Gift Guide

Differential Geometry of Curves and Surfaces: A Concise Guide / Edition 1

Paperback (Print)
Buy New
Buy New from
Used and New from Other Sellers
Used and New from Other Sellers
from $50.26
Usually ships in 1-2 business days
(Save 28%)
Other sellers (Paperback)
  • All (9) from $50.26   
  • New (7) from $50.26   
  • Used (2) from $88.79   


The study of curves and surfaces forms an important part of classical differential geometry. Differential Geometry of Curves and Surfaces: A Concise Guide presents traditional material in this field along with important ideas of Riemannian geometry. The reader is introduced to curves, then to surfaces, and finally to more complex topics. Standard theoretical material is combined with more difficult theorems and complex problems, while maintaining a clear distinction between the two levels.

Key topics and features:

* Covers central concepts including curves, surfaces, geodesics, and intrinsic geometry

* Substantive material on the Aleksandrov global angle comparison theorem, which the author generalized for Riemannian manifolds (a result now known as the celebrated Toponogov Comparison Theorem, one of the cornerstones of modern Riemannian geometry)

* Contains many nontrivial and original problems, some with hints and solutions

This rigorous exposition, with well-motivated topics, is ideal for advanced undergraduate and first-year graduate students seeking to enter the fascinating world of geometry.

Read More Show Less

Editorial Reviews

From the Publisher
From the reviews:

"This book by the late author covers … the subjects which are normally taught in a course on the differential geometry of curves and surfaces. … It can be recommended for first-year graduate students and also for use in the classroom. … the book is rich in geometry and concrete examples. … the book is very welcome since it is an original contribution in various aspects and gives a number of geometric insights … . Numerous illustrations make the reading enjoyable." (Wolfgang Kühnel, Mathematical Reviews, Issue 2006 m)

"Toponogov’s ‘concise guide’ to elementary differential geometry has the potential to be a useful reference and/or review book … ."(Fernando Q. Gouvèa, MathDL, March, 2006)

“This book, written by one of the Russian masters, offers a comprehensive introduction to classical differential geometry of curves and surfaces. Only a basic background in analysis and linear algebra is needed to follow the presentation. … A distinctive feature of the book is that, apart from a large number of exercises, it also contains many original problems with outlines of solutions.” (A. Cap, Monatshefte für Mathematik, Vol. 155 (1), September, 2008)

Read More Show Less

Product Details

  • ISBN-13: 9780817643843
  • Publisher: Birkhauser Verlag
  • Publication date: 12/5/2005
  • Edition description: 2006
  • Edition number: 1
  • Pages: 206
  • Sales rank: 1,261,873
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.47 (d)

Table of Contents

Chapter 1 Curves in a 3-dimensional Euclidean space and in the plane: Preliminaries.- Definition and methods of curves presentation.- Tangent line and an osculating plane.- Length of a curve.- Problems: plane convex curves.- Curvature of a curve.- Problems: curvature of plance curves.- Torsion of a curve.- Frenet formulas and the natural equation of a curve.- Problems: space curves- Phase length of a curve and Fenchel-Reshetnyak inequality.- Exercises Chapter 2 Extrinsic geometry of surfaces in a 3-dimensional Euclidean space.- Definition and methods of generating surfaces.- Tangent plane.- First fundamental form of a surface.- Second fundamental form of a surface.- The third fundamental form of a surface.- Classes of surfaces.- Some classes of curves on a surface.- The main equations of the surfaces theory.- Appendix: Indicatrix of a surface of revolution.- Exercises Chapter 3 Intrinsic geometry of surfaces.- Introducing notions.-Covariant derivative of a vector field.- Parallel translation of a vector along a curve on a surface.- Geodesics.- Shortest paths and geodesics.- Special coordinate system.- Gauss-Bonet theorem and comparison theorem for the angles of a triangle.- Local comparison theorems for triangle.- Alexandrov comparison theorem for the angles of a triangle.- Problems.- Bibliography.- Index

Read More Show Less

Customer Reviews

Be the first to write a review
( 0 )
Rating Distribution

5 Star


4 Star


3 Star


2 Star


1 Star


Your Rating:

Your Name: Create a Pen Name or

Barnes & Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at or to post customer reviews. Please see our Terms of Use for more details.

What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

Reviews should not contain any of the following:

  • - HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
  • - Time-sensitive information such as tour dates, signings, lectures, etc.
  • - Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
  • - Comments focusing on the author or that may ruin the ending for others
  • - Phone numbers, addresses, URLs
  • - Pricing and availability information or alternative ordering information
  • - Advertisements or commercial solicitation


  • - By submitting a review, you grant to Barnes & and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Terms of Use.
  • - Barnes & reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & also reserves the right to remove any review at any time without notice.
  • - See Terms of Use for other conditions and disclaimers.
Search for Products You'd Like to Recommend

Recommend other products that relate to your review. Just search for them below and share!

Create a Pen Name

Your Pen Name is your unique identity on It will appear on the reviews you write and other website activities. Your Pen Name cannot be edited, changed or deleted once submitted.

Your Pen Name can be any combination of alphanumeric characters (plus - and _), and must be at least two characters long.

Continue Anonymously

    If you find inappropriate content, please report it to Barnes & Noble
    Why is this product inappropriate?
    Comments (optional)