Differential Geometry: Cartan's Generalization of Klein's Erlangen Program (Graduate Texts in Mathematics, Vol. 166) (Graduate Texts in Mathematics (166))

This text presents the systematic and well motivated development of differential geometry leading to the global version of Cartan connections presented at a level accessible to a first year graduate student. The first four chapters provide a complete and economical development of the fundamentals of differential topology, foliations, Lie groups and homogeneous spaces. Chapter 5 studies Cartan geometries which generalize homogenous spaces in the same way that Riemannian geometry generalizes Euclidean geometry. One of the beautiful facets of Cartan Geometries is that curvature appears as an exact local measurement of "broken symmetry". The last three chapters study three examples: Riemannian geometry, conformal geometry and projective geometry. Some of the topics studied include: - a complete proof of the Lie group - Lie algebra correspondence - a classification of the Cartan space forms - a classification of submanifolds in conformal geometry - Cartan's "geometrization" of an ODE of the form y"=A(x,y)+B(x,y)y'+C(x,y)(y')^{2}+ D(x,y)(y')^{3} Topics included in the five appendices are a comparison of Cartan and Ehresmann connections, and the derivation of the divergence and curl operators from symmetry considerations. This text presents a systematic and well-motivated development of differential geometry leading to the global version of Cartan connections. The material is presented at a level accessible to a first-year graduate student. The first four chapters provide a complete development of the fundamentals of differential topology, foliations, Lie groups, and homogeneous spaces. Chapter 5 studies Cartan geometries which generalize homogeneous spaces in the same way that Riemannian geometry generalizes Euclidean geometry. One of the beautiful facets of Cartan geometries is that curvature appears as an exact local measurement of "broken symmetry." The last three chapters study Riemannian geometry, conformal geometry, and projective geometry. Topics included in the five appendices are a comparison of Cartan and Ehresmann connections, and the derivation of the divergence and curl operators from symmetry considerations.

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.

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.