Differential Geometry of Curves and Surfaces

by Manfredo P. Do Carmo

One of the most widely used texts in its field, this volume introduces the differential geometry of curves and surfaces in both local and global aspects. The presentation departs from the traditional approach with its more extensive use of elementary linear algebra and its emphasis on basic geometrical facts rather than machinery or random details. Many examples and exercises enhance the clear, well-written exposition, along with hints and answers to some of the problems.
The treatment begins with a chapter on curves, followed by explorations of regular surfaces, the geometry of the Gauss map, the intrinsic geometry of surfaces, and global differential geometry. Suitable for advanced undergraduates and graduate students of mathematics, this text's prerequisites include an undergraduate course in linear algebra and some familiarity with the calculus of several variables. For this second edition, the author has corrected, revised, and updated the entire volume.

Genres: Mathematics, Textbooks, Geometry, Science, Physics, Technical

851 pages, Kindle Edition

First published February 11, 1976

About the author

Manfredo Perdigão do Carmo (15 August 1928, Maceió – 30 April 2018, Rio de Janeiro) was a Brazilian mathematician. He spent most of his career at IMPA and is seen as the doyen of differential geometry in Brazil.

Reviews

[Mark Moon · Rating 2 out of 5]

This book features a rather nice collection of examples of interesting curves and surfaces. The treatment is rather old-fashioned; manifolds are always embedded in a euclidean space until the very end. It would be an ok book if not for the high frequency of confusing typos.

[Robert]

Took an undergraduate differential geometry course (M435) out of this book at Indiana University. Very clear introduction, but be warned: many of the problems and examples contain dangerously subtle typos. Many times I would be stuck on a problem for hours only to realize that there was an error in the description of the problem.

[Kai · Rating 5 out of 5]

quality textbook would recommend to a friend

[Tpinetz · Rating 4 out of 5]

I did a course on differential geometry and read this book as a guide and it worked well for that. A lot of additional exercises are included and it's not hard to follow along. However, I think its a shame that manifold theory is not included at all.

[C. Hinsley · Rating 3 out of 5]

Dr. Theodore Shifrin's introductory differential geometry text is better (more concise, prettier notation, less errata), but this book is good as a standalone or as supplement, covering some things that Shifrin's text does not. This book sacrifices some clarity and gains a bit of comprehensiveness.

[Sabrina · Rating 3 out of 5]

this is a decent differential geometry book.... I wish it had more detail than it did... I've found that sometimes certain chapters felt overly lack luster... this book had potential but it needed so much more than it had..

[Fan Wang · Rating 4 out of 5]

Very clear and detailed

[estrabismo · Rating 4 out of 5]

I went through the 4/5 chapters with auditing a class. The appendices made the book pretty self-contained and much appreciated. I liked the presentation of definitions, propositions, and proofs. I think the perspective was more or less friendly to people who have taken math classes geared towards science students (vector calc + linear algebra) with mild theory, although I have not read other textbooks on the same subject. As companion texts, I used Mary Boas's 'Mathematical Methods in the Physical Sciences,' for technical refreshing and David Hilbert's 'Geometry & the Imagination.' The latter is leisurely and gives intuition and visualization which is helpful, although topics are introduced in very different order than do Carmo. I also must give much credit to the professor who taught the course. She was fantastic.

[Gabriel Ferreira · Rating 4 out of 5]

Although I tried other textbooks and could not find one better than this, I wish this book (and the other ones I tried) had more explanations, including explanations on the intuition and solved exercises. Since I can't point a better book I am giving it 4 stars, instead of 3.
PS: maybe Tristan Needham book "Visual Differential Geometry and Forms" (I think it will be released in 2021) could complement this one nicely.

[DJ]

good advanced text for after Pressley

[Mark Hoyle · Rating 5 out of 5]

Excellent treatise on curves and surfaces with very clear exposition of the motivation behind many concepts in Riemannian Geometry.