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” (©Wikipedia) and laid on one side.

Meanwhile, the 20th century brought significant progress in representation theory, especially linear representations, which was requested by theoretical physics and supported by 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 [197]. 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 [172, 176].

EPAL provides a systematic tool for discovering hidden features, which previously escaped attention for various psychological reasons. In a sense [176], 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 [176].

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 [71] and Elementary Mathematics from an Advanced Standpoint [221, 222]. 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 notes¹. 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 [287], with many other successful examples following, e.g. [111, 164]. 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 Mouse in the margins, can be done through a Computer Algebra System (CAS). The DVD provided contains the full package and Appendix C 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. Usage of computer-supported proofs in geometry is already an established practice [275] and it is naturally to expect its further rapid growth.

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 C.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, July 2011

1

This is one of several features, which distinct our work from the excellent book [339], see also App. B.3. Kolmogorov [225] criticised Yaglom’s book for an insufficient number of exercises.