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Learn more about Scribd Membership Home. Much more than documents. Discover everything Scribd has to offer, including books and audiobooks from major publishers. Start Free Trial Cancel anytime. Differential Geometry of Curves and Surfaces - M. Do Carmo. Uploaded by Anton Perkov. Date uploaded May 08, Did you find this document useful? Is this content inappropriate? Report this Document. Flag for Inappropriate Content. Download Now. For Later. Related titles. Carousel Previous Carousel Next.

Viro, O. Ivanov, N. Netsvetaev And v. Gravitation, Gauge Theories and Differential Geometry. Geometry of Four-Manifolds, The - S. Milnor J. Topology From the Differentiable Viewpoint Princeton, Jump to Page. Search inside document. Prentice-Hall, Inc. Trches Indes 1, Gomer Dilleeaal 2. Cues 3. 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 fon 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 are Jength to derive some global properties of surfaces. Near the end of Chapter 5 Sec, , 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 mettics To maintain the proper balance between ideas and facts, we have presented a large number of examples that are computed in detail.

Further- more, a reasonable supply of exercises is provided. Some factual material of classical differential geometry found its place in these exercises. From calculus, a certain familiarity with calculus of several variables including the state- ment of the implicit function theorem is expected.

A large part of the translation was done by Leny Cavaleante, Tam 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 pro- grams for some beautiful drawings that appear in the book Figs, , , , , , and , 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. Rio de Jansiro Manfredo P. 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 ential 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 3: Secs, and —2 weeks.

Chapter 4: Secs. A more re- Jaxed 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 geodesies can then be defined as curves whose osculating planes contain the normals to the suriace. Second, we have used for parametrization a bold-faced x and that might become clumsy when writing on the black- board. Thus we have reserved the capital X as a suggested replacement.

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

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. For those wishing to go a bit further on the subject of curves, we have included Sees.

Our goal is to characterize certain subsets of? A natural way of defining such subsets is through different able functions. We say that a real function of a real variable is ciferentiable or smooth if thas, at all points, derivatives ofall orders which are automa- tically continuous.

A first definition of curve, not entirely satisfactory but sufficient for the purposes of this chapter, is the following. The variable fis called the parameter of the curve. The image set a 7 R? Notice that, 0. Example 5. The two distinct parametrized curves. Notice that the velocity vector of the second curve is the double of the first one Fig. Leto: 0, 2 — R? The trace of is called the sracir'x Fig, The clssold of Dioces. Figure 1.

The tracts, 1. Take the curve with the opposite orientation. They will be found useful in our later study of curves and surfaces. Itis convenient to begin by reviewing the notion of orientation of a vector space. Since the determinant of a change of basis is either positive or negative, there are only two such classes. Therefore, V has two orientations, and if we fix one of them arbitrarily, the other one is called the opposite orientation.

We aso say that a given ordered basis of R? It i immediate from the definition that 1, Jems es. Itis also very frequent to write w A vasu x wand refer to itas the cross product. The following properties can easily be checked actually they just express the usual properties of determinants : 1. In fact, we have the following. It follows im- mediately from Eq.

We shall review some of this material in the following exercises. Check whether the following bases are positive: a. A plane P contained in R? This is called the oriented area in R?.

Is this vector u uniquely deter- mined? If not, what is the most general solution? The Local Theory of Curves Parametrized by Arc Length This section contains the main results of curves which will be used in the later parts of the book.

This shows that f has the same trace asa and is parametrized by arc length. It is usual to say that f isa reparametriza- tion of aX by are length. This fact allows us to extend all local concepts previously defined to regular curves with an arbitrary parameter.

Thus, we say that the curvature oof a: I R? This is clearly independent of the choice of and shows that the restriction, made at the end of Sec.

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## Differential Geometry of Curves and Surfaces : Second Edition

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. Twice a Guggenheim Fellow and the winner of many prestigious national and international awards, the professor at the institute of Pure and Applied Mathematics in Rio de Janeiro is well known as the author of influential textbooks such as Differential Geometry of Curves and Surfaces. The area of differential geometry is the main focus of this selection, though it also contains do Carmo's own commentaries on his life as a scientist as well as assessment of the impact of his researches and a complete list of his publications. Aspects covered in the featured papers include relations between curvature and topology, convexity and rigidity, minimal surfaces, and conformal immersions, among others.

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## Differential Geometry of Curves and Surfaces - M.P. Do Carmo

The completed exams should be returned at the beginning of class the following Thursday, Oct. Envelopes containing the exam are also outside of HB You should work on the exam for 2 continuous hours. You may use books, notes, etc. The completed exams should be returned to my office HB or the Math.