Graduate Texts in Mathematics
149
Editorial Board
S. Axler K.A. Ribet
Graduate Texts in Mathematics
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
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TAKEUTI/ZARING. Introduction to
Axiomatic Set Theory. 2nd ed.
OXTOBY. Measure and Category. 2nd ed.
SCHAEFER. Topological Vector Spaces.
2nd ed.
HILTON/STAMMBACH. A Course in
Homological Algebra. 2nd ed.
MAC LANE. Categories for the Working
Mathematician. 2nd ed.
HUGHES/PIPER. Projective Planes.
J.-P. SERRE. A Course in Arithmetic.
TAKEUTI/ZARING. Axiomatic Set Theory.
HUMPHREYS. Introduction to Lie
Algebras and Representation Theory.
COHEN. A Course in Simple Homotopy
Theory.
CONWAY. Functions of One Complex
Variable I. 2nd ed.
BEALS. Advanced Mathematical Analysis.
ANDERSON/FULLER. Rings and
Categories of Modules. 2nd ed.
GOLUBITSKY/GUILLEMIN. Stable
Mappings and Their Singularities.
BERBERIAN. Lectures in Functional
Analysis and Operator Theory.
WINTER. The Structure of Fields.
ROSENBLATT. Random Processes. 2nd ed.
HALMOS. Measure Theory.
HALMOS. A Hilbert Space Problem
Book. 2nd ed.
HUSEMOLLER. Fibre Bundles. 3rd ed.
HUMPHREYS. Linear Algebraic Groups.
BARNES/MACK. An Algebraic
Introduction to Mathematical Logic.
GREUB. Linear Algebra. 4th ed.
HOLMES. Geometric Functional
Analysis and Its Applications.
HEWITT/STROMBERG. Real and Abstract
Analysis.
MANES. Algebraic Theories.
KELLEY. General Topology.
ZARISKI/SAMUEL. Commutative
Algebra. Vol. I.
ZARISKI/SAMUEL. Commutative
Algebra. Vol. II.
JACOBSON. Lectures in Abstract Algebra
I. Basic Concepts.
JACOBSON. Lectures in Abstract Algebra
II. Linear Algebra.
JACOBSON. Lectures in Abstract Algebra
III. Theory of Fields and Galois
Theory.
HIRSCH. Differential Topology.
34 SPITZER. Principles of Random Walk.
2nd ed.
35 ALEXANDER/WERMER. Several Complex
Variables and Banach Algebras. 3rd ed.
36 KELLEY/NAMIOKA et al. Linear
Topological Spaces.
37 MONK. Mathematical Logic.
38 GRAUERT/FRITZSCHE. Several Complex
Variables.
39 ARVESON. An Invitation to C*-Algebras.
40 KEMENY/SNELL/KNAPP. Denumerable
Markov Chains. 2nd ed.
41 APOSTOL. Modular Functions and
Dirichlet Series in Number Theory.
2nd ed.
42 J.-P. SERRE. Linear Representations of
Finite Groups.
43 GILLMAN/JERISON. Rings of
Continuous Functions.
44 KENDIG. Elementary Algebraic
Geometry.
45 LOÈVE. Probability Theory I. 4th ed.
46 LOÈVE. Probability Theory II. 4th ed.
47 MOISE. Geometric Topology in
Dimensions 2 and 3.
48 SACHS/WU. General Relativity for
Mathematicians.
49 GRUENBERG/WEIR. Linear Geometry.
2nd ed.
50 EDWARDS. Fermat's Last Theorem.
51 KLINGENBERG. A Course in Differential
Geometry.
52 HARTSHORNE. Algebraic Geometry.
53 MANIN. A Course in Mathematical Logic.
54 GRAVER/WATKINS. Combinatorics with
Emphasis on the Theory of Graphs.
55 BROWN/PEARCY. Introduction to
Operator Theory I: Elements of
Functional Analysis.
56 MASSEY. Algebraic Topology: An
Introduction.
57 CROWELL/FOX. Introduction to Knot
Theory.
58 KOBLITZ. p-adic Numbers, p-adic
Analysis, and Zeta-Functions. 2nd ed.
59 LANG. Cyclotomic Fields.
60 ARNOLD. Mathematical Methods in
Classical Mechanics. 2nd ed.
61 WHITEHEAD. Elements of Homotopy
Theory.
62 KARGAPOLOV/MERIZJAKOV.
Fundamentals of the Theory of Groups.
