This book, Differential Geometry: Foundations of Cauchy–Riemann and Pseudohermitian Geometry (Book I-C), is the third in a series of four books presenting a choice of topics, among fundamental and more advanced, in Cauchy–Riemann (CR) and pseudohermitian geometry, such as Lewy operators, CR structures and the tangential CR equations, the Levi form, Tanaka–Webster connections, sub-Laplacians, pseudohermitian sectional curvature, and Kohn–Rossi cohomology of the tangential CR complex. Recent results on submanifolds of Hermitian and Sasakian manifolds are presented, from the viewpoint of the geometry of the second fundamental form of an isometric immersion. The book has two souls, those of Complex Analysis versus Riemannian geometry, and attempts to fill in the gap among the two. The other three books of the series are:
Differential Geometry: Manifolds, Bundles, Characteristic Classes (Book I-A)
Differential Geometry: Riemannian Geometry and Isometric Immersions (Book I-B)
Differential Geometry: Advanced Topics in Cauchy–Riemann and Pseudohermitian Geometry (Book I-D)
The four books belong to an ampler book project “Differential Geometry, Partial Differential Equations, and Mathematical Physics”, by the same authors, and aim to demonstrate how certain portions of differential geometry (DG) and the theory of partial differential equations (PDEs) apply to general relativity and (quantum) gravity theory. These books supply some of the ad hoc DG and PDEs machinery yet do not constitute a comprehensive treatise on DG or PDEs, but rather authors’ choice based on their scientific (mathematical and physical) interests. These are centered around the theory of immersions—isometric, holomorphic, and 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.
Elisabetta Barletta is Professor of mathematical analysis at the department of mathematics, computer science, and economy, Universit a degli Studi della Basilicata (Potenza, Italy). She joined the university as Lecturer in 1979 and then became Associate Professor in 2003. She visited several institutes worldwide: Visiting Fellow at the University of Maryland (USA), from 1982 to 1983, to conduct research with Carlos A. Berenstein; Visiting Fellow at Indiana University (USA), from 1987 to 1988, to do research with Eric Bedford; and Visiting Professor at Tohoku University (Japan), in 2003, invited by Seiki Nishikawa. Her research interests include complex analysis of functions of several complex variables, reproducing kernel Hilbert spaces, the geometry of Levi flat Cauchy–Riemann manifolds, and proper holomorphic maps of pseudoconvex domains.
Sorin Dragomir is Professor of mathematical analysis at the Università degli Studi della, Basilicata, Potenza, Italy. He studied mathematics at the Universitatea din Bucuresti, Bucharest, under S. Ianus, D. Smaranda, I. Colojoara, M. Jurchescu, and K. 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 Kählerian geometry of pseudoconvex domains and the pseudohermitian geometry of their boundaries, the impact of subelliptic 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. 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 in 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; Universite 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 differential 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 differential 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 orientation 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.
