Foundations of Differential Geometry, 2 Volume Set
Foundations of Differential Geometry, Volume 1 (978-0-471-15733-5) and Foundations of Differential Geometry, Volume 2 (978-0-471-15732-8), both by Shoshichi Kobayashi and Katsumi Nomizu
This two-volume introduction to differential geometry, part of Wiley's popular Classics Library, lays the foundation for understanding an area of study that has become vital to contemporary mathematics. It is completely self-contained and will serve as a reference as well as a teaching guide. Volume 1 presents a systematic introduction to the field from a brief survey of differentiable manifolds, Lie groups and fibre bundles to the extension of local transformations and Riemannian connections. Volume 2 continues with the study of variational problems on geodesics through differential geometric aspects of characteristic classes. Both volumes familiarize readers with basic computational techniques.
Kobayashi's research spans the areas of differential geometry of real and complex variables, and his numerous resulting publications include several book: Foundations of Differential Geometry with K. Nomizu, Hyperbolic Complex Manifolds and Holomorphic Mappings and Differential Geometry of Complex Vector Bundles.