Conformal Geometry: A Publication of the Max-Planck-Institut Für Mathematik, Bonn

ISBN-13: 9783528089825 / Angielski / Miękka / 1988 / 246 str.

Ravi S. Kulkarni; Ravi S. Kulkarni

Opis

The contributions in this volume summarize parts of a seminar on conformal geometry which was held at the Max-Planck-Institut fur Mathematik in Bonn during the academic year 1985/86. The intention of this seminar was to study conformal structures on mani folds from various viewpoints. The motivation to publish seminar notes grew out of the fact that in spite of the basic importance of this field to many topics of current interest (low-dimensional topology, analysis on manifolds . . . ) there seems to be no coherent introduction to conformal geometry in the literature. We have tried to make the material presented in this book self-contained, so it should be accessible to students with some background in differential geometry. Moreover, we hope that it will be useful as a reference and as a source of inspiration for further research. Ravi Kulkarni/Ulrich Pinkall Conformal Structures and Mobius Structures Ravi S. Kulkarni* Contents 0 Introduction 2 1 Conformal Structures 4 2 Conformal Change of a Metric, Mobius Structures 8 3 Liouville's Theorem 12 n 4 The GroupsM(n) andM(E ) 13 5 Connection with Hyperbol ic Geometry 16 6 Constructions of Mobius Manifolds 21 7 Development and Holonomy 31 8 Ideal Boundary, Classification of Mobius Structures 35 * Partially supported by the Max-Planck-Institut fur Mathematik, Bonn, and an NSF grant. 2 O Introduction (0. 1) Historically, the stereographic projection and the Mercator projection must have appeared to mathematicians very startling.

Spis treści

Conformai Structures and Möbius Structures.- Conjugacy Classes in M(n).- Conformai Geometry from the Riemannian Viewpoint.- The Theorem of Lelong-Ferrand and Obata.- Conformai Transformations between Einstein Spaces.- Topics in the Theory of Quasiregular Mappings.- Conformai and Isometric Immersions of Conformally Flat Riemannian Manifolds into Spheres and Euclidean Spaces.- Compact Conformally Flat Hypersurfaces.