This book, Differential Geometry: Riemannian Geometry and Isometric Immersions (Book I-B), is the second in a captivating series of four books presenting a choice of topics, among fundamental and more advanced in differential geometry (DG). Starting with the basics of semi-Riemannian geometry, the book aims to develop the understanding of smooth 1-parameter variations of geodesics of, and correspondingly of, Jacobi fields. A few algebraic aspects required by the treatment of the Riemann–Christoffel four-tensor and sectional curvature are successively presented. Ricci curvature and Einstein manifolds are briefly discussed. The Sasaki metric on the total space of the tangent bundle over a Riemannian manifold is built, and its main properties are investigated. An important integration technique on a Riemannian manifold, related to the geometry of geodesics, is presented for further applications. The other three books of the series are
Differential Geometry 1: Manifolds, Bundle and Characteristic Classes (Book I-A)Differential Geometry 3: Foundations of Cauchy-Riemann and Pseudohermitian Geometry (Book I-C)Differential Geometry 4: Advanced Topics in Cauchy–Riemann and Pseudohermitian Geometry (Book I-D)
The four books belong to a larger book project (Differential Geometry, Partial Differential Equations, and Mathematical Physics) by the same authors, aiming to demonstrate how certain portions of DG and the theory of partial differential 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 scientific (mathematical and physical) interests. These are centered around the theory of immersions—isometric, holomorphic, 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.
Elisabetta Barletta is Associate Professor of Mathematical Analysis at the Department of Mathematics, Computer Science and Economy, Universitą degli Studi della Basilicata (Potenza, Italy). She studied mathematics at Universitą di Firenze, obtaining her degree in mathematics in 1979, under Giuseppe Tomassini. Assistant Professor at Universitą 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 Professor 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 flat Cauchy-Riemann manifolds, and proper holomorphic maps of pseudoconvex domains."
Sorin Dragomir is Professor of mathematical analysis at the Universitą degli Studi della, Basilicata, Potenza, Italy. He studied mathematics at the Universitatea din Bucureşti, Bucharest, under S. Ianuş, D. Smaranda, I. Colojoară, 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, 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 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 100 research papers and three contributed volumes published in Springer, 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.
