Geometry

274 Pages·2001·32.077 MB·English

by  V. V. Prasolov and V. M. Tikhomirov

Selected Title s i n This Serie s
200 V . V. Prasolov an d V. M. Tikhomirov, Geometry , 2001
199 Shigeyuk i Morita , Geometr y of characteristic classes, 2001
198 V . A. Smirnov, Simplicia l and operad methods in algebraic topology, 2001
197 Kenj i Ueno, Algebrai c geometry 2: Sheaves and cohomology, 2001
196 Yu . N. Lin'kov, Asymptoti c statistical methods for stochastic processes, 2001
195 Minor u Wakimoto, Infinite-dimensiona l Lie algebras, 2001
194 Valer y B. Nevzorov, Records : Mathematical theory, 2001
193 Toshi o Nishino, Functio n theory in several complex variables, 2001
192 Yu . P. Solovyov an d E. V. Troitsky, C*-algebra s and elliptic operators in differentia l
topology, 2001
191 Shun-ich i Amar i an d Hiroshi Nagaoka , Method s of information geometry, 2000
190 Alexande r N . Starkov , Dynamica l systems on homogeneous spaces, 2000
189 Mitsur u Ikawa, Hyperboli c partial differential equations and wave phenomena, 2000
188 V . V. Buldygin and Yu. V. Kozachenko, Metri c characterization of random variables
and random processes, 2000
187 A . V. Pursikov, Optima l control of distributed systems. Theory and applications, 2000
186 Kazuy a Kato, Nobushige Kurokawa , an d Takeshi Saito, Numbe r theory 1:
Fermat's dream, 2000
185 Kenj i Ueno, Algebrai c Geometry 1: From algebraic varieties to schemes, 1999
184 A . V. Mel'nikov, Financia l markets, 1999
183 Hajim e Sato , Algebrai c topology: an intuitive approach, 1999
182 I . S. Krasil'shchik an d A. M. Vinogradov, Editors , Symmetrie s and conservation
laws for differential equations of mathematical physics, 1999
181 Ya . G. Berkovich and E. M. Zhmud' , Character s of finite groups. Part 2, 1999
180 A . A. Milyutin and N. P. Osmolovskii, Calculu s of variations and optimal control,
1998
179 V . E. Voskresenskii, Algebrai c groups and their birational invariants, 1998
178 Mitsu o Morimoto, Analyti c functionals on the sphere, 1998
177 Sator u Igari, Rea l analysis—with an introduction to wavelet theory, 1998
176 L . M. Lerman and Ya. L. Umanskiy, Four-dimensiona l integrable Hamiltonian
systems with simple singular points (topological aspects), 1998
175 S . K. Godunov , Moder n aspects of linear algebra, 1998
174 Ya-Zh e Chen and Lan-Cheng Wu, Secon d order elliptic equations and elliptic
systems, 1998
173 Yu . A. Davydov, M . A. Lifshits, and N. V. Smorodina , Loca l properties of
distributions of stochastic functionals, 199 8
172 Ya . G. Berkovich and E. M. Zhmud' , Character s of finite groups. Part 1, 1998
171 E . M. Landis, Secon d order equations of elliptic and parabolic type, 1998
170 Vikto r Prasolov an d Yuri Solovyev, Ellipti c functions and elliptic integrals, 1997
169 S . K. Godunov , Ordinar y differential equations with constant coefficient, 199 7
168 Junjir o Noguchi, Introductio n to complex analysis, 1998
167 Masay a Yamaguti, Masayoshi Hata, and Jun Kigami, Mathematic s of fractals, 1997
166 Kenj i Ueno, A n introduction to algebraic geometry, 1997
165 V . V. Ishkhanov, B . B. Lur'e, and D. K. Faddeev, Th e embedding problem in
Galois theory, 1997
164 E . I. Gordon, Nonstandar d methods in commutative harmonic analysis, 1997
163 A . Ya. Dorogovtsev, D . S. Silvestrov, A . V. Skorokhod , an d M. I. Yadrenko,
Probability theory: Collection of problems, 1997
(Continued in the back of this publication)
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10.1090/mmono/200
Translations o f
MATHEMATICAL
MONOGRAPHS
Volume 20 0
Geometry
V. V. Prasolov
V. M. Tikhomirov
American Mathematical Societ y
Providence, Rhode Island
^AfDED
EDITORIAL COMMITTE E
AMS Subcommitte e
Robert D. MacPherson
Grigorii A. Margulis
James D. Stasheff (Chair )
ASL Subcommittee Steffe n Lempp (Chair)
IMS Subcommitte e Mar k I. Freidlin (Chair)
B. B . IIpacojiOB, B . M. THXOMHPO B
TEOMETPPM
MUHMO, MocKBa , 1997
Translated from the Russian by O. V. Sipacheva
Translation edited by A. B. Sossinski
2000 Mathematics Subject Classification. Primar y 51-01 , 51M04, 51M09.
ABSTRACT. Thi s book provides a systematic introduction to various geometries, including Eu-
clidean, afBne , projective , elliptic , hyperbolic, an d infinite-dimensional geometries . A unifor m
approach to different geometrie s is suggested, and the unified origins of different phenomen a are
traced. Al l geometric objects are studied from the point of view of duality theory. Th e theory
of conies and quadrics, including the theory of conies for non-Euclidean geometries, is thoroughly
developed. Th e book contains many striking geometric facts and solutions to plenty of beautiful
geometric problems.
