Manfredo P. do Carmo - Differential Geometry of Curves and Surfaces

DIFFERENTIAL
GEOMETRY

OF

CURVES &
SURFACES

DIFFERENTIAL
GEOMETRY

OF

CURVES &
SURFACES

REVISED & UPDATED

SECOND EDITION

MANFREDO P. DO CARMO

Instituto Nacional de Matemtica
Pura e Aplicada (IMPA)
Rio de Janeiro, Brazil

DOVER PUBLICATIONS, INC.
Mineola, New York

Copyright

Copyright 1976, 2016 by Manfredo P. do Carmo

All rights reserved.

Bibliographical Note

Differential Geometry of Curves and Surfaces: Revised & Updated Second Edition is a revised, corrected, and updated second edition of the work originally published in 1976 by Prentice-Hall, Inc., Englewood Cliffs, New Jersey. The author has also provided a new Preface for this edition.

International Standard Book Number

ISBN-13: 978-0-486-80699-0

ISBN-10: 0-486-80699-5

Manufactured in the United States by LSC Communications

806995012016

www.doverpublications.com

To Leny,
for her indispensable assistance
in all the stages of this book

In this edition, I have included many of the corrections and suggestions kindly sent to me by those who have used the book. For several reasons it is impossible to mention the names of all the people who generously donated their time doing that. Here I would like to express my deep appreciation and thank them all.

Thanks are also due to John Grafton, Senior Acquisitions Editor at Dover Publications, who believed that the book was still valuable and included in the text all of the changes I had in mind, and to the editor, James Miller, for his patience with my frequent requests.

As usual, my wife, Leny A. Cavalcante, participated in the project as if it was a work of her own; and I might say that without her this volume would not exist.

Finally, I would like to thank my son, Manfredo Jr., for helping me with several figures in this edition.

Manfredo P. do Carmo

September 20, 2016

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 and 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, , 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 readers convenience, we have tried to restrict our references to R. C. Buck, Advanced 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, Jrgen 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. ). 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 peopleand to Arthur Wester, Editor of Mathematics at Prentice-Hall, and Wilson Ges at IMPAI extend my sincere thanks.

Rio de Janeiro

Manfredo P. do Carmo

We tried to prepare this book so it could be used in more than one type of differential geometry course. Each chapter starts with an introduction that describes the material in the chapter and explains how this material will be used later in the book. For the readers convenience, we have used footnotes to point out the sections (or parts thereof) that can be omitted on a first reading.

Although there is enough material in the book for a full-year course (or a topics course), we tried to make the book suitable for a first course on differential geometry for students with some background in linear algebra and advanced calculus.

For a short one-quarter course (10 weeks), we suggest the use of the following material: (up to the local Gauss-Bonnet theorem; include applications (b) and (f))3 weeks.

The 10-week program above is on a pretty tight schedule. A more relaxed alternative is to allow more time for the first three chapters and to present survey lectures, on the last week of the course, on geodesics, the Gauss theorema egregium, and the Gauss-Bonnet theorem (geodesics can then be defined as curves whose osculating planes contain the normals to the surface).

).

Please also note that an asterisk attached to an exercise does not mean the exercise is either easy or hard. It only means that a solution or hint is provided at the end of the book. Second, we have used for parametrization a bold-faced x and that might become clumsy when writing on the blackboard. Thus we have reserved the capital X as a suggested replacement.

Where letter symbols that would normally be italic appear in italic context, the letter symbols are set in roman. This has been done to distinguish these symbols from the surrounding text.

The differential geometry of curves and surfaces has two aspects. One, which may be called classical differential geometry, started with the beginnings of calculus. Roughly speaking, classical differential geometry is the study of local properties of curves and surfaces. By local properties we mean those properties which depend only on the behavior of the curve or surface in the neighborhood of a point. The methods which have shown themselves to be adequate in the study of such properties are the methods of differential calculus. Because of this, the curves and surfaces considered in differential geometry will be defined by functions which can be differentiated a certain number of times.

The other aspect is the so-called global differential geometry. Here one studies the influence of the local properties on the behavior of the entire curve or surface. We shall come back to this aspect of differential geometry later in the book.

Perhaps the most interesting and representative part of classical differential geometry is the study of surfaces. However, some local properties of curves appear naturally while studying surfaces. We shall therefore use this first chapter for a brief treatment of curves.