63 BOLLOBAS. Graph Theory.
(continued after index)
John G. Ratcliffe
Foundations of Hyperbolic
Manifolds
Second Edition
John G. Ratcliffe
Department of Mathematics
Stevenson Center 1326
Vanderbilt University
Nashville, Tennessee 37240
Editorial Board:
S. Axler
Department of Mathematics
San Francisco State University
San Francisco, CA 94132
USA
K.A. Ribet
Department of Mathematics
University of California, Berkeley
Berkeley, CA 94720-3840
USA
Mathematics Subject Classification (2000): 57M50, 30F40, 51M10, 20H10
Library of Congress Control Number: 2006926460
ISBN-10: 0-387-33197-2
ISBN-13: 978-0387-33197-3
Printed on acid-free paper.
© 2006 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street,
New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly
analysis. Use in connection with any form of information storage and retrieval, electronic
adaptation, computer software, or by similar or dissimilar methodology now known or hereafter
developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not they
are subject to proprietary rights.
Printed in the United States of America.
9 8 7 6 5 4 3 2 1
springer.com
(MVY)
To Susan, Kimberly, and Thomas
Preface to the First Edition
This book is an exposition of the theoretical foundations of hyperbolic
manifolds. It is intended to be used both as a textbook and as a reference.
Particular emphasis has been placed on readability and completeness of argument. The treatment of the material is for the most part elementary and
self-contained. The reader is assumed to have a basic knowledge of algebra
and topology at the first-year graduate level of an American university.
The book is divided into three parts. The first part, consisting of Chapters 1-7, is concerned with hyperbolic geometry and basic properties of
discrete groups of isometries of hyperbolic space. The main results are the
existence theorem for discrete reflection groups, the Bieberbach theorems,
and Selberg’s lemma. The second part, consisting of Chapters 8-12, is devoted to the theory of hyperbolic manifolds. The main results are Mostow’s
rigidity theorem and the determination of the structure of geometrically
finite hyperbolic manifolds. The third part, consisting of Chapter 13, integrates the first two parts in a development of the theory of hyperbolic
orbifolds. The main results are the construction of the universal orbifold
covering space and Poincar´e’s fundamental polyhedron theorem.
This book was written as a textbook for a one-year course. Chapters
1-7 can be covered in one semester, and selected topics from Chapters 812 can be covered in the second semester. For a one-semester course on
hyperbolic manifolds, the first two sections of Chapter 1 and selected topics
from Chapters 8-12 are recommended. Since complete arguments are given
in the text, the instructor should try to cover the material as quickly as
possible by summarizing the basic ideas and drawing lots of pictures. If all
the details are covered, there is probably enough material in this book for
a two-year sequence of courses.
There are over 500 exercises in this book which should be read as part of
the text. These exercises range in difficulty from elementary to moderately
difficult, with the more difficult ones occurring toward the end of each set
of exercises. There is much to be gained by working on these exercises.
An honest effort has been made to give references to the original published sources of the material in this book. Most of these original papers
are well worth reading. The references are collected at the end of each
chapter in the section on historical notes.
This book is a complete revision of my lecture notes for a one-year course
on hyperbolic manifolds that I gave at the University of Illinois during 1984.
vii
viii
Preface to the First Edition
I wish to express my gratitude to:
(1) James Cannon for allowing me to attend his course on Kleinian
groups at the University of Wisconsin during the fall of 1980;
(2) William Thurston for allowing me to attend his course on hyperbolic
3-manifolds at Princeton University during the academic year 1981-82 and
for allowing me to include his unpublished material on hyperbolic Dehn
surgery in Chapter 10;
(3) my colleagues at the University of Illinois who attended my course
on hyperbolic manifolds, Kenneth Appel, Richard Bishop, Robert Craggs,
George Francis, Mary-Elizabeth Hamstrom, and Joseph Rotman, for their
many valuable comments and observations;
(4) my colleagues at Vanderbilt University who attended my ongoing
seminar on hyperbolic geometry over the last seven years, Mark Baker,
Bruce Hughes, Christine Kinsey, Michael Mihalik, Efstratios Prassidis,
Barry Spieler, and Steven Tschantz, for their many valuable observations
and suggestions;
(5) my colleagues and friends, William Abikoff, Colin Adams, Boris
Apanasov, Richard Arenstorf, William Harvey, Linda Keen, Ruth Kellerhals, Victor Klee, Bernard Maskit, Hans Munkholm, Walter Neumann,
Alan Reid, Robert Riley, Richard Skora, John Stillwell, Perry Susskind,
and Jeffrey Weeks, for their helpful conversations and correspondence;
(6) the library staff at Vanderbilt University for helping me find the
references for this book;
(7) Ruby Moore for typing up my manuscript;
(8) the editorial staff at Springer-Verlag New York for the careful editing
of this book.
I especially wish to thank my colleague, Steven Tschantz, for helping
me prepare this book on my computer and for drawing most of the 3dimensional figures on his computer.
Finally, I would like to encourage the reader to send me your comments
and corrections concerning the text, exercises, and historical notes.
