This book attempts to present a comprehensive survey of the geometry of CR-submanifolds. The theory of submanifolds is one of the most interesting topics in differential geometry. The topic is introduced by Aurel Bejancu as a generalization of holomorphic and totally real submanifolds of almost Hermitian manifolds, in 1978. Afterward, the study of CR-submanifolds became a very active research subject.
Organized into 22 chapters, the book starts with basic knowledge of Riemannian manifolds and submanifolds, almost Hermitian manifolds and their subclasses, Hopf fibration, symmetric spaces, and a general inequality for submanifolds in complex space forms (in Chaps. 1 and 2). Later, it presents the main results on CR-submanifolds in Kaehler manifolds, the basic inequalities associated with CR-submanifolds in Kaehler manifolds, and several theories and results related to Kaehler manifolds (in Chaps. 311). Further, the book discusses the basics of almost-contact metric manifolds and their subclasses, CR-submanifolds of Sasakian, trans-Sasakian and quasi-Sasakian manifolds, with a particular attention on the normal CR-submanifolds (in Chap. 12). It also investigates the contact CR-submanifolds of S-manifolds, the geometry of submersions of CR-submanifolds, and the results on contact CR-warped product submanifolds (in Chaps. 1618, 20). In Chapter 19, we discuss submersions of CR-submanifolds. The book also presents some recent results concerning CR-submanifolds of holomorphic statistical manifolds. In particular, it gives the classification of totally umbilical CR-statistical submanifolds in holomorphic statistical manifolds, as well as a ChenRicci inequality for such submanifolds (Chapter 21). In the last chapter, we present results on CR-submanifolds of indefinite Kaehler manifolds and their applications to physics.
Chapter 1 Basics on Manifolds and Submanifolds.- Chapter 2 Basics on
almost Hermitian manifolds and their subclasses.
Chapter 3 CR-submanifolds
of K¨ahler manifolds.
Chapter 4 Inequalities for CR-submanifolds in Kähler
manifolds.- Chapter 5 CR-warped products in Kähler manifolds.
Chapter 6
CR-submanifolds of locally conformal Kähler manifolds.
Chapter 7
CR-submanifolds of quaternion Kähler manifolds.
Chapter 8 CR-submanifolds of
nearly Kähler manifolds.
Chapter 9CR-submanifolds of quasi-Kähler
manifolds.
Chapter 10 Generic submanifolds of nearly Kähler manifolds.-
Chapter 11 Generic submanifold of locally conformal Kähler manifolds.-
Chapter 12 Basics of almost contact metric manifolds and their
subclasses.- Chapter 13 Contact CR-submanifolds of trans-Sasakian manifolds.-
Chapter 14 CContact CR-submanifolds of nearly Sasakian manifolds.- Chapter
15. Contact CR-submanifolds of nearly trans-Sasakian manifolds.
Chapter 16
Contact CR-submanifolds of quasi-Sasakian manifolds.- Chapter 17 Contact
CR-submanifolds of 𝑺-manifolds.
Chapter 18. Generic submanifolds of
manifolds equipped with almost contactmetric structures.
Chapter 19.
Submersion of CR-submanifolds.
Chapter 20 Contact CR-warped product
submanifolds.
Chapter 22CR-submanifolds of indefinite K¨ahler manifolds and
applications.
Bang-Yen Chen, a Taiwanese-American mathematician, is Distinguished Professor Emeritus at Michigan State University, USA, since 2012. He completed his Ph.D. degree at the University of Notre Dame, USA, in 1970, under the supervision of Prof. Tadashi Nagano. He received his M.Sc. degree from National Tsing Hua University, Hsinchu, Taiwan, in 1967, and B.Sc. degree from Tamkang University, Taipei, Taiwan, in 1965. Earlier at Michigan State University, he served as University Distinguished Professor (19902012), Full Professor (1976), Associate Professor (1972), and Research Associate (19701972). He taught at Tamkang University, Taiwan, from 1966 to 1968, and at National Tsing Hua University, Taiwan, during the academic year 19671968.
He is responsible for the invention of -invariants (also known as Chen invariants), Chen inequalities, Chen conjectures, and development of the theory of submanifolds of finite type, and he co-developed (M+, M)-theory. An author of 12 books and more than 500 research articles, Prof. Chen has been Visiting Professor at various universities, including the University of Notre Dame, USA; Science University of Tokyo, Japan; the University of Lyon, France; Katholieke Universiteit Leuven, Belgium; the University of Rome, Italy; National Tsing Hua University, Taiwan; and Tokyo Denki University, Japan.
Mohammad Hasan Shahid is Professor at the Department of Mathematics, Jamia Millia Islamia, New Delhi, India. He earned his Ph.D. in Mathematics from Aligarh Muslim University, India, on the topic On geometry of submanifolds in 1988 under (Late) Prof. Izhar Husain. Earlier, he served as Associate Professor at King Abdul Aziz University, Jeddah, Saudi Arabia, from 2001 to 2006. He was a recipient of the postdoctoral fellowship from the University of Patras, Greece, from October 1997 to April 1998. He has published more than 100 research articles in various national and international journals of repute. Recently, he was awarded the Sultana Nahar Distinguished Teacher Award of the Year 20172018 for his outstanding contribution to research. For research works and delivering talks, Prof. Shahid has visited several universities of the world: the University of Leeds, UK; the University of Montpellier, France; the University of Sevilla, Spain; Hokkaido University, Japan; Chuo University, Japan; and Manisa Celal Bayar University, Turkey.
Gabriel Eduard Vîlcu is Professor of Mathematics at the National University of Science and Technology Politehnica Bucharest, Romania. Additionally, he is a senior researcher at the Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy. He received his Ph.D. in Mathematics from the University of Bucharest, Romania, under the supervision of Prof. Stere Ianu, in the year 2007. Professor Vîlcu made important contributions to differential geometry, mathematical physics, mathematical economics as well as information geometry. He has authored more than 80 peer-reviewed
