Thoroughly revised and updated
Features new material on important topics such as hyperbolic geometry in higher dimensions and generalizations of hyperbolicity
The only genuinely introductory textbook devoted to this topic: it is self-contained and assumes very few prerequisites
Includes full solutions for all exercises – the only book on the subject to do so
Thoroughly updated, featuring new material on important topics such as hyperbolic geometry in higher dimensions and generalizations of hyperbolicity
Includes full solutions for all exercises
Successful first edition sold over 800 copies in North America
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This introductory text explores and develops the basic notions of geometry on the hyperbolic plane. 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 Poincar disc model, convex subsets of the hyperbolic plane, and the Gauss-Bonnet formula for the area of a hyperbolic polygon and its applications. Coverage provides readers with a firm grasp of the concepts and techniques of this beautiful area of mathematics.
The Basic Spaces.- The General Möbius Group.- Length and Distance in ?.- Planar Models of the Hyperbolic Plane.- Convexity, Area, and Trigonometry.- Nonplanar models.
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.
Thoroughly updated, featuring new material on important topics such as hyperbolic geometry in higher dimensions and generalizations of hyperbolicity
Includes full solutions for all exercises
Successful first edition sold over 800 copies in North America
Inhaltsverzeichnis
The Basic Spaces.- The General Möbius Group.- Length and Distance in ?.- Planar Models of the Hyperbolic Plane.- Convexity, Area, and Trigonometry.- Nonplanar models.
Klappentext
Thoroughly updated, featuring new material on important topics such as hyperbolic geometry in higher dimensions and generalizations of hyperbolicity
Includes full solutions for all exercises
Successful first edition sold over 800 copies in North America
This introductory text explores and develops the basic notions of geometry on the hyperbolic plane. 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 Poincar disc model, convex subsets of the hyperbolic plane, and the Gauss-Bonnet formula for the area of a hyperbolic polygon and its applications. Coverage provides readers with a firm grasp of the concepts and techniques of this beautiful area of mathematics.