Nashville, June, 1994
John G. Ratcliffe
Preface to the Second Edition
The second edition is a thorough revision of the first edition that embodies
hundreds of changes, corrections, and additions, including over sixty new
lemmas, theorems, and corollaries. The following theorems are new in the
second edition: 1.4.1, 3.1.1, 4.7.3, 6.3.14, 6.5.14, 6.5.15, 6.7.3, 7.2.2, 7.2.3,
7.2.4, 7.3.1, 7.4.1, 7.4.2, 10.4.1, 10.4.2, 10.4.5, 10.5.3, 11.3.1, 11.3.2, 11.3.3,
11.3.4, 11.5.1, 11.5.2, 11.5.3, 11.5.4, 11.5.5, 12.1.4, 12.1.5, 12.2.6, 12.3.5,
12.5.5, 12.7.8, 13.2.6, 13.4.1. It is important to note that the numbering
of lemmas, theorems, corollaries, formulas, figures, examples, and exercises
may have changed from the numbering in the first edition.
The following are the major changes in the second edition. Section 6.3,
Convex Polyhedra, of the first edition has been reorganized into two sections, §6.3, Convex Polyhedra, and §6.4, Geometry of Convex Polyhedra.
Section 6.5, Polytopes, has been enlarged with a more thorough discussion
of regular polytopes. Section 7.2, Simplex Reflection Groups, has been
expanded to give a complete classification of the Gram matrices of spherical, Euclidean, and hyperbolic n-simplices. Section 7.4, The Volume of a
Simplex, is a new section in which a derivation of Schlă
ais dierential formula is presented. Section 10.4, Hyperbolic Volume, has been expanded to
include the computation of the volume of a compact orthotetrahedron. Section 11.3, The Gauss-Bonnet Theorem, is a new section in which a proof
of the n-dimensional Gauss-Bonnet theorem is presented. Section 11.5,
Differential Forms, is a new section in which the volume form of a closed
orientable hyperbolic space-form is derived. Section 12.1, Limit Sets of Discrete Groups, of the first edition has been enhanced and subdivided into
two sections, §12.1, Limit Sets, and §12.2, Limit Sets of Discrete Groups.
The exercises have been thoroughly reworked, pruned, and upgraded.
There are over a hundred new exercises. Solutions to all the exercises in
the second edition will be made available in a solution manual.
Finally, I wish to express my gratitude to everyone that sent me corrections and suggestions for improvements. I especially wish to thank Keith
Conrad, Hans-Christoph Im Hof, Peter Landweber, Tim Marshall, Mark
Meyerson, Igor Mineyev, and Kim Ruane for their suggestions.
John G. Ratcliffe
Nashville, November, 2005
ix
Contents
Preface to the First Edition
vii
Preface to the Second Edition
ix
1 Euclidean Geometry
§1.1. Euclid’s Parallel Postulate . . . . . . .
§1.2. Independence of the Parallel Postulate
§1.3. Euclidean n-Space . . . . . . . . . . . .
§1.4. Geodesics . . . . . . . . . . . . . . . .
§1.5. Arc Length . . . . . . . . . . . . . . . .
§1.6. Historical Notes . . . . . . . . . . . . .
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2 Spherical Geometry
§2.1. Spherical n-Space . . .
§2.2. Elliptic n-Space . . . .
§2.3. Spherical Arc Length .
§2.4. Spherical Volume . . .
§2.5. Spherical Trigonometry
§2.6. Historical Notes . . . .
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35
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41
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47
52
3 Hyperbolic Geometry
§3.1. Lorentzian n-Space . . .
§3.2. Hyperbolic n-Space . . .
§3.3. Hyperbolic Arc Length .
§3.4. Hyperbolic Volume . . .
§3.5. Hyperbolic Trigonometry
§3.6. Historical Notes . . . . .
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54
54
61
73
77
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98
4 Inversive Geometry
§4.1. Reflections . . . . . . . . . . . . .
§4.2. Stereographic Projection . . . . .
Đ4.3. Mă
obius Transformations . . . . .
Đ4.4. Poincar´e Extension . . . . . . . .
§4.5. The Conformal Ball Model . . . .
§4.6. The Upper Half-Space Model . .
§4.7. Classification of Transformations
§4.8. Historical Notes . . . . . . . . . .
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100
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107
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131
136
142
x
Contents
xi
5 Isometries of Hyperbolic Space
§5.1. Topological Groups . . . . . .
§5.2. Groups of Isometries . . . . .
§5.3. Discrete Groups . . . . . . . .
§5.4. Discrete Euclidean Groups . .
§5.5. Elementary Groups . . . . . .
§5.6. Historical Notes . . . . . . . .
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144
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185
6 Geometry of Discrete Groups
§6.1. The Projective Disk Model . . .
