Inversion Theory and Conformal Mapping (Student Mathematical Library, V. 9) by David E. Blair (PDF)

Description

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 Caratheodory with the remarkable result that any circle-preserving transformation is necessarily a Mobius 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.

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⭐Liouville’s Theorem is rarely taught in school (graduate or advanced). It’s usually relegated to outside reading or independent study. It can be found in a number of graduate texts on Riemannian geometry, for instance, do Carmo’s text in chapter 8. Not difficult but involving a bit too much development for its proof explains this back burner treatment. The author assumes the reader has had enough analysis to get through or have done the inverse and implicit function theorems. Some knowledge of complex variables is also assumed. A smattering of differential geometry will also help though the book outlined explanation should suffice. Liouville’s Theorem is one of those miraculous subtle theorems which leads to conclusions which are not obvious. It has application in the field of optics and in fact justifies our techniques or way of doing geometrical optics. In chapter 3 of the famous Born and Wolf text, Maxwell’s Theorem on perfect imaging is proven showing the image to be conformal or a conformal mapping. Enter Liouville: the only conformal mappings in 3-space are inversions in a sphere and projectivities (includes similarities) and compositions of these. Pretty much we have geometrical optics in a nutshell. The development is as I said involved but if you force yourself to think about the need for the steps you’ll be rewarded. And with that little background, it’s self-contained!

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