Problems in Geometry

276 Pages·1984·3.61 MB·English

by  Berger

Problem Books in Mathematics
M. Berger
P.Pansu
F''
J.P. Berry
X. Saint-Raymond
F
Problems
in Geometry.
Springer-Verlag
NewYork Berlin Heidelberg

Marcel Berger
Pierre Pansu
Jean-Pic Berry
Xavier Saint-Raymond
Problems in Geometry
Translated by Silvio Levy
With 224 Illustrations
Springer-Verlag
New York Berlin Heidelberg Tokyo
Marcel Berger Pierre Pansu
U.E.R. de Mathematique Xavier Saint-Raymond
et Informatique Centre National de Ia Recherche
Université Paris VII Scientifique
75251 Paris, Cedex 05 Paris
France France
Jean-Pic Berry Silvio Levy (Translator)
P.U.K. Mathematics Department
Grenoble Princeton University
France Princeton, NJ 08540
U.S.A.
Editor
Paul R. Halmos
Department of Mathematics
Indiana University
Bloomington, IN 47405
U.S.A.
AMS Subject Classifications: 51-01, 52-01, 53-01, 00A07
Library of Congress Cataloging in Publication Data
G&,métrie, problêmes de géométrie et rédigés.
English.
Problems in geometry.
(Problem books in mathematics)
Translation of: Géometrie, problèmes de g&métrie
commentés et rédigés
Includes index.
2. (}eometry—Problems, exercises, etc. I. Berger,
Maivel 1927- II. Title. III. Series.
QA445.G44513 1984 516'.0076 84-5495
Original French edition: Problêmes de Commentés et Rédigés, CEDIC,
Paris, © 1982.
© 1984 by Marcel Berger.
All rights reserved. No part of this book may be translated or reproduced in any form
without written permission from the copyright holder.
Media conversion by Science Typographers, Medford, $
Printed and bound by R.R. Donnelley & Sons, Harris Virgintk.
Printed in the United States of America.
9208418
987654321
ISBN 0-387-90971-0 Springer-Verlag New York Berlin Heidelberg Tokyo
ISBN 3-540-90971-0 Springer-Verlag Berlin Heidelberg New York Tokyo
Preface
The textbook Geometry,published in French by CEDIC/Fernand Nathan and
in English by Springer-Verlag (scheduled for 1985) was very favorably re-
ceived. Nevertheless, many readers found the text too concise and the exercises
at the end of each chapter too difficult, and regretted the absence of any hints
for the solution of the exercises.
This book is intended to respond, at least in part, to these needs. The length
of the textbook (which will be referred to as [BJ throughout this book) and the
volume of the material covered in it preclude any thought of publishing an
expanded version, but we considered that it might prove both profitable and
amusing to some of our readers to have detailed solutions to some of the
exercises in the textbook.
At the same time, we planned this book to be independent, at least to a
certain extent, from the textbook; thus, we have provided summaries of each of
its twenty chapters, condensing in a few pages and under the same titles the
most important notions and results used in the solution of the problems. The
statement of the selected problems follows each summary, and they are
numbered in order, with a reference to the corresponding place in [BJ. These
references are not meant as indications for the solutions of the problems. In
the body of each summary there are frequent references to FBI, and these can
be helpful in elaborating a point which is discussed too cursorily in this book.
Following the summaries we included a number of suggestions and hints for
the solution of the problems; they may well be an intermediate step between
your personal solution and ours!
The bulk of the book is dedicated to a fairly detailed solution of each
problem, with references to both this book and the textbook. Following the
practice in [BI, we have made liberal use of illustrations throughout the text.
Finally, I would 111cc to express my heartfelt thanks to Springer-Verlag, for
including this work in their Problem Books in Mathematics series, and to Silvio
Levy, for his excellent and speedy translation.
Marcel Berger
v

Contents
Chapter 1. Groups Operating on a Set: Nomenclature, Examples,
Applications 1
Problems 9
Chapter 2. Affine Spaces 11
Problems 15
Chapter 3. Barycenters; the Universal Space 18
Problems 21
Chapter 4. Projective Spaces 23
Problems 28
Chapter 5. Affine-Projective Relationship: Applications 30
Problems 33
Chapter 6. Projective Lines, Homographies 35
Problems 38
Chapter 7. Complexifications 40
Problems 42
Chapter 8. More about Eucidean Vector Spaces 43
Problems 48
Cha$er 9. Eucidean Afline Spaces 51
Problems 56
Chapter 10. Triangles, Spheres, and Circles 58
Problems 62
Chapter 11. Convex Sets 66
Problems 67
Chapter 12. Polytopes; Compact Convex Sets 69
Problems 71
vii
Chapter 13. Quadratic Forms 74
Problems 77
Chapter 14. ProjectiveQuadrics 79
Problems 84
Chapter 15. Affine Quadrics 85
Problems 89
Chapter 16. Projective Conies 93
Problems 99
Chapter 17. Eucidean Conics 102
Problems 105
Chapter 18. The Sphere for Its Own Sake 106
Problems 110
Chapter 19. Elliptic and Hyperbolic Geometry 114
Problems 118
Chapter 20. The Space of Spheres 120
Problems 122
Suggestions and Hints 124
Solutions
Chapter 1 132
Chapter 2 143
Chapter 3 150
Chapter 4 156
Chapter 5 163
Chapter 6 166
Chapter 7 173
Chapter 8 173
Chapter 9 180
Chapter 10 188
Chapter 11 202
Chapter 12 207
Chapter 13 218
Chapter 14 220
Chapter 15 224
Chapter 16 232
Chapter 17 238
Chapter 18 244
Chapter 19 251
Chapter 20 258
Index 263
Chapter 1
Groups Operating on a Set:
Nomenclature, Examples, Applications
l.A Operation of a Group on a Set ([B, 1.1])
Let X be a set; we denote by the set of all bijections (permutations) f:
X —.X from X into itself, and we endow x with the law of composition
f
given by (f, g) —, o g. Thus x becomes a group, called the pennutation
group or symmetric group of X.
Now let G be a group and X a set; we say that G operates (or acts) on Xby
q if q is a homomorphism of G into i.e. if ci verifies the following
axioms:
p(g) E all g E G;
(gh)=.p(g)o.p(h),aJl g,hEG
(the law of composition of G is written multiplicatively). We will generally use
the abbreviation g(x) instead of
p on X is said to be transitive if for every pair (x, y) there
is a g such that g(x) =y. If G operates transitively on X, we say also that G
is a homogeneous space (for G).
1.C The Erlangen Program: Geometries
The situation where a group G operates transitively on a set X is a very
frequent one in geometry. This in fact led Felix Klein, in his famous Erlangen
program of 1872, to define a geometry as the study of the properties of a set
1