§6.2. Convex Sets . . . . . . . . . . .
§6.3. Convex Polyhedra . . . . . . . .
§6.4. Geometry of Convex Polyhedra
§6.5. Polytopes . . . . . . . . . . . .
§6.6. Fundamental Domains . . . . .
§6.7. Convex Fundamental Polyhedra
§6.8. Tessellations . . . . . . . . . . .
§6.9. Historical Notes . . . . . . . . .
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188
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261
7 Classical Discrete Groups
§7.1. Reflection Groups . . . . . . . . . . . .
§7.2. Simplex Reflection Groups . . . . . . .
§7.3. Generalized Simplex Reflection Groups
§7.4. The Volume of a Simplex . . . . . . . .
§7.5. Crystallographic Groups . . . . . . . .
§7.6. Torsion-Free Linear Groups . . . . . .
§7.7. Historical Notes . . . . . . . . . . . . .
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263
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332
8 Geometric Manifolds
§8.1. Geometric Spaces . . . . .
§8.2. Clifford-Klein Space-Forms
§8.3. (X, G)-Manifolds . . . . .
§8.4. Developing . . . . . . . . .
§8.5. Completeness . . . . . . .
§8.6. Curvature . . . . . . . . .
§8.7. Historical Notes . . . . . .
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334
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9 Geometric Surfaces
§9.1. Compact Surfaces . . . . . .
§9.2. Gluing Surfaces . . . . . . .
§9.3. The Gauss-Bonnet Theorem
§9.4. Moduli Spaces . . . . . . . .
§9.5. Closed Euclidean Surfaces .
§9.6. Closed Geodesics . . . . . .
§9.7. Closed Hyperbolic Surfaces .
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xii
Contents
§9.8. Hyperbolic Surfaces of Finite Area . . . . . . . . . . . . . 419
§9.9. Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . 432
10 Hyperbolic 3-Manifolds
§10.1. Gluing 3-Manifolds . . . . . . . . . . .
§10.2. Complete Gluing of 3-Manifolds . . . .
§10.3. Finite Volume Hyperbolic 3-Manifolds
§10.4. Hyperbolic Volume . . . . . . . . . . .
§10.5. Hyperbolic Dehn Surgery . . . . . . . .
§10.6. Historical Notes . . . . . . . . . . . . .
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435
435
444
448
462
480
505
11 Hyperbolic n-Manifolds
§11.1. Gluing n-Manifolds . . . . . . .
§11.2. Poincar´e’s Theorem . . . . . . .
§11.3. The Gauss-Bonnet Theorem . .
§11.4. Simplices of Maximum Volume .
§11.5. Differential Forms . . . . . . . .
§11.6. The Gromov Norm . . . . . . .
§11.7. Measure Homology . . . . . . .
§11.8. Mostow Rigidity . . . . . . . . .
§11.9. Historical Notes . . . . . . . . .
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508
508
516
523
532
543
555
564
580
597
12 Geometrically Finite n-Manifolds
§12.1. Limit Sets . . . . . . . . . . . . . . .
§12.2. Limit Sets of Discrete Groups . . . .
§12.3. Limit Points . . . . . . . . . . . . . .
§12.4. Geometrically Finite Discrete Groups
§12.5. Nilpotent Groups . . . . . . . . . . .
§12.6. The Margulis Lemma . . . . . . . . .
§12.7. Geometrically Finite Manifolds . . .
§12.8. Historical Notes . . . . . . . . . . . .
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600
600
604
617
627
644
654
666
677
13 Geometric Orbifolds
§13.1. Orbit Spaces . . . . .
§13.2. (X, G)-Orbifolds . . .
§13.3. Developing Orbifolds
§13.4. Gluing Orbifolds . . .
§13.5. Poincar´e’s Theorem .
§13.6. Historical Notes . . .
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681
681
691
701
724
740
743
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Bibliography
745
Index
768
CHAPTER 1
Euclidean Geometry
In this chapter, we review Euclidean geometry. We begin with an informal
historical account of how criticism of Euclid’s parallel postulate led to the
discovery of hyperbolic geometry. In Section 1.2, the proof of the independence of the parallel postulate by the construction of a Euclidean model of
the hyperbolic plane is discussed and all four basic models of the hyperbolic plane are introduced. In Section 1.3, we begin our formal study with
a review of n-dimensional Euclidean geometry. The metrical properties of
curves are studied in Sections 1.4 and 1.5. In particular, the concepts of
geodesic and arc length are introduced.
§1.1. Euclid’s Parallel Postulate
Euclid wrote his famous Elements around 300 B.C. In this thirteen-volume
work, he brilliantly organized and presented the fundamental propositions
of Greek geometry and number theory. In the first book of the Elements,
Euclid develops plane geometry starting with basic assumptions consisting
of a list of definitions of geometric terms, five “common notions” concerning
magnitudes, and the following five postulates:
(1) A straight line may be drawn from any point to any other point.
