Differential Geometry: Cartan's Generalization of Klein's Erlangen Program (Graduate Texts in Mathematics) Review

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Differential Geometry: Cartan's Generalization of Klein's Erlangen Program (Graduate Texts in Mathematics) ReviewSharpe'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.Differential Geometry: Cartan's Generalization of Klein's Erlangen Program (Graduate Texts in Mathematics) Overview

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