Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL2(R)

207 Pages·2012·1.35 MB·English

by  Vladimir V. Kisil

GEOMETRY OF MÖBIUS
TRANSFORMATIONS
Elliptic, Parabolic and
Hyperbolic Actions of SL (R)
2
P835.9781860945540-tp.indd 2 16/5/12 3:45 PM
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GEOMETRY OF MÖBIUS
TRANSFORMATIONS
Elliptic, Parabolic and
Hyperbolic Actions of SL (R)
2
Vladimir V. Kisil
University of Leeds, UK
Imperial College Press
ICP
P835.9781860945540-tp.indd 1 16/5/12 3:45 PM
Published by
Imperial College Press
57 Shelton Street
Covent Garden
London WC2H 9HE
Distributed by
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UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
GEOMETRY OF MÖBIUS TRANSFORMATIONS
Elliptic, Parabolic and Hyperbolic Actions of SL(RRRRR)
2
(with DVD-ROM)
Copyright © 2012 by Imperial College Press
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN-13 978-1-84816-858-9
ISBN-10 1-84816-858-6
Printed in Singapore.
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Dedicated to my parents
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Preface
Everythingnewisold...understoodagain.
Yu.M.Polyakov
The idea proposedby Sophus Lie and Felix Klein was that geometry is the
theoryof invariants of atransitivetransformation group. Itwasusedasthe
main topic of F. Klein’s inauguration lecture for professorship at Erlangen
in 1872 and, thus, become known as the Erlangen programme (EP). As
with any great idea, it was born ahead of its time. It was only much later
when the theory of groups, especially the theory of group representations,
was able to make a serious impact. Therefore,the EP had been markedas
‘producing only abstract returns’ (cWikipedia) and laid on one side.
(cid:13)
Meanwhile,the 20thcenturybroughtsignificantprogressinrepresenta-
tion theory, especially linear representations, which was closely connected
to achievements in functional analysis. Therefore, a ‘study of invariants’
becomes possible in the linear spaces of functions and associated algebras
of operators, e.g. the main objects of modern analysis. This is echoed in
the saying which Yu.I. Manin attributed to I.M. Gelfand:
Mathematics of any kind is a representation theory.
This attitude can be encoded as the Erlangen programme at large
(EPAL). In this book, we will systematically apply it to construct geo-
metry of two-dimensional spaces. Further development shall extend it to
analyticfunctiontheoriesonsuchspacesandtheassociatedco-andcontra-
variant functional calculi with relevant spectra [69]. Functional spaces are
naturallyassociatedwithalgebrasofcoordinatesonageometrical(orcom-
mutative) space. An operator (non-commutative) algebra is fashionably
treated as a non-commutative space. Therefore,EPAL plays the same rˆole
for non-commutative geometry as EP for commutative geometry [59,60].
EPALprovidesa systematic tool for discoveringhidden features,which
previously escaped attention for various psychological reasons. In a
vii
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viii Geometry of M¨obius Transformations
sense [60], EPAL works like the periodic table of chemical elements dis-
covered by D.I. Mendeleev: it allows us to see which cells are still empty
and suggest where to look for the corresponding objects [60].
Mathematicaltheoremsonceproved,remaintrueforever. However,this
does not mean we should not revise the corresponding theories. Excellent
examples are given in Geometry Revisited [23] and Elementary Mathema-
tics from an Advanced Standpoint [71,72]. Understanding comes through
comparison and there are many excellent books about the Lobachevsky
half-plane which made their exposition through a contrast with Euclidean
geometry. Our book offers a different perspective: it considers the Lo-
bachevsky half-plane as one of three sister conformal geometries–elliptic,
parabolic, and hyperbolic–on the upper half-plane.
Exercises are an integral part of these notes. If a mathematical state-
ment is presented as an exercise, it is not meant to be peripheral, unim-
portant or without further use. Instead, the label ‘Exercise’ indicates that
demonstration of the result is not very difficult and may be useful for un-
derstanding. Presentation of mathematical theory through a suitable col-
lection of exercises has a long history, starting from the famous Polya and
Szeg˝obook[92],withmanyothersuccessfulexamplesfollowing,e.g.[31,55].
Mathematics is among those enjoyable things which are better to practise
yourself rather than watch others doing it.
Forsomeexercises,Iknowonlyabrute-forcesolution,whichiscertainly
Í
undesirable. Fortunately, all of them, marked by the symbol in the
margins, can be done through a Computer Algebra System (CAS). The
DVD provided contains the full package and Appendix B describes initial
instructions. Computer-assistedexercisesalsoformatestcaseforourCAS,
which validates both the mathematical correctness of the library and its
practical usefulness.
All figures in the book are printed in black and white to reduce costs.
The coloured versions of all pictures are enclosed on the DVD as well–see
Appendix B.1 to find them. The reader will be able to produce even more
illustrations him/herself with the enclosed software.
There are many classical objects, e.g. pencils of cycles, or power of
a point, which often re-occur in this book under different contexts. The
detailed index will help to trace most of such places.
Chapter1servesasanoverviewandagentleintroduction,sowedonot
give a description of the book content here. The reader is now invited to
start his/her journey into M¨obius-invariant geometries.
Odessa, January 2012
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Contents
Preface vii
List of Figures xiii
1. Erlangen Programme: Preview 1
1.1 Make a Guess in Three Attempts . . . . . . . . . . . . . . 2
1.2 Covariance of FSCc . . . . . . . . . . . . . . . . . . . . . 5
1.3 Invariants: Algebraic and Geometric . . . . . . . . . . . . 8
1.4 Joint Invariants: Orthogonality . . . . . . . . . . . . . . . 9
1.5 Higher-order Joint Invariants: Focal Orthogonality . . . . 11
1.6 Distance, Length and Perpendicularity . . . . . . . . . . . 12
1.7 The ErlangenProgramme at Large . . . . . . . . . . . . . 15
2. Groups and Homogeneous Spaces 17
2.1 Groups and Transformations . . . . . . . . . . . . . . . . 17
2.2 Subgroups and Homogeneous Spaces . . . . . . . . . . . . 20
2.3 Differentiation on Lie Groups and Lie Algebras . . . . . . 23
3. Homogeneous Spaces from the Group SL (R) 29
2
3.1 The Affine Group and the Real Line . . . . . . . . . . . . 29
3.2 One-dimensional Subgroups of SL (R) . . . . . . . . . . . 30
2
3.3 Two-dimensional Homogeneous Spaces . . . . . . . . . . . 32
3.4 Elliptic, Parabolic and Hyperbolic Cases . . . . . . . . . . 35
3.5 Orbits of the Subgroup Actions . . . . . . . . . . . . . . . 37
3.6 Unifying EPH Cases: The First Attempt. . . . . . . . . . 39
3.7 Isotropy Subgroups . . . . . . . . . . . . . . . . . . . . . . 40
ix