(2) A finite straight line may be extended continuously in a straight line.
(3) A circle may be drawn with any center and any radius.
(4) All right angles are equal.
(5) If a straight line falling on two straight lines makes the interior angles
on the same side less than two right angles, the two straight lines, if
extended indefinitely, meet on the side on which the angles are less
than two right angles.
1
2
1. Euclidean Geometry
α
β
Figure 1.1.1. Euclid’s parallel postulate
The first four postulates are simple and easily grasped, whereas the fifth
is complicated and not so easily understood. Figure 1.1.1 illustrates the
fifth postulate. When one tries to visualize all the possible cases of the
postulate, one sees that it possesses an elusive infinite nature. As the sum
of the two interior angles α + β approaches 180◦ , the point of intersection
in Figure 1.1.1 moves towards infinity. Euclid’s fifth postulate is equivalent
to the modern parallel postulate of Euclidean geometry:
Through a point outside a given infinite straight line there is
one and only one infinite straight line parallel to the given line.
From the very beginning, Euclid’s presentation of geometry in his Elements was greatly admired, and The Thirteen Books of Euclid’s Elements
became the standard treatise of geometry and remained so for over two
thousand years; however, even the earliest commentators on the Elements
criticized the fifth postulate. The main criticism was that it is not sufficiently self-evident to be accepted without proof. Adding support to this
belief is the fact that the converse of the fifth postulate (the sum of two
angles of a triangle is less than 180◦ ) is one of the propositions proved by
Euclid (Proposition 17, Book I). How could a postulate, whose converse
can be proved, be unprovable? Another curious fact is that most of plane
geometry can be proved without the fifth postulate. It is not used until
Proposition 29 of Book I. This suggests that the fifth postulate is not really
necessary.
Because of this criticism, it was believed by many that the fifth postulate
could be derived from the other four postulates, and for over two thousand
years geometers attempted to prove the fifth postulate. It was not until
the nineteenth century that the fifth postulate was finally shown to be
independent of the other postulates of plane geometry. The proof of this
independence was the result of a completely unexpected discovery. The
denial of the fifth postulate leads to a new consistent geometry. It was
Carl Friedrich Gauss who first made this remarkable discovery.
§1.1. Euclid’s Parallel Postulate
3
Gauss began his meditations on the theory of parallels about 1792. After
trying to prove the fifth postulate for over twenty years, Gauss discovered
that the denial of the fifth postulate leads to a new strange geometry, which
he called non-Euclidean geometry. After investigating its properties for over
ten years and discovering no inconsistencies, Gauss was fully convinced of
its consistency. In a letter to F. A. Taurinus, in 1824, he wrote: “The
assumption that the sum of the three angles (of a triangle) is smaller than
180◦ leads to a geometry which is quite different from our (Euclidean)
geometry, but which is in itself completely consistent.” Gauss’s assumption
that the sum of the angles of a triangle is less than 180◦ is equivalent to the
denial of Euclid’s fifth postulate. Unfortunately, Gauss never published his
results on non-Euclidean geometry.
Only a few years passed before non-Euclidean geometry was rediscovered
independently by Nikolai Lobachevsky and J´
anos Bolyai. Lobachevsky
published the first account of non-Euclidean geometry in 1829 in a paper
entitled On the principles of geometry. A few years later, in 1832, Bolyai
published an independent account of non-Euclidean geometry in a paper
entitled The absolute science of space.
The strongest evidence given by the founders of non-Euclidean geometry for its consistency is the duality between non-Euclidean and spherical
trigonometries. In this duality, the hyperbolic trigonometric functions play
the same role in non-Euclidean trigonometry as the ordinary trigonometric
functions play in spherical trigonometry. Today, the non-Euclidean geometry of Gauss, Lobachevsky, and Bolyai is called hyperbolic geometry,
and the term non-Euclidean geometry refers to any geometry that is not
Euclidean.
Spherical-Hyperbolic Duality
Spherical and hyperbolic geometries are oppositely dual geometries. This
duality begins with the opposite nature of the parallel postulate in each
geometry. The analogue of an infinite straight line in spherical geometry
is a great circle of a sphere. Figure 1.1.2 illustrates three great circles on
a sphere. For simplicity, we shall use the term line for either an infinite
straight line in hyperbolic geometry or a great circle in spherical geometry.
In spherical geometry, the parallel postulate takes the form:
Through a point outside a given line there is no line parallel to
the given line.
The parallel postulate in hyperbolic geometry has the opposite form:
Through a point outside a given line there are infinitely many
lines parallel to the given line.
