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Geometry of Möbius transformations. Elliptic, parabolic and hyperbolic actions of SL\(_2(\mathbb R)\). With DVD-ROM. (English) Zbl 1254.30001
Hackensack, NJ: World Scientific (ISBN 978-1-84816-858-9/hbk; 978-1-84816-859-6/ebook). xiv, 192 p. (2012).
This book provides a systematic and complete presentation of the geometries associated with Möbius transformations of the hypercomplex plane. The author presents surprising new results about the geometry of circles, parabolas and hyperbolas applying the Erlangen Programme (EP) of E. Klein to construct the two-dimensional spaces. Further development extends this geometry to analytic function theories on such spaces and associated co- and contra-variant functional calculi with relevant spectra. The functional spaces are naturally associated with algebras of coordinates on a geometrical (or commutative) space. The operator (non-commutative) algebra is treated as a non-commutative space. It is emphasized that the EP at large, where mathematics of any kind is a representation theory, provides a systematic tool for discovering hidden features, which previously escaped attention for various reasons. Even, it allows us to see which cells are still empty and suggests where to look for the corresponding objects. The introductive notions of the group transformations and homogeneous spaces from the group \(\mathrm{SL}_2(\mathbb{R})\) are reviewed and then the invariant properties of the cycles under the Möbius transformations are discussed. The connections between cycles and vector space structure are investigated. The metrical quantities from cycles are derived in a way which is Möbius-invariant. Some physical examples from optics, classical and quantum mechanics, or relativity of space-time are illustrated. Invariant metrics and geodesics which are preserved by Möbius transformations are obtained and the construction of parabolic and hyperbolic models is given considering the compact unit disk rather than the unbounded upper half plane. The exercises are an integral part of the book, many of them being solved through a computer algebra system. An attached DVD contains the full package and the instructions are described in an appendix. All figures in the book are printed in black and white, but their coloured versions are enclosed on the DVD. It is emphasized that further works on the ideas of this book will reveal more supporting evidence from representation theory, analytic functions and operator theory.

30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
51-02 Research exposition (monographs, survey articles) pertaining to geometry
51N25 Analytic geometry with other transformation groups
53A40 Other special differential geometries
30C20 Conformal mappings of special domains
30G35 Functions of hypercomplex variables and generalized variables
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
22E30 Analysis on real and complex Lie groups
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