PDF Download Differential Geometry: Cartan's Generalization of Klein's Erlangen Program (Graduate Texts in Mathematics, Vol. 166), by R.W. Sharpe
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Differential Geometry: Cartan's Generalization of Klein's Erlangen Program (Graduate Texts in Mathematics, Vol. 166), by R.W. Sharpe
PDF Download Differential Geometry: Cartan's Generalization of Klein's Erlangen Program (Graduate Texts in Mathematics, Vol. 166), by R.W. Sharpe
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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.
- Sales Rank: #1761129 in Books
- Published on: 2000-11-21
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x 1.06" w x 6.14" l, 1.64 pounds
- Binding: Hardcover
- 426 pages
Most helpful customer reviews
38 of 40 people found the following review helpful.
Everything via principal bundles.
By A Customer
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.
0 of 0 people found the following review helpful.
A unique introduction to differential geometry that emphasizes the role of symmetry
By Geometer
This is a truly unique introduction to differential geometry. The role of Lie groups in differential geometry, which is often somewhat ambiguous in other introductory texts, is emphasized. Following the philosophy that geometry is determined by the symmetries we are interested in, Cartan geometry is a framework that encompasses nearly all differential geometric structures of interest, including Riemannian and semi-Riemannian geometry (the geometry of Relativity), CR geometry, conformal geometry, projective geometry, and many others. This is the best introductory text to Cartan geometry. A good second text on Cartan geometry once you are finished with this one is Cap and Slovak's Parabolic Geometry.
5 of 20 people found the following review helpful.
Great Book
By Kar-fai Andy Toh
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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