This book, Di erential 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 di erential geometry (DG), such as manifolds and tensor calculus, di 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:

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

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

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

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

Autorentext

Elisabetta Barletta is Associate Professor of Mathematical Analysis at the Department of Mathematics, Computer Science and Economy, Universita degli Studi della Basilicata (Potenza, Italy). She studied mathematics at Universita di Firenze, obtaining her degree in mathematics in 1979, under Giuseppe Tomassini. Assistant Professor at Universit`a della Basilicata since 1986, she became Associate Professor in 2003. Visiting Fellow at the University of Maryland, USA (1982-1983, working with Carlos A. Berenstein); Visiting Fellow at Indiana University, USA (1987-1988, working with Eric Bedford); Visiting Pro-fessor at Tohoku University, Japan, (2003, invited by Seiki Nishikawa). Author of over 60 research papers and of the AMS Monograph Foliations in Cauchy-Riemann geometry (2007), her research interests include complex analysis of functions of several complex variables, reproducing kernel Hilbert spaces, the geometry of Levi at Cauchy-Riemann manifolds, and proper holomorphic maps of pseudoconvex domains.

Sorin Dragomir is Professor of Mathematical Analysis at the Universit`a degli Studi della, Basilicata, Potenza, Italy. He studied mathematics at the Universitatea din Bucure sti, Bucharest, under Stere Ianüs, Dumitru Smaranda, Ion Colojoar a, Martin Jurchescu, and Kostache Teleman, and earned his Ph.D. at Stony Brook University, New York, in 1992, under Denson C. Hill. His research interests are in the study of the tangential Cauchy-Riemann (CR) equations, the interplay between the K ahlerian geometry of pseudoconvex domains and the pseudohermitian geometry of their boundaries, the impact of subel-liptic theory on CR geometry, and the applications of CR geometry to space-time physics. With more than 140 research papers and 4 monographs, his wider interests regard the development and dissemination of both Western and Eastern mathematical sciences. An Italian citizen since 1991, he was born in Romania, and has solid cultural roots in Romanian mathematics, while his mathematical orientation over the last 10 years strongly owes to H. Urakawa (Sendai, Japan), E. Lanconelli (Bologna, Italy), J.P. D'Angelo (Urbana-Champaign, USA), and H. Jacobowitz (Camden, USA). He is Member of Unione Matematica Italiana, American Mathematical Society, and Mathematical Society of Japan.

Mohammad Hasan Shahid is former Professor at the Department of Mathematics, Jamia Millia Islamia (New Delhi, India). He also served at King Abdul Aziz University (Jeddah, Kingdom of Saudi Arabia) as Associate Professor, from 2001 to 2006. He earned his Ph.D. degree from Aligarh Muslim University (Aligarh, India), in 1988. His areas of research are the geometry of CR-submanifolds, Riemannian submersions, and tangent bundles. Author of more than 60 research papers, he has visited several world universities including, but not limited to, the University of Patras (Greece) (from 1997 to 1998) under postdoctoral scholarship from State Scholarship Foundation (Greece); the University of Leeds (England), in 1992, to deliver lectures; Ecole Polytechnique (Paris), in 2015; Universit e De Montpellier (France), in 2015; and Universidad De Sevilla (Spain), in 2015. He is Member of the Industrial Mathematical Society and the Indian Association for General Relativity.

Falleh R. Al-Solamy is Professor of di erential geometry at King Abdulaziz University (Jeddah, Saudi Arabia). He studied mathematics at King Abdulaziz University and earned his Ph.D. at the University of Wales Swansea (Swansea, UK), in 1998, under Edwin Beggs. His research interests concern the study of the geometry of submanifolds in Riemannian and semi-Riemannian manifolds, Einstein manifolds, and applications of di erential geometry in physics. With more than 54 research papers to his credit and coedited 1 book titled, Fixed Point Theory, Variational Analysis, and Optimization, his mathematical ori-entation over the last 10 years strongly owes to S. Deshmukh (Riyadh, Saudi Arabia), Mohammad Hasan Shahid (New Delhi, India), and V.A. Khan (Aligarh, India). He is Member of the London Mathematical Society, the Institute of Physics, the Saudi Association for Mathematical Sciences, the Tensor Society, the Saudi Computer Society, and the American Mathematical Society.