Differential Geometry of Curves and Surfaces

Differential Geometry of Curves and Surfaces

(Parte 1 de 5)

pifferential Geometry of Curves and Surfaces

Mantredo P. do ~rmo Instituto de Matematica Pura e Aplicada (IMP:4)

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~ UP COLLEGE OF SCIENCE 4· ilililililllll ~ UDSCB0013255

Prentice-Ha~lnc., Englewood Cliffs, New Jersey

Carmo, Manfredo Perdigao do. Differential geometry of curves and surfaces

"A free translation, with additional material, of a book and a set of notes, both published originally in Portuguese. " Bibliography: p.

I neludes index. 1. Geometry, Differential. 2. Curves. 3. Surfaces.

T. Title QA641.C33 ISBN: 0-13-212589-7

To Leny

© 1976 -by Prentice-Hall, Inc. Englewood Cliffs, New Jersey

All rights reserved. No part of this book may be reproduced in any form, Or by any means, without permission in writing from the pUblisher

Current printing: 10 9 8 7 6 5 4 3 2

Printed in the United States of America

Prentice-Hall International, Inc., London Prentice-Hall of Australia Pty., Limited, Sydney Prentice-Hall of Canada, Ltd., Toronto Prentice-Hall of India Private Ltd., New Delhi

Prentice-Hall of Japan, Inc., Tokyo Prentice-Hall of Southeast Asia Private Limited, Singapore


Preface v

Some Remarks on Using this Book vii

1. Curves 1

1-1 Introduction 1 1-2 Parametrized Curves 2 1-3 Regular Curves; Arc Length 5 1-4 The Vector Product in R3 1

1-5 The Local Theory of Curves Parametrized by Arc Length 16 1-6 The Local Canonical Form 27 1-7 Global Properties of Plane Curves 30

2. Regular Surfaces 51

2-1 Introduction 51 2-2 Regular Surfaces; Inverse Images of Regular Values 52 2-3 Change of Parameters; Differential Functions on Surfaces 69 2-4 The Tangent Plane; the Differential of a Map 83 2-5 The First Fundamental Form; Area 92 2-6 Orientation of Surfaces 102 2-7 A Characterizat;on of Compact Orientable Surfaces 109 2-8 A Geometric Definition of Area 114

Appendix: A Brief Review on Continuity and Differenfiability 118 iv Contents

3. The Geometry of the Gauss Map 134

3-1 Introduction 134

3-2 The Definition of the Gauss Map and

Its Fundamental Properties 135 3-3 The Gauss Map in Local Coordinates 153 3-4 Vector Fields 175 3-5 Ruled Surfaces and Minimal Surfaces 188 Appendix: Self-Adjoint Linear Maps and Quadratic Forms 214

4. The Intrinsic Geometry of Surfaces 217

4-1 Introduction 217 4-2 Isometries; Conformal Maps 218 4-3 The Gauss Theorem and the Equations of Compatibility 231 4-4 Parallel Transport; Geodesics 238 4-5 The Gauss-Bonnet Theorem and its Applications 264

4-6 The Exponential Map. Geodesic Polar Coordinates 283

4-7 Further Properties of Geodesics. Convex Neighborhoods 298

Appendix: Proofs of the Fundamental Theorems of The Local Theory of Curves and Surfaces 309

5. Global Differential Geometry 315

5-1 Introduction 315 5-2 The Rigidity of the Sphere 317 5-3 Complete Surfaces. Theorem of Hopf-Rinow 325 5-4 First and Second Variations of the Arc Length;

Bonnet's Theorem 339 5-5 Jacobi Fields and Conjugate Points 357 5-6 Covering Spaces; the Theorems of Hadamard 371 5-7 Global Theorems for Curves; the Fary-Milnor Theorem 380

5-8 . Surfaces of Zero Gaussian Curvature 408 5-9 Jacobi's Theorems 415 5-10 Abstract Surfaces; Further Generalizations 425 5-1 Hilbert's Theorem 446 Appendix: Point-Set Topology of Euciidean Spaces 456

Bibliography and Comments 471 Hints and Answers to Some Exercises 475

Index 497


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 hasicgeometrical 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 R3; 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 R3. 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 vi Preface 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: Mc Gra w-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 IMP A for their comments and support. In particular, Elan Lima read part of the Portuguese version and made valuable comments.

Robert Gardner, liirgen Kern, Blaine Lawson, and Nolan Wallach read critically the English manuscript and helped me to avoid several mistakes, both in English and Mathematics. Ray Ogawa prepared the computer programs for some beautiful drawings that appear in the book (Figs. 1-3, 1-8, 1-9,1-10,1-1,3-45 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 Arthur Wester, Editor of Mathematics at Prentice-Hall, and Wilson G6es at U"IPA-I extend my sincere thanks.

Rio de Janeiro Manfredo P. do Carmo

Some Remarks on Using This Book

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 reader's 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: Chapter 1: Secs. 1-2, 1-3, 1-4, 1-5 and one topic of Sec. 1-7-2 weeks. Chapter 2: Secs. 2-2 and 2-3 (omit the proofs), Secs. 2-4 and 2-5-3 weeks. Chapter 3: Secs. 3-2 and 3-3-2 weeks. Chapter 4: Secs. 4-2 (omit conformal maps and Exercises 4, 13-18, 20), 4-3 (up to Gauss theorema egregium), 4-4 (up to Prop. 4; omit Exercises 12, 13, 16, 18-21), 4-5 (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).

