Hyperbolic Geometry

282 Pages·2005·1.76 MB·English

by  Anderson J.

Springer Undergraduate Mathematics Series
Advisory Board
M.A.J.Chaplain UniversityofDundee
K.Erdmann OxfordUniversity
A.MacIntyre UniversityofLondon
L.C.G.Rogers CambridgeUniversity
E.Su¨li OxfordUniversity
J.F.Toland UniversityofBath
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James W. Anderson
Hyperbolic Geometry
Second Edition
With 21 Figures
James W. Anderson, BA, PhD
School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK
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MathematicsSubjectClassification(2000):51-01
Anderson,JamesW.,1964–
Hyberbolicgeometry.—2nded.—(Springerundergraduate
mathematicsseries)
1.Geometry,Hyperbolic
I.Title
516.9
ISBN1852339349
LibraryofCongressControlNumber:2005923338
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Contents
Preamble to the Second Edition ................................ vii
Preamble to the First Edition .................................. ix
1. The Basic Spaces ........................................... 1
1.1 A Model for the Hyperbolic Plane .......................... 1
1.2 The Riemann Sphere C ................................... 8
1.3 The Boundary at Infinity of H ............................. 18
2. The General M¨obius Group ................................ 23
2.1 The Group of Mo¨bius Transformations ...................... 23
+
2.2 Transitivity Properties of Mo¨b ............................ 30
2.3 The Cross Ratio.......................................... 36
2.4 Classification of Mo¨bius Transformations .................... 39
2.5 A Matrix Representation .................................. 42
2.6 Reflections............................................... 48
2.7 The Conformality of Elements of Mo¨b....................... 53
2.8 Preserving H............................................. 56
2.9 Transitivity Properties of Mo¨b(H) .......................... 62
2.10 The Geometry of the Action of Mo¨b(H) ..................... 65
v
vi Hyperbolic Geometry
3. Length and Distance in H................................... 73
3.1 Paths and Elements of Arc-length .......................... 73
3.2 The Element of Arc-length on H ........................... 80
3.3 Path Metric Spaces ....................................... 88
3.4 From Arc-length to Metric................................. 92
3.5 Formulae for Hyperbolic Distance in H...................... 99
3.6 Isometries ...............................................103
3.7 Metric Properties of (H,dH) ...............................108
4. Planar Models of the Hyperbolic Plane.....................117
4.1 The Poincar´e Disc Model..................................117
4.2 A General Construction ...................................130
5. Convexity, Area, and Trigonometry.........................145
5.1 Convexity ...............................................145
5.2 Hyperbolic Polygons ......................................154
5.3 The Definition of Hyperbolic Area ..........................164
5.4 Area and the Gauss–Bonnet Formula .......................169
5.5 Applications of the Gauss–Bonnet Formula ..................174
5.6 Trigonometry in the Hyperbolic Plane.......................181
6. Nonplanar models ..........................................189
6.1 The Hyperboloid Model of the Hyperbolic Plane .............189
6.2 Higher Dimensional Hyperbolic Spaces ......................209
Solutions to Exercises ..........................................217
References......................................................265
List of Notation ................................................269
Index...........................................................273
Preamble to the Second Edition
WelcometothesecondeditionofHyperbolic Geometry,andthanksforreading.
I have tried to keep the basic structure of the book relatively unchanged, so
that it can still be used by the reader either for self-study or as a classroom
text. I have also tried to maintain the self-contained aspect of the book. A
few new exercises and small bits of new material have been added to most
chapters, and a few exercises and small bits of material have been removed.
Overall, Chapters 1, 2, 3, and 5 are essentially the same as they were in the
first edition.
In addition to this tinkering with exercises and material, there have been two
major changes from the first edition of this book.
First,IhavetightenedthefocusofChapter4tojustplanarmodelsofhyperbolic
planethatarisefromcomplexanalysis.Thishasresultedintheintroductionof
somemoreadvancedmaterialfromcomplexanalysis,butnotsomuchthatthe
self-contained aspect of the book is seriously threatened. I have tried to make
more clear the connections between planar hyperbolic geometry and complex
analysis.
Second, I have changed Chapter 6 completely. Gone is the material on discrete
subgroups of Mo¨b(H). In its place is an introduction to the hyperboloid model
of the hyperbolic plane. Unfortunately, I did not feel that I had space to do
justice to the Klein model as well, and so I haven’t built the bridge from the
Poincar´e disc model to the hyperboloid model via the Klein model, but this
has been done elsewhere by others. I close Chapter 6, and the book, with a
very brief look at higher dimensional hyperbolic geometry.
vii
viii Hyperbolic Geometry
The prerequisites for reading the book haven’t significantly changed from the
first edition. The book is written primarily for a third- or fourth-year under-
graduate student who has encountered some calculus (univariate and multi-
variate), particularly the definition of arc-length, integration over regions in
