Geometry from a Differentiable Viewpoint

322 Pages·1995·5.766 MB·English

by  John McCleary

John McCleary
Geometry from a Differentiable Viewpoint

Geometry
from a Differentiable Viewpoint
JOHN McCLEARY
Vassar College
CAMBRIDGE
UNIVERSITY PRESS
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
The Pitt Budding, Trumpington Street, Cambridge CB2 IRP, United Kingdom
CAMBRIDGE UNIVERSITY PRESS
The Edinburgh Building, Cambridge CB2 2RU, United Kingdom
40 West 20th Street, New York, NY 10011-4211, USA
10 Stamford Road, Oakleigh, Melbourne 3166, Australia
® Cambridge University Press 1994
This book is in copyright. Subject to statutory exception and
to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1994
Reprinted 1996, 1997
Printed in the United States of America
Typeset in Garamond
A catalogue record for tlus book is available from the British Library
Library of Congress Cataloguing-in-Publication Data is available
ISBN 0-521-41430-X hardback
ISBN 0-521-42480-1 paperback
To my sisters
Marv Ann, Denise, and Rose

Contents
Introduction page ix
PART A
Prelude and themes: Synthetic methods and results
I. Spherical geometry 3
2. Euclid 10
Euclid's theory of parallels 16
Appendix. The Elements: Book 1 21
3. The theory of parallels 24
Uniqueness of parallels 24
Equidistance and boundedness of parallels 26
On the angle sum of a triangle 28
Similarity of triangles 31
4. Non-Euclidean geometry 1 34
The work of Saccheri 34
The work of Gauss, Bolyai, and Lobachevskii 39
5. Non-Euclidean geometry 11 45
The circumference of a circle 56
PART B
Development: Differential geometry
6. Curves
Early work on plane curves (Huygens, Leibniz. Newton, Euler)
The tractrix
Directed curvature
Digression: Involutes and evolutes
7. Curves in space
Appendix: On Euclidean rigid motions
8. Surfaces
The tangent plane
The first fundamental form
Area
8°i. Map projections
Stereographic projection
Vii
viii Contents
Central projection
123
Mercator projection
124
Lambert's cylindrical projection
126
Azimuthal projection
126
Sample map projections
127
9. Curvature for surfaces
131
Euler's work on surfaces
131
The Gauss map
134
10. Metric equivalence of surfaces 145
Special coordinates
151
11. Geodesics 157
Euclid revisited I: The Hopf-Rinow theorem
165
12. The Gauss-Bonnet theorem
171
Euclid revisited I1: Uniqueness of lines
175
Compact surfaces
176
A digression on curves 180
13. Constant-curvature surfaces 186
Euclid revisited III: Congruences
191
The work of Minding
192
PART C
Recapitulation and coda
14. Abstract surfaces 201
Hilbert's theorem 203
Abstract surfaces 206
15. Modeling the non-Euclidean plane 217
The Beltrami disk 220
The Poincar6 disk 224
The Poincar6 half-plane 227
16. Epilog: Where from here? 242
Manifolds (differential topology) 243
Vector and tensor fields 247
Metrical relations (Riemannian manifolds) 249
Curvature 252
Covariant differentiation 261
Riemann's Habilitationsvortrag: On the hypotheses which lie at the foundations
of geometry 269
Appendix: Notes on selected exercises 279
Bibliography 297
Symbol index 303
Name index 304
Subject index 305
Introduction
ArEaMETPHTOE MHAEIE E!ElTn.
Over the entrance to Plato's Academy
One of the many roles of history is to tell a story. The history of the Parallel Postulate is a
great story - it spans more than two millennia, stars an impressive cast of characters, and
contains some of the most beautiful results in all of mathematics. My immodest goal for
this book is to tell this story.
Another role of history is to focus our attention and so to provide a thread of unity through
a parade of events, people, and ideas. My goal grows small and quite modest before all of
Geometry, especially its recent history. A more modest goal then is to provide a focus in
which to view the standard tools of differential geometry, and in so doing offer an exposition,
motivated by the history, that prepares the reader for the modern, global foundations of the
subject.
In recent years, it has become a luxury to offer a course in differential geometry in an
undergraduate curriculum. When such a course exists, its students often arrive with a modern
introduction to analysis, but without having seen geometry since high school. In the United
States geometry taught in high schools is generally elementary Euclidean geometry based
on Hilbert's axiom scheme. The beautiful world of non-Euclidean geometry is relegated to
a footnote, enrichment material, or a "cultural" essay. This is also the case in most current
introductions to differential geometry. The modern subject turns on problems that have
emerged from the new foundations that are far removed from the ancient roots of geometry.
When we teach the new and cut off the past, students are left to find their own way to
a meaning of geometry in differential geometry, or to identify their activity as something
different, unconnected.
This book is an attempt to carry the reader from the familiar Euclid to the state of
development of differential geometry at the beginning of the twentieth century. One narrow
thread that runs through this vast historical period is the search for a proof of Euclid's
Postulate V. the Parallel Postulate, and the eventual emergence of a new and non-Euclidean
geometry. In the course of spinning out this tale, another theme enters - the importance of
properties of a surface that are intrinsic, that is, independent of the manner in which the
surface is embedded in space. This idea, emphasized by Gauss, provides an analytic key
concerning the properties that are really geometric, and it introduces new realms to explore.
The book is written in sonata-allegro form. Pan A opens with a prelude - a small dose
of spherical geometry, in which some of the important ideas of non-Euclidean geometry
are touched on. One of the main themes of the sonata is given in Chapters 2 and 3, which
ix