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Graduate Texts in Mathematics #166
Differential Geometry: Cartan's Generalization of Klein's Erlangen Program
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.
446 pages, Hardcover
First published June 12, 1997
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October 20, 2016
This is a great basic-level differential geometry book from high perspective. The book origins from Felix Klein's Erlangen program, stating that geometry should be regarded as a study of invariant properties of a homogeneous space under certain transformations. Also, Elie Cartan generalized the idea so as to allow Klein geometry to be not flat. That is, Cartan geometry looks locally like Klein geometry.
Following this guideline, the author firstly introduces basics differential geometry's stuff such as manifold, fibre bundles, tangent vector, differential forms and Lie group. Then the author shows the basic theory of foliations and distributions. This naturally reaches integrability condition, which is core making homogeneous space flat, and violation of which will lead to Cartan's generalization. Next, The book talks about Maurer-Cartan form, structural equation and Darboux derivative, which were deeply studied by Cartan in his theory of moving frame. And later these become the fundamental theorems of calculus in terms of exterior differential forms. Till now, all essences for Klein geometry have been gently introduced. And I say, wow, amazing. Nothing comes from nowhere, everything is concise and elegant.
In the following part of the book, Klein geometry and its properties are properly covered. The gauge view of Klein geometry is unique and fantastic, because the gauge theory, which was independently discovered by theoretical physicists, taking them so much hard work, seem so natural in eyes of mathematicians. One can say, gauge theory is exactly the theory of principal bundle. In 1975, CN Yang visited SS Chern and said: "this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere." Chern protested immediately: "No, no. These concepts were not dreamed up. They were natural and real."
Cartan's idea to geometry is that the left side of structural equation, defined as curvature form, doesn't have to vanish. The real fun origins from this. The book develops ideas of Cartan's, talking about curvature, parallel transport, covariant derivative, holonomy and space forms. Readers who know about Riemannian geometry will find this familiar because Cartan is also the generalization of Riemannian geometry.
In the last part of this book, the author discusses three special and useful models of Cartan geometries, Riemannian, Mobius and projective. With the knowledge of Cartan geometry, these geometries immediately become easy specialization, and also concise and elegant.
A last word, the book is really fantastic worthy of reading multiple times.
Following this guideline, the author firstly introduces basics differential geometry's stuff such as manifold, fibre bundles, tangent vector, differential forms and Lie group. Then the author shows the basic theory of foliations and distributions. This naturally reaches integrability condition, which is core making homogeneous space flat, and violation of which will lead to Cartan's generalization. Next, The book talks about Maurer-Cartan form, structural equation and Darboux derivative, which were deeply studied by Cartan in his theory of moving frame. And later these become the fundamental theorems of calculus in terms of exterior differential forms. Till now, all essences for Klein geometry have been gently introduced. And I say, wow, amazing. Nothing comes from nowhere, everything is concise and elegant.
In the following part of the book, Klein geometry and its properties are properly covered. The gauge view of Klein geometry is unique and fantastic, because the gauge theory, which was independently discovered by theoretical physicists, taking them so much hard work, seem so natural in eyes of mathematicians. One can say, gauge theory is exactly the theory of principal bundle. In 1975, CN Yang visited SS Chern and said: "this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere." Chern protested immediately: "No, no. These concepts were not dreamed up. They were natural and real."
Cartan's idea to geometry is that the left side of structural equation, defined as curvature form, doesn't have to vanish. The real fun origins from this. The book develops ideas of Cartan's, talking about curvature, parallel transport, covariant derivative, holonomy and space forms. Readers who know about Riemannian geometry will find this familiar because Cartan is also the generalization of Riemannian geometry.
In the last part of this book, the author discusses three special and useful models of Cartan geometries, Riemannian, Mobius and projective. With the knowledge of Cartan geometry, these geometries immediately become easy specialization, and also concise and elegant.
A last word, the book is really fantastic worthy of reading multiple times.
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