Geometry of Mobius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL_2(R)

Vladimir V. Kisil
Imperial College Press
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This book is an exposition of geometries associated with Möbius transformations of the plane, based on properties of the group $SL_2({\mathbb R})$. The presentation is self-contained, starting from elementary facts in group theory, and unveiling surprising new results about the geometry of circles, parabolas and hyperbolas. The treatment of elliptic, parabolic and hyperbolic Möbius transformations is provided in a uniform way. This is possible due to an appropriate usage of complex, dual and double numbers, which represent all non-isomorphic commutative associative two-dimensional algebras over the real numbers.

MSC main category: 
51 Geometry

This book is a deep analysis of Möbius transformations from an unusual point of view. The approach is based on the Erlangen programme of Felix Klein, who defined geometry as a study of invariants under a transitive group action. The book focuses on the group $SL_2({\mathbb R})$ and its action by Möbius transformations: $x \mapsto \frac{ax+b}{cx+d}$. This acts on the complex plane, but it also acts on the plane of dual numbers and on the plane of double numbers. Actually, these are the three possible non-isomorphic commutative associative two-dimensional algebras over the real numbers, which are ${\mathbb R}[\sigma]$, with $\sigma^2=-1,0,+1$. The corresponding actions are called elliptic, parabolic and hyperbolic Möbius transformations. The three geometries correspond to the homogeneous spaces with group $SL_2({\mathbb R})$ for the three possible one-dimensional subgroups.

The book studies in depth the geometry associated to the "cycles" in these spaces (circles in the first case, parabolas with horizontal directrix in the second, and equilateral hyperbolas in the third). There is a three dimensional real projective space parametrising such cycles, and a corresponding action of $SL_2({\mathbb R})$ on it. Moreover, there is a naturally defined (indefinite) quadratic form on the space of cycles which serves to recover the initial geometric space, and the usual geometric transformations on it. Then the books moves on to analyse many geometric properties of cycles. This is completed with several aside considerations: the relationship with the physics of Minkowski and Galilean space-time, the more classical point of view of (semi)riemannian geometry, questions on conformal geometry, and more far away subjects like optics or tropical algebra.

The book is accompanied by a DVD with a program which runs under linux (also freely available in internet) which serves the reader to perform computations that appear along the book. This is used often in the book to complete some proofs, which are done by brute force calculation. However, the use of the program requires some knowledge of programming, as the interface is not very user-friendly. This is useful to the reader to complete the arguments and get convinced that the results are true. However, I would have preferred at some points to read a concise and theoretic proof, much more appealing than checking a calculation.

The book is addressed to undergraduate and graduate students in the areas of geometry and algebra. The presentation is basically self-contained. There are many exercises scattered along the book for the interested reader. On the one hand, the point of view is not classical, so a student trying to learn basic properties of Möbius transformations and the relation with complex/Kähler geometry may not get totally satisfied. On the other hand, I think that the author has been successful in transmitting the idea (as he confess in the epilogue that this was his intention) that the three geometries: elliptic, parabolic and hyperbolic deserve to be treated on an equal footing, and that all of them are very rich.

Vicente Muñoz


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