Differential Geometry

This book, Diï¬erential Geometry: Manifolds, Bundles and Characteristic Classes (Book I-A), is the ï¬rst in a captivating series of four books presenting a choice of topics, among fundamental and more advanced, in diï¬erential geometry (DG), such as manifolds and tensor calculus, diï¬erentiable 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:

Diï¬erential Geometry: Riemannian Geometry and Isometric Immersions (Book I-B)

Diï¬erential Geometry: Foundations of Cauchy-Riemann and Pseudohermitian Geometry (Book I-C)

Diï¬erential Geometry: Advanced Topics in Cauchy-Riemann and Pseudohermitian Geometry (Book I-D)

The four books belong to an ampler book project (Diï¬erential Geometry, Partial Diï¬erential Equations, and Mathematical Physics, by the same authors) and aim to demonstrate how certain portions of DG and the theory of partial diï¬erential 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 scientiï¬c (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.