Euclidean space, and the change of variables theorem; some analysis, particu-
larlycontinuity,openandclosedsetsintheplane,andinfimumandsupremum;
and some basic complex analysis, such as the arithmetic of the complex num-
bers C and the basics of holomorphic functions.
I would like to close this introduction by adding some acknowledgements to
thelistgiveninthePreambletotheFirstEdition.IwouldliketothankKaren
Borthwick at Springer for giving me the opportunity to write this second edi-
tion, and for being patient with me. I have continued to teach the course of
Hyperbolic Geometry at the University of Southampton on which this book
is based, and I would like to thank the students who have been a part of the
course over the past several years, and who have pointed out the occasional
mistake. The errors that remain are of course mine. And I would like to thank
my wife Barbara, who once again put up with me through the final stages of
writing.
Preamble to the First Edition
What you have in your hands is an introduction to the basics of planar hyper-
bolic geometry. Writing this book was difficult, not because I was at any point
at a loss for topics to include, but rather because I continued to come across
topicsthatIfeltshouldbeincludedinanintroductorytext.Ibelievethatwhat
has emerged from the process of writing gives a good feel for the geometry of
the hyperbolic plane.
This book is written to be used either as a classroom text or as more of a
self-studybook,perhapsaspartofadirectedreadingcourse.Forthatreason,I
have included solutions to all the exercises. I have tried to choose the exercises
to give reasonable coverage to the sorts of calculations and proofs that inhabit
the subject. The reader should feel free to make up their own exercises, both
proofs and calculations, and to make use of other sources.
Ihavealsotriedtokeeptheexpositionasself-containedaspossible,andtomake
as little use of mathematical machinery as possible. The book is written for a
third or fourth year student who has encountered some Calculus, particularly
the definition of arc-length, integration over regions in Euclidean space, and
the change of variables theorem; some Analysis, particularly continuity, open
and closed sets in the plane, and infimum and supremum; has a familiarity
with Complex Numbers, as most of the book takes place in the complex plane
C, but need not have taken a class in Complex Analysis; and some Abstract
Algebra, as we make use of some of the very basics from the theory of groups.
Non-Euclidean geometry in general, and hyperbolic geometry in particular, is
anareaofmathematicswhichhasaninterestinghistoryandwhichisstillbeing
activelystudiedbyresearchersaroundtheworld.Onereasonforthecontinuing
interest in hyperbolic geometry is that it touches on a number of different
ix
x Hyperbolic Geometry
fields, including but not limited to Complex Analysis, Abstract Algebra and
Group Theory, Number Theory, Differential Geometry, and Low-dimensional
Topology.
This book is not written as an encyclopedic introduction to hyperbolic geom-
etry but instead offers a single perspective. Specifically, I wanted to write a
hyperbolic geometry book in which very little was assumed, and as much as
possible was derived from following Klein’s view that geometry, in this case
hyperbolic geometry, consists of the study of those quantities invariant under
a group. Consequently, I did not want to write down, without what I felt to be
reasonable justification, the hyperbolic element of arc-length, or the group of
hyperbolicisometries,butinsteadwantedthemtoariseasnaturallyaspossible.
And I think I have done that in this book.
ThereisalargenumberoftopicsIhavechosennottoinclude,suchasthehyper-
boloidandKleinmodelsofthehyperbolicplane.Also,Ihaveincludednothing
of the history of hyperbolic geometry and I have not taken the axiomatic ap-
proach to define the hyperbolic plane. One reason for these omissions is that
therearealreadyanumberofexcellentbooksonboththehistoryofhyperbolic
geometry and on the axiomatic approach, and I felt that I would not be able
to add anything of note to what has already been done. There is an extensive
literature on hyperbolic geometry. The interested reader is directed to the list
of sources for Further Reading at the end of the book.
And now, a brief outline of the approach taken in this book. We first develop
a model of the hyperbolic plane, namely the upper half-plane model H, and
definewhatwemeanbyahyperboliclineinH.Wethentrytodeterminearea-
sonablegroupoftransformationsofHthattakeshyperboliclinestohyperbolic
+
lines, which leads us to spend some time studying the group Mo¨b of Mo¨bius
transformations and the general Mo¨bius group Mo¨b.
After determining the subgroup Mo¨b(H) of Mo¨b preserving H, we derive an
invariant element of arc-length on H. That is, we derive a means of calculating
thehyperboliclengthofapathf :[a,b]→Hinsuchawaythatthehyperbolic
length of a path is invariant under the action of Mo¨b(H), which is to say that
the hyperbolic length of a path f :[a,b]→H is equal to the hyperbolic length
of its translate γ ◦f : [a,b] → H for any element γ of Mo¨b(H). We are then
able to define a natural metric on H in terms of the shortest hyperbolic length
of a path joining a pair of points.
After exploring calculations of hyperbolic length, we move onto a discussion of
convexity and of hyperbolic polygons, and then to the trigonometry of poly-
gons in the hyperbolic plane and the three basic laws of trigonometry in the
hyperbolicplane.Wealsodeterminehowtocalculatehyperbolicarea,andstate