Numerous picture s hel p gain a clearer understandin g o f the presented geometri c theorems .
Almost all chapters include problem sections, which allows the book to be used as a textbook.
The majority of problems are supplied with answers and hints, some with complete solutions.
The purpos e o f the book is to contribute t o the development o f research in geometry and
improvement o f mathematical education. Th e book is intended for college undergraduate and
graduate students, high school mathematics teachers, and researchers in mathematics and physics.
Library of Congress Cataloging-in-Publicatio n Dat a
Prasolov, V. V. (Viktor Vasil'evich)
[Geometriia. English ]
Geometry / V.V. Prasolov, V.M. Tikhomirov ; [translated from the Russian by O.V. Sipacheva].
p. cm. — (Translations of mathematical monographs, ISSN 0065-9282 ; v. 200)
Includes bibliographical references and index.
ISBN 0-8218-2038-9 (acid-free paper )
1. Geometry. I . Tikhomirov, V. M. (Vladimir Mikhailovich), 1934- II . Title. III . Series.
QA445.P7313 200 1
516—dc21 200102206 3
© 200 1 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
@ Th e paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Information on copying and reprinting can be found in the back of this volume.
Visit the AMS home page at URL: http://www.ams.org/
10 9 8 7 6 5 4 3 2 1 0 6 05 04 03 02 01
Preface
Once, Israil' Gelfand said that mathematics has three parts: analysis, geom-
etry, an d combinatorics. "Wha t is combinatorics?"—the listeners asked. Th e
answer was: "This is a science not yet created ... "
It is natural to ask: What is geometry?
... I n one of his interviews (published in the popular science journal Kvant),
I. M. Gelfand recollect s his past. H e spent hi s childhood in a tiny town not fa r
from Odessa. Very early, he began to think about mathematical problems—alone,
because he had nobody to ask: there were neither learned people nor books in the
town. H e had to invent and devise things on his own. H e came to the conclusion
that there exists two mathematics separated by an abyss: algebra (actually, algebra
and analysis; the second word had not been familiar to the boy at that time) an d
geometry. Polynomials are algebra, while sines are quite different, they have to do
with geometry.
Gelfand sai d in his interview that, man y times in his life, he had been ver y
lucky. Fo r instance, he asked his parents to buy him a book on mathematics, but
his request was declined because of lack of money. But (oh, luck!) the boy had had
an attack of appendicitis, and it was necessary to go to the city for surgery. Then ,
he said that he would not go anywhere if his parents did not buy the book. An d
so the book was bought.
It wa s an ordinary provincia l calculu s textbook, where , say , function s wer e
classified into continuous, differentiable, and empirical. But, among other stuff, the
book contained the Maclaurin formula and the series
sin x = x - x s/6 + x5/120 .
This led to a revolution i n the boy's mind: h e understood tha t mathematic s i s
indivisible!
So, what is the place of geometry in this indivisible mathematics?
About two decades ago, an interesting discovery in physiology was made: Pro-
fessor Roger Sperry of the California Institute of Technology proved, by inspecting
patients with cut "callou s body" (whic h joins the two cerebral hemispheres), that
the functions o f the hemispheres are somewhat asymmetric . (Fo r this discovery,
R. Sperry won the 1981 Nobel Prize in biology and medicine.)
At present, i t is accepted as proved that th e two hemispheres of the huma n
cerebrum have different functions: the left hemisphere (for right-handers) is respon-
sible for logical analysis. I t also controls speech, writing, and other "algorithmic "
procedures. Thi s is, so to say, the "algebraic-analytical " hemisphere . Th e righ t
hemisphere is, on the contrary, synthetic, it "governs" everything "imaginational, "
such as vision, image perception, imagery ... Thu s we can say that the wrorld of
the right hemisphere is the world of Geometry!