4
1. Euclidean Geometry
C
A
B
Figure 1.1.2. A spherical equilateral triangle ABC
The duality between spherical and hyperbolic geometries is further evident in the opposite shape of triangles in each geometry. The sum of the
angles of a spherical triangle is always greater than 180◦ , whereas the sum
of the angles of a hyperbolic triangle is always less than 180◦ . As the sum
of the angles of a Euclidean triangle is 180◦ , one can say that Euclidean
geometry is midway between spherical and hyperbolic geometries. See Figures 1.1.2, 1.1.3, and 1.1.5 for an example of an equilateral triangle in each
geometry.
C
A
B
Figure 1.1.3. A Euclidean equilateral triangle ABC
§1.1. Euclid’s Parallel Postulate
5
Curvature
Strictly speaking, spherical geometry is not one geometry but a continuum
of geometries. The geometries of two spheres of different radii are not metrically equivalent; although they are equivalent under a change of scale.
The geometric invariant that best distinguishes the various spherical geometries is Gaussian curvature. A sphere of radius r has constant positive
curvature 1/r2 . Two spheres are metrically equivalent if and only if they
have the same curvature.
The duality between spherical and hyperbolic geometries continues. Hyperbolic geometry is not one geometry but a continuum of geometries. Curvature distinguishes the various hyperbolic geometries. A hyperbolic plane
has constant negative curvature, and every negative curvature is realized
by some hyperbolic plane. Two hyperbolic planes are metrically equivalent
if and only if they have the same curvature. Any two hyperbolic planes
with different curvatures are equivalent under a change of scale.
For convenience, we shall adopt the unit sphere as our model for spherical
geometry. The unit sphere has constant curvature equal to 1. Likewise,
for convenience, we shall work with models for hyperbolic geometry whose
constant curvature is −1. It is not surprising that a Euclidean plane is of
constant curvature 0, which is midway between −1 and 1.
The simplest example of a surface of negative curvature is the saddle
surface in R3 defined by the equation z = xy. The curvature of this surface
at a point (x, y, z) is given by the formula
κ(x, y, z) =
−1
.
(1 + x2 + y 2 )2
(1.1.1)
In particular, the curvature of the surface has a unique minimum value of
−1 at the saddle point (0, 0, 0).
There is a well-known surface in R3 of constant curvature −1. If one
starts at (0, 0) on the xy-plane and walks along the y-axis pulling a small
wagon that started at (1, 0) and has a handle of length 1, then the wagon
would follow the graph of the tractrix (L. trahere, to pull) defined by the
equation
y = cosh−1
1
x
−
1 − x2 .
(1.1.2)
This curve has the property that the distance from the point of contact
of a tangent to the point where it cuts the y-axis is 1. See Figure 1.1.4.
The surface S obtained by revolving the tractrix about the y-axis in R3 is
called the tractroid. The tractroid S has constant negative curvature −1;
consequently, the local geometry of S is the same as that of a hyperbolic
plane of curvature −1. Figure 1.1.5 illustrates a hyperbolic equilateral
triangle on the tractroid S.
6
1. Euclidean Geometry
y
x
1
Figure 1.1.4. Two tangents to the graph of the tractrix
C
A
B
Figure 1.1.5. A hyperbolic equilateral triangle ABC on the tractroid
§1.2. Independence of the Parallel Postulate
7
§1.2. Independence of the Parallel Postulate
After enduring twenty centuries of criticism, Euclid’s theory of parallels was
fully vindicated in 1868 when Eugenio Beltrami proved the independence
of Euclid’s parallel postulate by constructing a Euclidean model of the hyperbolic plane. The points of the model are the points inside a fixed circle,
in a Euclidean plane, called the circle at infinity. The lines of the model
are the open chords of the circle at infinity. It is clear from Figure 1.2.1
that Beltrami’s model has the property that through a point P outside a
line L there is more than one line parallel to L. Using differential geometry,
Beltrami showed that his model satisfies all the axioms of hyperbolic plane
geometry. As Beltrami’s model is defined entirely in terms of Euclidean
plane geometry, it follows that hyperbolic plane geometry is consistent if
Euclidean plane geometry is consistent. Thus, Euclid’s parallel postulate
is independent of the other postulates of plane geometry.
In 1871, Felix Klein gave an interpretation of Beltrami’s model in terms
of projective geometry. In particular, Beltrami and Klein showed that the
congruence transformations of Beltrami’s model correspond by restriction
to the projective transformations of the extended Euclidean plane that
leave the model invariant. For example, a rotation about the center of
the circle at infinity restricts to a congruence transformation of Beltrami’s
model. Because of Klein’s interpretation, Beltrami’s model is also called
Klein’s model of the hyperbolic plane. We shall take a neutral position and
call this model the projective disk model of the hyperbolic plane.
