of Toronto just when his book was published. His highly abstract thinking is very impressive and I have enjoyed immensely his first chapter on differential topology, which is my specialized area. Though his book branches off into realms that don't particularly suit me, the beginnings of his book had given me great inspiration in my discipline in differential topology
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First Sentence:
Although several mathematicians, especially C.F. Gauss, studied the notion of a smooth manifold in special cases, the idea of an abstract manifold of arbitrary finite dimension seems to be due to Riemann. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
ambient geometry, normal submodule, submanifold topology, foliated atlas, flat normal bundle, geometric isomorphism, isoparametric submanifolds, rolling map, unique smooth structure, isoparametric hypersurfaces, infinitesimal connection, geometry localized, umbilic points, regular submanifold, generalized circles, smooth fiber bundle, trivial holonomy, oriented atlas, bundle automorphism, principal bundle, integral submanifold, monodromy representation, projective connection, leaf space, smooth bundle Key Phrases - Capitalized Phrases (CAPs): (learn more)
Elie Cartan, Immersed Surfaces, The Geometry of Geodesics, The Principal Bundle Hidden, The Projective Connection New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Although several mathematicians, especially C.F. Gauss, studied the notion of a smooth manifold in special cases, the idea of an abstract manifold of arbitrary finite dimension seems to be due to Riemann. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
ambient geometry, normal submodule, submanifold topology, foliated atlas, flat normal bundle, geometric isomorphism, isoparametric submanifolds, rolling map, unique smooth structure, isoparametric hypersurfaces, infinitesimal connection, geometry localized, umbilic points, regular submanifold, generalized circles, smooth fiber bundle, trivial holonomy, oriented atlas, bundle automorphism, principal bundle, integral submanifold, monodromy representation, projective connection, leaf space, smooth bundle Key Phrases - Capitalized Phrases (CAPs): (learn more)
Elie Cartan, Immersed Surfaces, The Geometry of Geodesics, The Principal Bundle Hidden, The Projective Connection New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 100
books:
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- Introduction to Lie Algebras and Representation Theory (Graduate Texts in Mathematics) (v. 9) by J.E. Humphreys in Front Matter, and Back Matter
- Functions of One Complex Variable (Graduate Texts in Mathematics - Vol 11) (v. 1) by John B Conway in Front Matter, and Back Matter
- Elliptic Curves: Diophantine Analysis (Grundlehren der mathematischen Wissenschaften) by S. Lang in Back Matter (1), and Back Matter (2)
- The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics) (v. 106) by Joseph H. Silverman in Back Matter (1), and Back Matter (2)
- Galois Theory (Graduate Texts in Mathematics) by Harold M. Edwards in Back Matter
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