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Curved Spaces
From Classical Geometries to
Elementary Differential Geometry
P. M. H. Wilson
Department of Pure Mathematics, University of Cambridge,
and Trinity College, Cambridge
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
Information on this title: 0521886291
© P. M. H. Wilson 2008
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2007
ISBN-13 978-0-511-37757-0 eBook (EBL)
ISBN-13 978-0-521-88629-1 hardback
ISBN-13 978-0-521-71390-0 paperback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
For Stanzi, Toby and Alexia,
in the hope that one day
they might understand what is written herein,
and to Sibylle.
Contents
Preface page ix
1 Euclidean geometry 1
Euclidean space 1
Isometries 4
The group O(3, R) 9
Curves and their lengths 11
pactness 15
Polygons in the Euclidean plane 17
Exercises 22
2 Spherical geometry 25
Introduction 25
Spherical triangles 26
Curves on the sphere 29
Finite groups of isometries 31
Gauss– and spherical polygons 34
Möbius geometry 39
The double cover of SO(3) 42
Circles on S2 45
Exercises 47
3 Triangulations and Euler numbers 51
Geometry of the torus 51
Triangulations 55
Polygonal positions 59
Topology of the g-holed torus