In a one-semester course, the first alternative could be taught more vii viii some Remarks on Using this Book leisurely and the instructor could probably include additional material (for instance, Secs. 5-2 and 5-10 (partially), or Secs. 4-6, 5-3 and 5-4).

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.

1 Curves

1-1. Introduction

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 differen- tial 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.

The chapter has been organized in such a way that a reader interested mostly in surfaces can read only Secs. 1-2 through 1-5. Sections 1-2 through 1-4 contain essentially introductory material (parametrized curves, arc length, vector product), which will probably be known from other courses and is included here for completeness. Section 1-5 is the heart of the chapter

2 Curves and contains the material of curves needed for the study of surfaces. For those wishing to go a bit further on the subject of curves, we have included Secs. 1-6 and 1-7.

1-2. Parametrized Curves

We denote by R3 the set of triples (x, y, z) of real numbers. Our goal is to characterize certain subsets of R3 (to be called curves) that are, in a certain sense, one-dimensional and to which the methods of differential calculus can be applied. A natural way of defining such subsets is through differentiable functions. We say that a real function of a real variable is differentiable (or smooth) if it has, at all points, derivatives of all orders (which are automatically continuous). A first definition of curve, not entirely satisfactory but sufficient for the purposes of this chapter, is the following.

DEFINITION. A parametrized differentiable curve is a differentiable map rt,: I ---4 R3 of an open interval I = (a, b) of the real line R into R3.t

The word differentiable in this definition means that rt, is a correspondence which maps each t E I into a point rt,(t) = (x(t), yet), z(t)) E R3 in such a way that the functions x(t), yet), z(t) are differentiable. The variable t is called the parameter of the curve. The word interval is taken in a generalized sense, so that we do not exclude the cases a = -0, b = + 0.

If we denote by x'(t) the first derivative of x at the point t and use similar notations for the functions y and z, the vector (x'(t), y'(t), z'(t)) = rt,'(t) E R3 is called the tangent vector (or velocity vector) of the curve rt, at t. The image set rt,(I) c R3 is called the trace of rt,. As illustrated by Example 5 below, one should carefully distinguish a parametrized curve, which is a map, from its trace, which is a subset of R3.

A warning about terminology. Many people use the term "infinitely differentiable" for functions which have derivatives of all orders and reserve the word "differentiable" to mean that only the existence of the first derivative is required. We shall not follow this usage.

Example 1. The parametrized differentiable curve given by rt,(t) = (a cos t, a sin t, bt), t E R, has as its trace in R3 a helix of pitch 21tb on the cylinder x2 + y2 = a2• The parameter t here measures the angle which the x axis makes with the line joining the origin 0 to the projection of the point rt,(t) over the xy plane (see

Fig. 1-1).

tIn italic context, letter symbols will not be italicized so they will be clearly distinguished from the surrounding text.

Parametrized Curves 3

------------~o+-------------x x

Figure 1-1 Figure 1-2

Example 2. The map rt,: R --> R2 given by rt,(t) = (13, t2), t E R, is a parametrized differentiable curve which has Fig. 1-2 as its trace. Notice that rt,'(0) = (0, 0); that is, the velocity vector is zero for t = O.

Example 3. The map rt,: R --> R2 given by rt,(t) = (t3 -4t, t2 -4), t E R, is a parametrized differentiable curve (see Fig. 1-3). Notice that rt,(2) = rt,( -2) = (0, 0); that is, the map rt, is not one-ta-one.

_ ~~------------~.x y


Figure 1-3 Figure 1-4

Example 4. The map rt,: R --> R2 given by rt,(t) = (I, I t I), t E R, is not a parametrized differentiable curve, since I t I is not differentiable at t = 0 (Fig. 1-4).

Example 5. The two distinct parametrized curves rt,(t) = (cos t, sin t), pet) = (cos 2t, sin 2t),

4 Curves where t E (0 -f, 2n + f), f > 0, have the same trace, namely, the circle x2 + y2 = 1. Notice that the velocity vector of the second curve is the double of the first one (Fig. 1-5).

Figure 1-5

We shall now recall briefly some properties of the inner (or dot) product of vectors in R3. Let U = (ul, U2, u3) E R3 and define its norm (or length) by

1 U 1 = ~ui + + ui.

Geometrically, 1 U 1 is the distance from the point (ul, U2, u3) to the origin

° = (0, 0, 0). Now, let u = (ul, u2, u3) and v = (VI> v2, V3) belong to R3, and let (), ° < () < n, be the angle formed by the segments Ou and Ov. The inner product U· v is defined by (Fig. 1-6)

U·v = 1 U 1 v 1 cos ().

.. y x Figure 1-6

The following properties hold:

1. Assume that U and v are nonzero vectors. Then U·V = ° if and only if u is orthogonal to v.

Regular Curves: Arc Length 5

2. U·V = V·U.

3. A(U'V) = AU'V = U·AV. 4. u·(v + w) = u,v + u·w.

A useful expression for the inner product can be obtained as follows.

Let e1 = (1, 0, 0), e2 = (0, 1,0), and e3 = (0, 0, 1). It is easily checked that e/·ej = 1 if i = j and that e/·ej = ° if i j, where i,j = 1,2,3. Thus, by writing and using properties 3 and 4, we obtain

(Parte 1 de 5)