Inversion Theory and Conformal Mapping

by Blair, David E.

• ISBN13: 9780821826362
• ISBN10: 0821826360
• Format: Paperback
• Copyright: 2000-09-01
• Publisher: Amer Mathematical Society
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Summary

It is rarely taught in undergraduate or even graduate curricula that the only conformal maps in Euclidean space of dimension greater than two are those generated by similarities and inversions in spheres. This is in stark contrast to the wealth of conformal maps in the plane. This fact is taught in most complex analysis courses. The principal aim of this text is to give a treatment of this paucity of conformal maps in higher dimensions. The exposition includes both an analytic proof, due to Nevanlinna, in general dimension and a differential geometric proof in dimension three. For completeness, enough complex analysis is developed to prove the abundance of conformal maps in the plane. In addition, the book develops inversion theory as a subject, along with the auxiliary theme of circle-preserving maps. A particular feature is the inclusion of a paper by Carathéodory with the remarkable result that any circle-preserving transformation is necessarily a Möbius transformation--not even the continuity of the transformation is assumed. The text is at the level of advanced undergraduates and is suitable for a capstone course, topics course, senior seminar or as an independent study text. Students and readers with university courses in differential geometry or complex analysis bring with them background to build on, but such courses are not essential prerequisites.

Table of Contents

Preface	ix
Classical Inversion Theory in the Piane	1	(26)
Definition and basic properties	1	(8)
Cross ratio	9	(5)
Applications	14	(3)
Miquel's Theorem	17	(4)
Feuerbach's Theorem	21	(6)
Linear Fractional Transformations	27	(36)
Complex numbers	27	(2)
The extended complex plane and stereographic projection	29	(5)
Linear fractional transformations	34	(3)
Cross ratio	37	(2)
Some special linear fractional transformations	39	(4)
Extended Mobius transformations	43	(9)
The Poincare models of hyperbolic geometry	52	(7)
A distortion theorem	59	(4)
Advanced Calculus and Conformal Maps	63	(12)
Review of advanced calculus	63	(7)
Inner products	70	(3)
Conformal maps	73	(2)
Conformal Maps in the Plane	75	(8)
Complex function theory	75	(3)
Abundance of conformal maps	78	(5)
Conformal Maps in Euclidean Space	83	(12)
Inversion in spheres	83	(4)
Conformal maps in Euclidean space	87	(5)
Sphere preserving transformations	92	(3)
The Classical Proof of Liouville's Theorem	95	(12)
Surface theory	95	(8)
The classical proof	103	(4)
When Does Inversion Preserve Convexity?	107	(8)
Curve theory and convexity	107	(3)
Inversion and convexity	110	(4)
The problem for convex bodies	114	(1)
Bibliography	115	(2)
Index	117