# Manfredo P. Do Carmo Differential Geometry Of Curves And Surfaces Pdf

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- Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition
- Manfredo P. do Carmo – Selected Papers
- MATH 320A: Differential Geometry

In mathematics , the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically , relating to their embedding in Euclidean space and intrinsically , reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature , first studied in depth by Carl Friedrich Gauss , [1] who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces naturally arise as graphs of functions of a pair of variables , and sometimes appear in parametric form or as loci associated to space curves.

## Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition

By Manfredo P. The author has also provided a new Preface for this edition. 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. 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. 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. 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.

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. 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. 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.

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. Chapter 2: Secs. Chapter 3: Secs. Chapter 4: Secs. The 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 leisurely and the instructor could probably include additional material for instance, Secs.

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. The chapter has been organized in such a way that a reader interested mostly in surfaces can read only Secs.

Sections through 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 is the heart of the chapter 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. 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. The variable t is called the parameter of the curve. 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 2. Example 3. Example 4. Notice that the velocity vector of the second curve is the double of the first one Fig.

Assume that u and v are nonzero vectors. A useful expression for the inner product can be obtained as follows. Thus, by writing. For the study of the differential geometry of a curve it is essential that there exists such a tangent line at every point. From now on we shall consider only regular parametrized differentiable curves and, for convenience, shall usually omit the word differentiable.

It can happen that the parameter t is already the arc length measured from some point. To simplify our exposition, we shall restrict ourselves to curves parametrized by arc length; we shall see later see Sec. It is convenient to set still another convention. We say, then, that these two curves differ by a change of orientation. A circular disk of radius 1 in the plane xy rolls without slipping along the x axis.

The figure described by a point of the circumference of the disk is called a cycloid Fig. Compute the arc length of the cycloid corresponding to a complete rotation of the disk. If we rotate r about 0, the point p will describe a curve called the cissoid of Diocles.

By taking 0 A as the x axis and 0 Y as the y axis, prove that. Show that. The length of the segment of the tangent of the tractrix between the point of tangency and the y axis is constantly equal to 1.

Take the curve with the opposite orientation. Thus, the curve in Example 3 of Sec. Draw a sketch of the curve and its tangent vectors. For every partition. The norm P of a partition P is defined as.

Exercise 7. A Nonrectifiable Curve. Straight Lines as Shortest. They will be found useful in our later study of curves and surfaces.

## Manfredo P. do Carmo – Selected Papers

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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. For some years now, I, as well as a number of other contributors to this column, have on occasion expressed appreciation to Dover Publications for the service it provides to the mathematical community by re-issuing classic textbooks and making them available to a new generation at an affordable price. Of late, however, it seems to me based on anecdotal evidence garnered from a highly unscientific survey that not as many departments offer such a course. Yet, there must still be some market for books like this, because several have recently appeared, including a second edition of Differential Geometry of Curves and Surfaces by Banchoff and Lovett and another book with the same title by Kristopher Tapp. Most books with titles like this offer similar content.

## MATH 320A: Differential Geometry

By Manfredo P. The author has also provided a new Preface for this edition. 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.

It seems that you're in Germany. We have a dedicated site for Germany. This volume of selected academic papers demonstrates the significance of the contribution to mathematics made by Manfredo P.

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