ix
Contents
Preface i x
Introduction 1
Chapter 1. Th e Euclidean World 5
1.1. Th e Euclidean line and plane 5
Cartesian model of the Euclidean straight line and plane 7
The Euclidean plane and complex numbers 1 1
Some problems 1 3
1.2. n-dimensiona l Euclidean space 1 7
The vector space Rn 1 7
The affine space An and the Euclidean space En 2 0
1.3. Introductio n to the multidimensional world
of Euclidean geometry 2 1
Affine varieties 2 2
Determinants and volumes 2 3
Simplices and balls 2 7
Problems 2 7
Chapter 2. Th e Affine World 3 3
2.1. Th e affine line and the affine plane 3 3
Arithmetical model of the affine line 3 3
Arithmetical model of the affine plane 3 3
Linear equations on the plane 3 4
Convex geometry on the plane and the theory of linear inequalities 3 6
The fundamental theorem of affine geometry 3 9
2.2. Affin e space. Theory of linear equations and inequalities 4 2
2.3. Introductio n to finite-dimensional conve x geometry 4 3
The Caratheodory and Radon lemmas 4 3
Helly's theorem 4 4
Problems 4 5
Chapter 3. Th e Projective World 4 7
3.1. Th e projective line and the projective plane 4 7
A model and some facts of the geometry of projective line 4 7
The projective plane 5 2
Pappus' and Desargues' theorems 5 3
3.2. Projectiv e n-space 5 7
Problems 5 9
Chapter 4. Conie s and Quadrics 6 1
vi CONTENT S
4.1. Plan e curves of the second order 6 1
Metric, affine, and projective classification of second-order curves 6 1
The ellipse, hyperbola, and parabola 6 3
The ellipse, hyperbola, and parabola as conic sections 6 8
4.2. Additiona l remarks 6 9
Fourth-degree equations 6 9
The theorem about the conic passing through five points 6 9
The theorem about the pencil of conies passing through four points 7 0
The butterfly problem 7 0
Hyperbolas with perpendicular asymptotes 7 0
Pascal's theorem 7 1
Common chords of two conies inscribed in the same conic 7 2
4.3. Som e properties of quadrics 7 3
Two families of straight lines on a quadric 7 5
Problems 7 7
Chapter 5. Th e World of Non-Euclidean Geometries 8 3
5.1. Th e circle and the two-dimensional sphere: one- and two-dimensional
Riemannian geometries 8 3
The circle and the sphere 8 3
Elementary spherical geometry 86
Geometry of the n-sphere 8 8
Riemannian, or elliptic, geometry 8 8
5.2. Lobachevsk y geometry 8 9
The Klein model of Lobachevsky geometry 8 9
Linear-fractional transformation s and stereographic projections 9 3
Other models of Lobachevsky geometry 9 5
Elementary hyperbolic geometry 10 0
5.3. Isometrie s in the three geometries 10 4
Isometries of Euclidean space 10 5
Isometries of the sphere 11 0
Three types of proper motions of the Lobachevsky plane 11 3
Problems 11 6
Chapter 6. Th e Infinite-Dimensional World 12 3
6.1. Basi c definitions 12 3
6.2. Statement s of theorems 12 6
6.3. Proof s of the theorems 12 8
6.4. Concludin g comments 13 2
Addendum 13 5
1. Geometr y and physics 13 5
Projectiles move along parabolas (Galileo and Newton) 13 5
The planets move along ellipses, and the asteroids, along second-order
curves (Kepler and Newton) 13 6
Geometry and special relativity (Einstein and Minkowski) 13 7
2. Polyhedr a and polygons 14 0
Convex polyhedra 14 0
The Euler-Poincare formula for the alternating sum of the numbers of
faces (of various dimensions) of a convex polyhedron 14 0
CONTENTS vi i
Dual polyhedra 14 2
The Gram-Sommerville formula for the alternating sum of solid angles
at the faces of a convex polyhedron 14 3
The Gauss-Bonnet theorem 14 5
The Minkowski theorem 14 6
The Cauchy theorem on rigid convex polyhedra 14 8
Regular polyhedra 15 1
The Cauchy formula 15 4
The Steiner-Minkowski formula 15 5
Polygons in Rm 15 6
3. Additiona l questions of projective geometry 15 8
The complex projective space CPn 15 8
The polar line of a point with respect to a curve in CP2 16 0
Projective duality 16 3
Fixed points of projective transformations o f the line and Steiner' s
construction 16 6
Projective involutions and harmonic quadruples of points and lines 16 8
Problems 17 1
4. Specia l properties of conies and quadrics 17 4
Confocal conies and quadrics 17 4
Rational parametrizations of conies 17 8
Poncelet's theorem and the zigzag theorem 17 9
The cross ratio of four points on a conic 18 2
Problems 18 5
5. Additiona l topics of non-Euclidean geometries 18 5
Paving the sphere, the plane, and the Lobachevsky plane by triangles 18 5
Fundamental domains of the modular group 18 7
Poincare's theorem about fundamental polygon s 19 0
The Lobachevsky space 19 3
The quaternion model 19 6
About the axiomatic approach to Euclidean and non-Euclidean geometries 196
A brief excursion into the history of non-Euclidean geometry 20 0
Conic sections in spherical and Lobachevsky geometries 20 1
Parabolic mirrors in Lobachevsky geometry 20 5
The volume of a simplex with vertices on the absolute 20 7
Problems 20 9
Solutions, Hints, and Answers 21 1
Chapter 1 21 1
Chapter 2 21 8
Chapter 3 21 9
Chapter 4 22 1
Chapter 5 23 0
Addendum 24 4
Bibliography 25 1
Author Index 25 3
Subject Inde x 255
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