Description
This book, DiÃâ€Ã‚¨Ã„erential Geometry: Manifolds, Bundles and Characteristic Classes (Book I-A), is the Ãâ€Ã‚¨Ã…rst in a captivating series of four books presenting a choice of topics, among fundamental and more advanced, in diÃâ€Ã‚¨Ã„erential geometry (DG), such as manifolds and tensor calculus, diÃâ€Ã‚¨Ã„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: DiÃâ€Ã‚¨Ã„erential Geometry: Riemannian Geometry and Isometric Immersions (Book I-B) DiÃâ€Ã‚¨Ã„erential Geometry: Foundations of Cauchy-Riemann and Pseudohermitian Geometry (Book I-C) DiÃâ€Ã‚¨Ã„erential Geometry: Advanced Topics in Cauchy–Riemann and Pseudohermitian Geometry (Book I-D) The four books belong to an ampler book project (DiÃâ€Ã‚¨Ã„erential Geometry, Partial DiÃâ€Ã‚¨Ã„erential Equations, and Mathematical Physics, by the same authors) and aim to demonstrate how certain portions of DG and the theory of partial diÃâ€Ã‚¨Ã„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 scientiÃâ€Ã‚¨Ã…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.