Ebook: Hyperbolic Geometry

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, M?bius transformations, the general M?bius group, and their subgroups preserving the upper half-plane, hyperbolic arc-length and distance as quantities invariant under these subgroups, the Poincar? 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.

 

 

 

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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, M?bius transformations, the general M?bius group, and their subgroups preserving the upper half-plane, hyperbolic arc-length and distance as quantities invariant under these subgroups, the Poincar? 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.

 

 

 

Content:
Front Matter....Pages i-ix
The Basic Spaces....Pages 1-18
The General M?bius Group....Pages 19-56
Length and Distance in ?....Pages 57-94
Other Models of the Hyperbolic Plane....Pages 95-110
Convexity, Area, and Trigonometry....Pages 111-151
Groups Acting on ?....Pages 153-178
Back Matter....Pages 179-230

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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, M?bius transformations, the general M?bius group, and their subgroups preserving the upper half-plane, hyperbolic arc-length and distance as quantities invariant under these subgroups, the Poincar? 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.

 

 

 

Content:
Front Matter....Pages i-ix
The Basic Spaces....Pages 1-18
The General M?bius Group....Pages 19-56
Length and Distance in ?....Pages 57-94
Other Models of the Hyperbolic Plane....Pages 95-110
Convexity, Area, and Trigonometry....Pages 111-151
Groups Acting on ?....Pages 153-178
Back Matter....Pages 179-230
....