Differential Geometry. - Cartan's Generalization of Klein's Erlangen Program

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 firstyear 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. 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 Tepic inclue in the five appendices are a comparison of Cartan and Ehresmann connections, and the derivation of the divergence and curl operators from symmetry considerations.