Differential Geometry
Subtitle: Cartan's Generalization of Klein's Erlangen Program (Graduate Texts in Mathematics)
Contributors: S.S. Chern (Preface)
Publisher: Springer
Published: November 21, 2000
Language: eng
Number of pages: 421
ISBN-13: 9780387947327
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.
Description: 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.