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基本信息
- 原书名：Differential Geometry of Curves and Surfaces
- 原出版社： Prentice Hall/Pearson
编辑推荐
《曲线与曲面的微分几何》（英文版）是一本关于曲线和曲面微分几何的导论，介绍微分几何这两个方面的局部特性与整体特性。同传统的微分几何英亚注册不同，本书更广泛地应用初等线性代数的知识，并把重点放在基本的几何论据上。
内容简介
目录
Some Remarks on Using this Book vii
Curves 1
Introduction I
Parametrized Curves 2
Regular Curves; Arc Length 5
The Vector Product in Ra3 11
The Local Theory of Curves Parametrized by Arc Length 16
The Local Canonical Form 27
Global Properties of Plane Curves 30
Regular Surfaces 51
Introduction 51
Regular Surfaces; Inverse Images of Regular Values 52
Change of Parameters; Differential Functions on Surfaces 69
The Tangent Plane; the Differential of a Map 83
The First Fundamental Form; Area 92
Orientation of Surfaces 102
A Characterization of Compact Orientable Surfaces 109
A Geometric Definition of Area 114
Appendix: A Brief Review on Continuity
and Differentiability 118
Curves 1
Introduction I
Parametrized Curves 2
Regular Curves; Arc Length 5
The Vector Product in Ra3 11
The Local Theory of Curves Parametrized by Arc Length 16
The Local Canonical Form 27
Global Properties of Plane Curves 30
Regular Surfaces 51
Introduction 51
Regular Surfaces; Inverse Images of Regular Values 52
Change of Parameters; Differential Functions on Surfaces 69
The Tangent Plane; the Differential of a Map 83
The First Fundamental Form; Area 92
Orientation of Surfaces 102
A Characterization of Compact Orientable Surfaces 109
A Geometric Definition of Area 114
Appendix: A Brief Review on Continuity
and Differentiability 118
前言
This book is an introduction to the differential geometry of curves and surfaces, both in its local and global aspects. The presentation differs from the traditional ones by a more extensive use of elementary linear algebra add by a certain emphasis placed on basic geometrical facts, rather than on machinery or random details.
We have tried to build each chapter of the book around some simple and fundamental idea. Thus, Chapter 2 develops around the concept of a regular surface in Rs; when this concept is properly developed, it is probably the best model for differentiable manifolds. Chapter 3 is built on the Gauss normal map and contains a large amount of the local geometry of
surfaces in Ra. Chapter 4 unifies the intrinsic geometry of surfaces around the concept of covariant derivative; again, our purpose was to prepare the reader for the basic notion of connection in Riemannian geometry. Finally, in Chapter 5, we use the first and second variations of arc length to derive some global properties of surfaces. Near the end of Chapter 5 (Sec. 5-10),we show how questions on surface theory, and the experience of Chapters 2 and 4, lead naturally to the consideration of differentiable manifolds and Riemannian metrics.
To maintain the proper balance between ideas and facts, we have presented a large number of examples that are computed in detail. Furthermore, a reasonable supply of exercises is provided. Some factual material of classical differential geometry found its place in these exercises. Hints or answers are given for the exercises that are starred.
The prerequisites for reading this book are linear algebra and calculus.
From linear algebra, only the most basic concepts are needed, and a standard undergraduate course on the subject should suffice. From calculus,a certain familiarity with calculus of several variables (including the statement of the implicit function theorem) is expected. For the reader's convenience, we have tried to restrict our references to R. C. Buck, Advancd
Calculus, New York: McGraw-Hill, 1965 (quoted as Buck, Advanced Calculus). A certain knowledge of differential equations will be useful but it is not required.
This book is a free translation, with additional material, of a book and a set of notes, both published originally in Portuguese. Were it not for the enthusiasm and enormous help of Blaine Lawson, this book would not have come into English. A large part of the translation was done by Leny Cavalcante. I am also indebted to my colleagues and students at IMPA for their comments and support. In particular, Elon Lima read part of the Portuguese version and made valuable comments.
Robert Gardner, Jiirgen Kern, Blaine Lawson, and Nolan Wallach read critically the English manuscript and helped me to avoid several mistakes,both in English and Mathematics. Roy Ogawa prepared the computer programs for some beautiful drawings that appear in the book (Figs. 1-3, 1-8,1-9, 1-10, 1-11, 3-45 and 4-4). Jerry Kazdan devoted his time generously and literally offered hundreds of suggestions for the improvement of the manuscript. This final form of the book has benefited greatly from his advice. To all these people--and to A'rthur Wester, Editor of Mathematics at Prentice-Hall, and Wilson G6es at IMPA--I extend my sincere thanks.
Rio de Janeiro
Manfredo P. do Carmo
We have tried to build each chapter of the book around some simple and fundamental idea. Thus, Chapter 2 develops around the concept of a regular surface in Rs; when this concept is properly developed, it is probably the best model for differentiable manifolds. Chapter 3 is built on the Gauss normal map and contains a large amount of the local geometry of
surfaces in Ra. Chapter 4 unifies the intrinsic geometry of surfaces around the concept of covariant derivative; again, our purpose was to prepare the reader for the basic notion of connection in Riemannian geometry. Finally, in Chapter 5, we use the first and second variations of arc length to derive some global properties of surfaces. Near the end of Chapter 5 (Sec. 5-10),we show how questions on surface theory, and the experience of Chapters 2 and 4, lead naturally to the consideration of differentiable manifolds and Riemannian metrics.
To maintain the proper balance between ideas and facts, we have presented a large number of examples that are computed in detail. Furthermore, a reasonable supply of exercises is provided. Some factual material of classical differential geometry found its place in these exercises. Hints or answers are given for the exercises that are starred.
The prerequisites for reading this book are linear algebra and calculus.
From linear algebra, only the most basic concepts are needed, and a standard undergraduate course on the subject should suffice. From calculus,a certain familiarity with calculus of several variables (including the statement of the implicit function theorem) is expected. For the reader's convenience, we have tried to restrict our references to R. C. Buck, Advancd
Calculus, New York: McGraw-Hill, 1965 (quoted as Buck, Advanced Calculus). A certain knowledge of differential equations will be useful but it is not required.
This book is a free translation, with additional material, of a book and a set of notes, both published originally in Portuguese. Were it not for the enthusiasm and enormous help of Blaine Lawson, this book would not have come into English. A large part of the translation was done by Leny Cavalcante. I am also indebted to my colleagues and students at IMPA for their comments and support. In particular, Elon Lima read part of the Portuguese version and made valuable comments.
Robert Gardner, Jiirgen Kern, Blaine Lawson, and Nolan Wallach read critically the English manuscript and helped me to avoid several mistakes,both in English and Mathematics. Roy Ogawa prepared the computer programs for some beautiful drawings that appear in the book (Figs. 1-3, 1-8,1-9, 1-10, 1-11, 3-45 and 4-4). Jerry Kazdan devoted his time generously and literally offered hundreds of suggestions for the improvement of the manuscript. This final form of the book has benefited greatly from his advice. To all these people--and to A'rthur Wester, Editor of Mathematics at Prentice-Hall, and Wilson G6es at IMPA--I extend my sincere thanks.
Rio de Janeiro
Manfredo P. do Carmo
