نظریه وارونگی و نگاشت هم‌ریخت (کتابخانه ریاضی دانشجویی، جلد ۹)

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. INVERSION THEORY AND CONFORMAL MAPPING......Page 1 Title Page......Page 5 Copyright Page......Page 0 Dedication......Page 7 Contents......Page 9 Preface......Page 11 1.1. Definition and basic properties......Page 13 1.2. Cross ratio......Page 21 1.3. Applications......Page 26 1.4. Miquel's Theorem......Page 29 1.5. Feuerbach's Theorem......Page 33 2.1. Complex numbers......Page 39 2.2. The extended complex plane and stereographic projection......Page 41 2.3. Linear fractional transformations......Page 46 2.4. Cross ratio......Page 49 2.5. Some special linear fractional transformations......Page 51 2.6. Extended Möbius transformations......Page 55 "The Most General Transformations of Plane Regions Which Transform Circles into Circles," by Constantin Carathéodory......Page 57 2.7. The Poincaré models of hyperbolic geometry......Page 64 2.8. A distortion theorem......Page 71 3.1. Review of advanced calculus......Page 75 3.2. Inner products......Page 82 3.3. Conformal maps......Page 85 4.1. Complex function theory......Page 87 4.2. Abundance of conformal maps......Page 90 5.1. Inversion in spheres......Page 95 5.2. Conformal maps in Euclidean space......Page 99 5.3. Sphere preserving transformations......Page 104 6.1. Surface theory......Page 107 6.2. The classical proof......Page 115 7.1. Curve theory and convexity......Page 119 7.2. Inversion and convexity......Page 122 7.3. The problem for convex bodies......Page 126 Bibliography......Page 127 Index......Page 129 Back Cover......Page 132

It is rarely taught in an undergraduate or even graduate curriculum 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. 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 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 independent study. 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.

It is rarely taught in an undergraduate or even graduate curriculum 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. 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 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 independent study. 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. "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 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 advanced undergraduate level and is suitable for a capstone course, topics course, senior seminar or independent study. 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."--Page 4 de la couverture "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 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 advanced undergraduate level and is suitable for a capstone course, topics course, senor seminar or independent study. 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."--Jacket