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

244 Pages·1988·4.412 MB·German

by  Ravi S. Kulkarni (auth.), Ravi S. Kulkarni, Ulrich Pinkall (eds.)

Ravi S. Kulkarni, Ulrich Pinkall (Eds.)
Conformal Geometry
Aspects of Ma1hematics
Aspekte Mathematik
der
Editor: Klas Diederich
All volumes of the series are listed on pages 237-238.
Ravi S. Kulkarni, Ulrich Pinkall (Eds.)
Conformal
Geometry
A Publication of the Max-Planck-Institut fUr Mathematik, Bonn
Adviser: Friedrich Hirzebruch
Friedr. Vieweg & Sohn Braunschweig I Wiesbaden
CIP-Titelaufnahme der Deutschen Bibliothek
Conformal geometry:
a pub I. of the Max-Planck-Inst. fur Mathematik,
Bonn/Ravi S. Kulkarni; Ulrich Pinkall (ed.).
Advisor: Friedrich Hirzebruch. - Braunschweig;
Wiesbaden: Vieweg, 1988
(Aspects of mathematics: E; Vol. 12)
ISBN 978-3-528-08982-5 ISBN 978-3-322-90616-8 (eBook)
DOl 10.1007/978-3-322-90616-8
NE: Kulkarni, Ravi S. [Hrsg.J; Max-Planck
Institut fur Mathematik (Bonn); Aspects of
mathematics 1E
AMS Subject Classification: 53 A 30
Vieweg is a subsidiary company of the Bertelsmann Publishing Group.
All rights reserved
© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1988
No part of this publication may be reproduced, stored in a retrieval
system or transmitted, mechanical, photocopying or otherwise, without
prior permission of the copyright holder.
Produced by W. Langeluddecke, Braunschweig
Contents
Authors of this Volume VI
Preface VII
Conformal Structures and Mobius Structures
Ravi S. Kulkarni
Conjugacy Classes in M (n) 41
Ravi S. Kulkarni
Conformal Geometry from the Riemannian Viewpoint 65
Jacques Lafontaine
The Theorem of Lelong-Ferrand and Obata 93
Jacques Lafontaine
Conformal Transformations between Einstein Spaces 105
Wolfgang Kuhnel
Topics in the Theory of Quasiregular Mappings 147
Seppo Rickman
Conformal and Isometric I mmersions of Conformaliy Flat Riemannian Manifolds
into Spheres and Euclidean Spaces 191
Hans-Bert Rademacher
Compact Conformally Flat Hypersurfaces 217
Ulrich Pinkall
The reader can find a detailed table of contents at the beginning of each article.
Authors of this Volume
Wolfgang Kuhnel
Fachbereich Mathematik
Universitiit Duisburg
D-4100 Duisburg 1
FRG
Ravi S. Kulkarni
Department of Mathematics
City University of New York,
Queens College,
Flushing, NY 11367
USA
Jacques Lafontaine
U.E.R. de Mathematique
Universite Paris 7
F-75251 Paris Cedex 05
France
Ulrich Pinkall
Fachbereich Mathematik
TU Berlin
StraBe des 17. Juni 135
D -1 000 Berlin 12
Hans-Bert Rademacher
Mathematisches I nstitut
Universitat Bonn
Wegelerstr. 10
D-5300 Bonn 1
FRG
Seppo Rickman
Department of Mathematics
University of Helsinki
SF-00100 Helsinki
Finland
Preface
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
§4 The GroupsM(n) andM(En) 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. It was an indication that the
conformal maps among the surfaces have far more flexibility than for example the
isometries among surfaces, or line-preserving maps among planar domains. This
was confirmed by Gauss in his A general solution to the problem of mapping parts
of a given surface onto another surface such that the image and the mapped parts
are similar in the smallest parts. This is esentially the existence of "isothermal
co-ordinates" in the CW case. It is interesting to note that this study preceded and
partially motivated Gauss's later foundational work on the notion of curvature. For
an account of this interesting history see Dombrowski [DJ, pp 127-130.
(0.2) Another equally startling discovery is the connection of isothermal coor
dinates to an entirely different idea, namely that of a holomorphic function of a
complex variable. For example, the Mercator projection is essentially the holomor
phic map z 1--+ log z ! The global aspect of this is the theory of Riemann surfaces.
(0.3) Still another startling fact is that a compact Riemann surface is essentially
the same as a complex projective algebraic curve.
(0.4) These deep local and global connections set up one of the natural goals
of conformal geometry: namely, to understand the differential-geometric under
pinnings of these classical theories-i.e., to separate the analytic aspects from the
topological ones as clearly as possible, and relate them to the more primitive geo
metric notions of distance, angle, area, straight line ... , and equally importantly, to
isolate the role of symmetry. The symmetry considerations here mainly concern the
groups of Mobius transformations-in particular, the classical theories of Fuchsian
and Kleinian groups. Due to certain isomorphisms such as Mo(1) ~ PSL2(R) ~
SOo(2, 1), Mo(2) ~ PSL2(C) ~ SOo(3, 1)··· the conformal considerations have
turned out to be basic for 3- and 4-dimensional topology, and in the physics of
relativity.
(0.5) In the first seven sections of this chapter we have striven to bring together
the traditional differential-geometric, and the traditional Kleinian-group theoretic
viewpoints. These two traditions have developed (until quite recently) quite inde
pendently. Their mutual interaction should prove to be fruitful. We have isolated
the notion of a "Mobius structure" from the general conformal structure. A geome
ter would find it useful to keep in mind the following facts.
3
i) A 2-dimensional conformal structure is always integrable, but is ambient to sev
eral Mobius structures. (Example: the stereographic projection is circle-preserving,
i.e., a Mobius map, whereas the Mercator projection preserves angles but not cir
cles, i.e., is not a Mobius map.)
ii) In dimensions 2: 3, a conformal structure is not always integrable, but an inte
grable one is ambient to a unique Mobius structure. The latter is essentially the
Liouville's theorem. The proof of Liouville's theorem given here is quite elemen
tary and works in the C2 case.! In general, the "Mobius" arguments, provided they
work at all, are aften simpler and give better results.
The last section indicates some recent developments in the joint work with U.
Pinkall.
The author is thankful to L. Mansfield for his considerable help in drawing and
inserting the figures in this paper.
! The reduction from C3 to C2 was pointed out by Dombrowski.