GEOMETRY OF MÖBIUS TRANSFORMATIONS

Transcription

1 GEOMETRY OF MÖBIUS TRANSFORMATIONS Elliptic, Parabolic and Hyperbolic Actions of SL 2 (R) Vladimir V. Kisil University of Leeds, UK ICP Imperial College Press

2 Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ 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 2 (R) (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, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN ISBN Printed in Singapore.

3 Moim roditel m posv waets Dedicated to my parents

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5 Preface Everything new is old... understood again. Yu.M. Polyakov The idea proposed by Sophus Lie and Felix Klein was that geometry is the theory of invariants of a transitive transformation group. It was used as the 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 marked as producing only abstract returns ( c Wikipedia) and laid on one side. Meanwhile, the 20th century brought significant progress in representation 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 geometry of two-dimensional spaces. Further development shall extend it to analytic function theories on such spaces and the associated co- and contravariant functional calculi with relevant spectra [69]. Functional spaces are naturally associated with algebras of coordinates on a geometrical (or commutative) space. An operator (non-commutative) algebra is fashionably treated as a non-commutative space. Therefore, EPAL plays the same rôle for non-commutative geometry as EP for commutative geometry [59, 60]. EPAL provides a systematic tool for discovering hidden features, which previously escaped attention for various psychological reasons. In a vii

6 viii Geometry of Möbius Transformations sense [60], EPAL works like the periodic table of chemical elements discovered by D.I. Mendeleev: it allows us to see which cells are still empty and suggest where to look for the corresponding objects [60]. Mathematical theorems once proved, remain true forever. However, this does not mean we should not revise the corresponding theories. Excellent examples are given in Geometry Revisited [23] and Elementary Mathematics 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 Lobachevsky 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 statement is presented as an exercise, it is not meant to be peripheral, unimportant or without further use. Instead, the label Exercise indicates that demonstration of the result is not very difficult and may be useful for understanding. Presentation of mathematical theory through a suitable collection of exercises has a long history, starting from the famous Polya and Szegő book[92], with many other successful examples following, e.g.[31, 55]. Mathematics is among those enjoyable things which are better to practise yourself rather than watch others doing it. For some exercises, I know only a brute-force solution, which is certainly 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-assisted exercises also form a test case for our CAS, 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. Chapter 1 serves as an overview and a gentle introduction, so we do not give a description of the book content here. The reader is now invited to start his/her journey into Möbius-invariant geometries. Odessa, January 2012

7 Contents Preface List of Figures 1. Erlangen Programme: Preview Make a Guess in Three Attempts Covariance of FSCc Invariants: Algebraic and Geometric Joint Invariants: Orthogonality Higher-order Joint Invariants: Focal Orthogonality Distance, Length and Perpendicularity The Erlangen Programme at Large Groups and Homogeneous Spaces Groups and Transformations Subgroups and Homogeneous Spaces Differentiation on Lie Groups and Lie Algebras vii xiii 3. Homogeneous Spaces from the Group SL 2 (R) The Affine Group and the Real Line One-dimensional Subgroups of SL 2 (R) Two-dimensional Homogeneous Spaces Elliptic, Parabolic and Hyperbolic Cases Orbits of the Subgroup Actions Unifying EPH Cases: The First Attempt Isotropy Subgroups ix

8 x Geometry of Möbius Transformations 4. The Extended Fillmore Springer Cnops Construction Invariance of Cycles Projective Spaces of Cycles Covariance of FSCc Origins of FSCc Projective Cross-Ratio Indefinite Product Space of Cycles Cycles: An Appearance and the Essence Cycles as Vectors Invariant Cycle Product Zero-radius Cycles Cauchy Schwarz Inequality and Tangent Cycles Joint Invariants of Cycles: Orthogonality Orthogonality of Cycles Orthogonality Miscellanea Ghost Cycles and Orthogonality Actions of FSCc Matrices Inversions and Reflections in Cycles Higher-order Joint Invariants: Focal Orthogonality Metric Invariants in Upper Half-Planes Distances Lengths Conformal Properties of Möbius Maps Perpendicularity and Orthogonality Infinitesimal-radius Cycles Infinitesimal Conformality Global Geometry of Upper Half-Planes Compactification of the Point Space (Non)-Invariance of The Upper Half-Plane Optics and Mechanics Relativity of Space-Time Invariant Metric and Geodesics Metrics, Curves Lengths and Extrema

9 Contents xi 9.2 Invariant Metric Geodesics: Additivity of Metric Geometric Invariants Invariant Metric and Cross-Ratio Conformal Unit Disk Elliptic Cayley Transforms Hyperbolic Cayley Transform Parabolic Cayley Transforms Cayley Transforms of Cycles Unitary Rotations Unitary Rotations An Algebraic Approach Unitary Rotations A Geometrical Viewpoint Rebuilding Algebraic Structures from Geometry Invariant Linear Algebra Linearisation of the Exotic Form Conformality and Geodesics Epilogue: About the Cover 155 Appendix A Supplementary Material 157 A.1 Dual and Double Numbers A.2 Classical Properties of Conic Sections A.3 Comparison with Yaglom s Book A.4 Other Approaches and Results A.5 FSCc with Clifford Algebras Appendix B How to Use the Software 163 B.1 Viewing Colour Graphics B.2 Installation of CAS B.3 Using the CAS and Computer Exercises B.4 Library for Cycles B.5 Predefined Objects at Initialisation Bibliography 173 Index 181

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11 List of Figures 1.1 Actions of the subgroups A and N by Möbius transformations Action of the subgroup K K-orbits as conic sections Decomposition of an arbitrary Möbius transformation Cycle implementations, centres and foci Different σ-implementations of the same σ-zero-radius cycles Orthogonality of the first kind in the elliptic point space Focal orthogonality for circles Radius and distance for parabolas The perpendicular as the shortest route to a line Actions of isotropy subgroups Linear spans of cycle pairs in EPH cases Positive and negative cycles Relation between centres and radii of orthogonal circles Orthogonal pencils of cycles Nine types of cycle orthogonality Three types of inversions of the rectangular grid Focal orthogonality of cycles Zero-radius cycles and the phase transition Compactification and stereographic projections Continuous transformation from future to the past Hyperbolic objects in the double cover Double cover of the hyperbolic space xiii

12 xiv Geometry of Möbius Transformations 8.5 Some elementary optical systems and their transfer matrices Lobachevsky geodesics and extrema of curves lengths Region where the triangular inequality fails Geodesics and equidistant orbits in EPH geometries Action of the isotropy subgroups under the Cayley transform Cayley transforms in elliptic, parabolic and hyperbolic spaces EPH unit disks and actions of one-parameter subgroups Rotations of algebraic wheels Geodesics as spokes A.1 Classical definitions of conic sections A.2 The correspondence between complex, dual and double numbers 161 A.3 Comparison with the Yaglom book B.1 Initial screens of software start up