This book, Dierential Geometry: Manifolds, Bundles and Characteristic Classes (Book I-A) , is the rst in a captivating series of four books presenting a choice of topics, among fundamental and more advanced, in dierential geometry (DG), such as manifolds and tensor calculus, dierentiable actions and principal bundles, parallel displacement and exponential mappings, holonomy, complex line bundles and characteristic classes. The inclusion of an appendix on a few elements of algebraic topology provides a didactical guide towards the more advanced Algebraic Topology literature. The subsequent three books of the series are:

Dierential Geometry: Riemannian Geometry and Isometric Immersions (Book I-B)

Dierential Geometry: Foundations of Cauchy-Riemann and Pseudohermitian Geometry (Book I-C)

Dierential Geometry: Advanced Topics in CauchyRiemann and Pseudohermitian Geometry (Book I-D)

The four books belong to an ampler book project (Dierential Geometry, Partial Dierential Equations, and Mathematical Physics, by the same authors) and aim to demonstrate how certain portions of DG and the theory of partial dierential equations apply to general relativity and (quantum) gravity theory. These books supply some of the ad hoc DG machinery yet do not constitute a comprehensive treatise on DG, but rather Authors' choice based on their scientic (mathematical and physical) interests. These are centered around the theory of immersions - isometric, holomorphic, and Cauchy-Riemann (CR) -and pseudohermitian geometry, as devised by Sidney Martin Webster for the study of nondegenerate CR structures, themselves a DG manifestation of the tangential CR equations.

Discusses the linear connections: parallel translation, exponential mappings, normal coordinates, and holonomy groups Reviews Sobolev inequalities in the Euclidean setting and how they carry over to Riemannian manifolds Presents the advanced topics in DG: characteristic classes and holonomy theory to the geometry of characteristic classes

Inhalt

Chapter 1 Manifolds and Tensor Calculus.- Chapter 2 Differentiable Actions and Principal Bundles.- Chapter 3 Infinite dimensional Differential Geometry.