The projective disk model has the advantage that its lines are straight,
but it has the disadvantage that its angles are not necessarily the Euclidean
angles. This is best illustrated by examining right angles in the model.
P
L
Figure 1.2.1. Lines passing through a point P parallel to a line L
8
1. Euclidean Geometry
L
P
L
Figure 1.2.2. Two perpendicular lines L and L of the projective disk model
Let L be a line of the model which is not a diameter, and let P be the
intersection of the tangents to the circle at infinity at the endpoints of L
as illustrated in Figure 1.2.2. Then a line L of the model is perpendicular
to L if and only if the Euclidean line extending L passes through P . In
particular, the Euclidean midpoint of L is the only point on L at which the
right angle formed by L and its perpendicular is a Euclidean right angle.
We shall study the projective disk model in detail in Chapter 6.
The Conformal Disk Model
There is another model of the hyperbolic plane whose points are the points
inside a fixed circle in a Euclidean plane, but whose angles are the Euclidean angles. This model is called the conformal disk model, since its
angles conform with the Euclidean angles. The lines of this model are the
open diameters of the boundary circle together with the open circular arcs
orthogonal to the boundary circle. See Figures 1.2.3 and 1.2.4. The hyperbolic geometry of the conformal disk model is the underlying geometry
of M.C. Escher’s famous circle prints. Figure 1.2.5 is Escher’s Circle Limit
IV. All the devils (angels) in Figure 1.2.5 are congruent with respect to the
underlying hyperbolic geometry. Some appear larger than others because
the model distorts distances. We shall study the conformal disk model in
detail in Chapter 4.
The projective and conformal disk models both exhibit Euclidean rotational symmetry with respect to their Euclidean centers. Rotational symmetry is one of the two basic forms of Euclidean symmetry; the other is
translational symmetry. There is another conformal model of the hyperbolic plane which exhibits Euclidean translational symmetry. This model
is called the upper half-plane model.
§1.2. Independence of the Parallel Postulate
9
Figure 1.2.3. Asymptotic parallel lines of the conformal disk model
C
A
B
Figure 1.2.4. An equilateral triangle ABC in the conformal disk model
10
1. Euclidean Geometry
Figure 1.2.5. M. C. Escher: Circle Limit IV
c 2006 The M.C. Escher Company - Holland. All rights reserved.
The Upper Half-Plane Model
The points of the upper half-plane model are the complex numbers above
the real axis in the complex plane. The lines of the model are the open rays
orthogonal to the real axis together with the open semicircles orthogonal
to the real axis. See Figures 1.2.6 and 1.2.7. The orientation preserving
congruence transformations of the upper half-plane model are the linear
fractional transformations of the form
az + b
with a, b, c, d real and ad − bc > 0.
φ(z) =
cz + d
In particular, a Euclidean translation τ (z) = z + b is a congruence transformation. The upper half-plane model exhibits Euclidean translational
symmetry at the expense of an unlimited amount of distortion. Any magnification µ(z) = az, with a > 1, is a congruence transformation. We shall
study the upper half-plane model in detail in Chapter 4.
§1.2. Independence of the Parallel Postulate
11
Figure 1.2.6. Asymptotic parallel lines of the upper half-plane model
C
A
B
Figure 1.2.7. An equilateral triangle ABC in the upper half-plane model
12
1. Euclidean Geometry
The Hyperboloid Model
All the models of the hyperbolic plane we have described distort distances.
Unfortunately, there is no way we can avoid distortion in a useful Euclidean
model of the hyperbolic plane because of a remarkable theorem of David
Hilbert that there is no complete C2 surface of constant negative curvature
in R3 . Hilbert’s theorem implies that there is no reasonable distortion-free
model of the hyperbolic plane in Euclidean 3-space.
Nevertheless, there is an analytic distortion-free model of the hyperbolic
plane in Lorentzian 3-space. This model is called the hyperboloid model of
the hyperbolic plane. Lorentzian 3-space is R3 with a non-Euclidean geometry (described in Chapter 3). Even though the geometry of Lorentzian
3-space is non-Euclidean, it still has physical significance. Lorentzian 4space is the model of space-time in the theory of special relativity.
The points of the hyperboloid model are the points of the positive sheet
(x > 0) of the hyperboloid in R3 defined by the equation
x2 − y 2 − z 2 = 1.
(1.2.1)
A line of the model is a branch of a hyperbola obtained by intersecting
the model with a Euclidean plane passing through the origin. The angles
in the hyperboloid model conform with the angles in Lorentzian 3-space.
In Chapter 3, we shall adopt the hyperboloid model as our basic model of
hyperbolic geometry because it most naturally exhibits the duality between
spherical and hyperbolic geometries.
