This book, Diferential Geometry: Manifolds, Bundles and Characteristic Classes (Book I-A), is the frst in a captivating series of four books presenting a choice of topics, among fundamental and more advanced, in diferential geometry (DG), such as manifolds and tensor calculus, diferentiable 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:

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

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

Diferential Geometry: Advanced Topics in Cauchy-Riemann and Pseudohermitian Geometry (Book I-D)

The four books belong to an ampler book project (Diferential Geometry, Partial Diferential Equations, and Mathematical Physics, by the same authors) and aim to demonstrate how certain portions of DG and the theory of partial diferential 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 scientifc (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.