Hyperbolic Geometry (Springer Undergraduate Mathematics Series) [Paperback]

James W Anderson (Author)

Book Description

ISBN-10: 1852331569 | ISBN-13: 978-1852331566 | Publication Date: October 18, 1999 | Edition: 1

The geometry of the hyperbolic plane has been an active and fascinating field of mathematical inquiry for most of the past two centuries. This book provides a self-contained introduction to the subject, taking the approach that hyperbolic geometry consists of the study of those quantities invariant under the action of a natural group of transformations. Topics covered include the upper half-space model of the hyperbolic plane, Möbius transformations, the general Möbius group and the subgroup preserving path length in the upper half-space model, arc-length and distance, the Poincaré disc model, convex subsets of the hyperbolic plane, and the Gauss-Bonnet formula for the area of a hyperbolic polygon and its applications. The style and level of the book, which assumes few mathematical prerequisites, make it an ideal introduction to this subject, providing the reader with a firm grasp of the concepts and techniques of this beautiful area of mathematics.

Editorial Reviews

From the Back Cover

The geometry of the hyperbolic plane has been an active and fascinating field of mathematical inquiry for most of the past two centuries. This book provides a self-contained introduction to the subject, suitable for third or fourth year undergraduates. The basic approach taken is to define hyperbolic lines and develop a natural group of transformations preserving hyperbolic lines, and then study hyperbolic geometry as those quantities invariant under this group of transformations. Topics covered include the upper half-plane model of the hyperbolic plane, Möbius transformations, the general Möbius group, and their subgroups preserving the upper half-plane, hyperbolic arc-length and distance as quantities invariant under these subgroups, the Poincaré disc model, convex subsets of the hyperbolic plane, hyperbolic area, the Gauss-Bonnet formula and its applications. This updated second edition also features: an expanded discussion of planar models of the hyperbolic plane arising from complex analysis; the hyperboloid model of the hyperbolic plane; brief discussion of generalizations to higher dimensions; many new exercises. The style and level of the book, which assumes few mathematical prerequisites, make it an ideal introduction to this subject and provides the reader with a firm grasp of the concepts and techniques of this beautiful part of the mathematical landscape.       --This text refers to an alternate Paperback edition.

Product Details

• Paperback: 230 pages
• Publisher: Springer; 1 edition (October 18, 1999)
• Language: English
• ISBN-10: 1852331569
• ISBN-13: 978-1852331566
• Product Dimensions: 9.2 x 6.7 x 0.6 inches
• Shipping Weight: 12.8 ounces
• Average Customer Review: 5.0 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #3,138,588 in Books (See Top 100 in Books)

More About the Author

Discover books, learn about writers, read author blogs, and more.

Customer Reviews

Most Helpful Customer Reviews

10 of 10 people found the following review helpful:

5.0 out of 5 stars Excellent book, August 22, 2007

By 

mathematician - See all my reviews

This review is from: Hyperbolic Geometry (Springer Undergraduate Mathematics Series) (Paperback)

This is an excellent introduction to hyperbolic geometry. It assumes knowledge of euclidean geometry, trigonometry, basic complex analysis, basic abstract algebra, and basic point set topology. That material is very well presented, and the exercises shed more light on what is being discussed. Plus, solutions to all the exercises are at the end of the book.

8 of 8 people found the following review helpful:

5.0 out of 5 stars Very good introduction, August 3, 2007

By 

Justin Curry - See all my reviews
(REAL NAME)   

This review is from: Hyperbolic Geometry (Springer Undergraduate Mathematics Series) (Paperback)

I used this text along with Tristan Needham's "Visual Complex Analysis" to get a full dose of the geometric beauty inherent in studying complex variables. I found it to be a nice complement to the second year course in geometry at Cambridge University. Anderson does a wonderful job of working out in detail lots of examples so that you can get the algorithmic practice of solving problems. However this is not merely a cookbook. Rather, core elements of the theory are presented from the ground up, with plenty of time spent on understanding the group structure of Mobius transformations in various settings. Disc and upper-half plane models are treated as well as more general models. I recommend you buy both this book and Needham's if you want to appreciate the world of complex numbers.

11 of 14 people found the following review helpful:

5.0 out of 5 stars great book, January 28, 2004

By A Customer

This review is from: Hyperbolic Geometry (Springer Undergraduate Mathematics Series) (Paperback)

this is a really great introduction to hyperbolic geometry. especially if you want to study gammas acting on the upper half plane. it starts at a much lower level then any other text.