Bang-yen Chen

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.

The book aims to present a comprehensive survey on biharmonic submanifolds and maps from the viewpoint of Riemannian geometry. It provides some basic knowledge and tools used in the study of the subject as well as an overall picture of the development of the subject with most up-to-date important results. Biharmonic submanifolds are submanifolds whose isometric immersions are biharmonic maps, thus biharmonic submanifolds include minimal submanifolds as a subclass. Biharmonic submanifolds also appeared in the study of finite type submanifolds in Euclidean spaces. Biharmonic maps are maps between Riemannian manifolds that are critical points of the bienergy. They are generalizations of harmonic maps and biharmonic functions which have many important applications and interesting links to many areas of mathematics and theoretical physics. Since 2000, biharmonic submanifolds and maps have become a vibrant research field with a growing number of researchers around the world, with many interesting results have been obtained. This book containing basic knowledge, tools for some fundamental problems and a comprehensive survey on the study of biharmonic submanifolds and maps will be greatly beneficial for graduate students and beginning researchers who want to study the subject, as well as researchers who have already been working in the field.

A warped product manifold is a Riemannian or pseudo-Riemannian manifold whose metric tensor can be decomposed into a Cartesian product of the y geometry and the x geometry — except that the x-part is warped, that is, it is rescaled by a scalar function of the other coordinates y. The notion of warped product manifolds plays very important roles not only in geometry but also in mathematical physics, especially in general relativity. In fact, many basic solutions of the Einstein field equations, including the Schwarzschild solution and the Robertson-Walker models, are warped product manifolds. The first part of this volume provides a self-contained and accessible introduction to the important subject of pseudo-Riemannian manifolds and submanifolds. The second part presents a detailed and up-to-date account on important results of warped product manifolds, including several important spacetimes such as Robertson-Walker's and Schwarzschild's. The famous John Nash's embedding theorem published in 1956 implies that every warped product manifold can be realized as a warped product submanifold in a suitable Euclidean space. The study of warped product submanifolds in various important ambient spaces from an extrinsic point of view was initiated by the author around the beginning of this century. The last part of this volume contains an extensive and comprehensive survey of numerous important results on the geometry of warped product submanifolds done during this century by many geometers.

During the last four decades, there were numerous important developments on total mean curvature and the theory of finite type submanifolds. This unique and expanded second edition comprises a comprehensive account of the latest updates and new results that cover total mean curvature and submanifolds of finite type. The longstanding biharmonic conjecture of the author's and the generalized biharmonic conjectures are also presented in details. This book will be of use to graduate students and researchers in the field of geometry.

During the last four decades, there were numerous important developments on total mean curvature and the theory of finite type submanifolds. This unique and expanded second edition comprises a comprehensive account of the latest updates and new results that cover total mean curvature and submanifolds of finite type. The longstanding biharmonic conjecture of the author's and the generalized biharmonic conjectures are also presented in details. This book will be of use to graduate students and researchers in the field of geometry.

The first part of this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian manifolds and their non-degenerate submanifolds, only assuming from the reader some basic knowledge about manifold theory. A number of recent results on pseudo-Riemannian submanifolds are also included. The second part of this book is on d-invariants, which was introduced in the early 1990s by the author. The famous Nash embedding theorem published in 1956 was aimed for, in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, this hope had not been materialized as pointed out by M Gromov in his 1985 article published in Asterisque. The main reason for this is the lack of control of the extrinsic invariants of the submanifolds by known intrinsic invariants. In order to overcome such difficulties, as well as to provide answers for an open question on minimal immersions, the author introduced in the early 1990s new types of Riemannian invariants, known as d-invariants, which are very different in nature from the classical Ricci and scalar curvatures. At the same time he was able to establish general optimal relations between d-invariants and the main extrinsic invariants. Since then many new results concerning these d-invariants have been obtained by many geometers. The second part of this book is to provide an extensive and comprehensive survey over this very active field of research done during the last two decades.

The purpose of this book is to introduce the reader to two interesting topics in geometry which have developed over the last fifteen years, namely, total mean curvature and submanifolds of finite type. The theory of total mean curvature is the study of the integral of the n-th power of the mean curvature of a compact n-dimensional submanifold in a Euclidean m-space and its applications to other branches of mathematics. The relation of total mean curvature to analysis, geometry and topology are discussed in detail. Motivated from these studies, the author introduces and studies submanifolds of finite type in the last chapter. Some applications of such submanifolds are also given. This book is self-contained. The author hopes that the reader will be encouraged to pursue his studies beyond the confines of the present book.