Short description:

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. 3?11). 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. 16?18, 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 Chen?Ricci 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.

Long description:

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. 3?11). 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. 16?18, 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 Chen?Ricci 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.

Table of Contents:

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 -----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.