Exercise 1.2
1. Let P be a point outside a line L in the projective disk model. Show that
there exists two lines L1 and L2 passing through P parallel to L such that
every line passing through P parallel to L lies between L1 and L2 . The two
lines L1 and L2 are called the parallels to L at P . All the other lines passing
through P parallel to L are called ultraparallels to L at P . Conclude that
there are infinitely many ultraparallels to L at P .
2. Prove that any triangle in the conformal disk model, with a vertex at the
center of the model, has angle sum less than 180◦ .
3. Let u, v be distinct points of the upper half-plane model. Show how to
construct the hyperbolic line joining u and v with a Euclidean ruler and
compass.
4. Let φ(z) = az+b
with a, b, c, d in R and ad − bc > 0. Prove that φ maps the
cz+d
complex upper half-plane bijectively onto itself.
5. Show that the intersection of the hyperboloid x2 − y 2 − z 2 = 1 with a
Euclidean plane passing through the origin is either empty or a hyperbola.
§1.3. Euclidean n-Space
13
§1.3. Euclidean n-Space
The standard analytic model for n-dimensional Euclidean geometry is the
n-dimensional real vector space Rn . A vector in Rn is an ordered n-tuple
x = (x1 , . . . , xn ) of real numbers. Let x and y be vectors in Rn . The
Euclidean inner product of x and y is defined to be the real number
x · y = x1 y1 + · · · + xn yn .
(1.3.1)
The Euclidean inner product is the prototype for the following definition:
Definition: An inner product on a real vector space V is a function from
V × V to R, denoted by (v, w) → v, w , such that for all v, w in V ,
(1) v,
and
, w are linear functions from V to R (bilinearity);
(2) v, w = w, v (symmetry); and
(3) if v = 0, then there is a w = 0 such that v, w = 0 (nondegeneracy).
The Euclidean inner product on Rn is obviously bilinear and symmetric.
Observe that if x = 0 in Rn , then x · x > 0, and so the Euclidean inner
product is also nondegenerate.
An inner product , on a real vector space V is said to be positive
definite if and only if v, v > 0 for all nonzero v in V . The Euclidean inner
product on Rn is an example of a positive definite inner product.
Let , be a positive definite inner product on V . The norm of v in V ,
with respect to , , is defined to be the real number
v = v, v
1
2
.
(1.3.2)
The norm of x in R , with respect to the Euclidean inner product, is called
the Euclidean norm and is denoted by |x|.
n
Theorem 1.3.1. (Cauchy’s inequality) Let , be a positive definite inner
product on a real vector space V . If v, w are vectors in V , then
| v, w | ≤ v
w
with equality if and only if v and w are linearly dependent.
Proof: If v and w are linearly dependent, then equality clearly holds.
Suppose that v and w are linearly independent. Then tv − w = 0 for all t
in R, and so
0 <
tv − w
2
=
tv − w, tv − w
2
= t v
2
− 2t v, w + w 2 .
The last expression is a quadratic polynomial in t with no real roots, and
so its discriminant must be negative. Thus
4 v, w
2
−4 v
2
w
2
< 0.
14
1. Euclidean Geometry
Let x, y be nonzero vectors in Rn . By Cauchy’s inequality, there is a
unique real number θ(x, y) between 0 and π such that
x · y = |x| |y| cos θ(x, y).
(1.3.3)
The Euclidean angle between x and y is defined to be θ(x, y).
Two vectors x, y in Rn are said to be orthogonal if and only if x · y = 0.
As cos(π/2) = 0, two nonzero vectors x, y in Rn are orthogonal if and only
if θ(x, y) = π/2.
Corollary 1. (The triangle inequality) If x and y are vectors in Rn , then
|x + y| ≤ |x| + |y|
with equality if and only if x and y are linearly dependent.
Proof: Observe that
|x + y|2
= (x + y) · (x + y)
= |x|2 + 2x · y + |y|2
≤
=
|x|2 + 2|x| |y| + |y|2
(|x| + |y|)2
with equality if and only if x and y are linearly dependent.
Metric Spaces
The Euclidean distance between vectors x and y in Rn is defined to be
dE (x, y) = |x − y|.
(1.3.4)
The distance function dE is the prototype for the following definition:
Definition: A metric on a set X is a function d : X × X → R such that
for all x, y, z in X,
(1) d(x, y) ≥ 0 (nonnegativity);
(2) d(x, y) = 0 if and only if x = y (nondegeneracy);
(3) d(x, y) = d(y, x) (symmetry); and
(4) d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
The Euclidean distance function dE obviously satisfies the first three
axioms for a metric on Rn . By Corollary 1, we have
|x − z| = |(x − y) + (y − z)| ≤ |x − y| + |y − z|.
Therefore dE satisfies the triangle inequality. Thus dE is a metric on Rn ,
called the Euclidean metric.
