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PREFACE
The theory of submanifolds as a field of differential geometry is as old as differential geometry itself, beginning with the theory of curves and surfaces. However, the theory of submanifolds given in this book is relatively new in the realm of contemporary differential geometry.
The reader is assumed to be somewhat familiar with general theory of differential geometry as can be found, for example, in Kobayashi-Nomizu's Foundations of Differential Geometry. Most of the required background material is collected in the first two chapters. In Chapter 1, we have given a brief survey of Riemannian geometry and in Chapter 2, we have given a brief survey of the general theory of submanifolds.
In Chapter 3, minimal submanifolds are studied. Results in this chapter include classical results on the first variation of the volume integral and Bernstein's theorem as well as some recent results of Calabi, do Carmo, Chern, Kobayashi, Lawson, Simons, Takahashi, Wallach, Yano, and the author.
In Chapter 4, submanifolds with parallel mean curvature vector are studied. The theory of analytic functions is applied to the case of surfaces and give a powerful method which was used by Hopf. The main results of this chapter include recent works of Erbacher, Ferus, Hoffman, Ishihara, Klotz, Leung, Ludden, Nomizu, Osserman, Ruh, Smyth, Wolf, Yano, Yau, and the author.
In Chapter 5, conformally flat submanifolds are studied. The results in this chapter were mostly obtained by Yano and the author.
In Chapter 6, submanifolds umbilical with respect to a normal direction are studied. The normal connection of submanifolds play an important role in this chapter. Most results of this chapter were obtained by Nomizu, Smyth, Yano, and the author.
In the last chapter, geometric inequalities of submanifolds are given. Some elementary results in Morse theory are collected in the first section. These results have many important applications to the later sections. Results of Chern and Lashof on total absolute curvature are given in the second section. In sections 3 through 6, the total mean curvature of a submanifolds is studied. The results in these four sections include recent work of Shiohama, Takagi, Willmore, and the author. In the last section, stable hypersurfaces with respect to the total mean curvature are studied.
At the end of each chapter, some problems are given. These problems can be regarded as supplements to the text.
Since this book is based primarily on the author's recent work on real submanifolds, omissions of important results are inevitable.
In concluding the preface, the author would like to thank Professor S.S. Chern and Professor S. Kobayashi, who invited the author to undertake this project. The author also likes to thank Professor S.S. Chern, Professor T. Nagano, and Professor T. Otsuki for their constant encouragement and guidance. Finally, the author is greatly indebted to his colleagues, Dr. D.E. Blair, Dr. G.D. Ludden, and Dr. K. Ogiue for their help, which resulted in many improvements of both of the content and the presentation. Finally, the author wishes to thank Mrs. Mary Reynolds, who typed the manuscript, for her patience and cooperation.
Bang-Yen Chen
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Excerpted from "Geometry of Submanifolds"
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Copyright © 2019 Bang-Yen Chen.
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