Two-Dimensional Spaces, Volume 3 NON-EUCLIDEAN GEOMETRY ... · Two-Dimensional Spaces, Volume 3 NON-EUCLIDEAN GEOMETRY AND CURVATURE 10.1090/mbk/110

Two-Dimensional Spaces, Volume 3

NON-EUCLIDEAN GEOMETRY AND CURVATURE James W. Cannon


NON-EUCLIDEAN GEOMETRY AND CURVATURE

10.1090/mbk/110

A M E R I C A N M A T H E M A T I C A L S O C I E T Y

P r o v i d e n c e , R h o d e I s l a n d


NON-EUCLIDEAN GEOMETRY AND CURVATURE James W. Cannon

2010 Mathematics Subject Classification. Primary 51M10, 53A35, 53A05.

For additional information and updates on this book, visitwww.ams.org/bookpages/mbk-110

Library of Congress Cataloging-in-Publication Data

Names: Cannon, James W., author.Title: Two-dimensional spaces / James W. Cannon.Description: Providence, Rhode Island : American Mathematical Society, [2017] | Includes bibli-

ographical references.Identifiers: LCCN 2017024690 | ISBN 9781470437145 (v. 1) | ISBN 9781470437152 (v. 2) | ISBN

9781470437169 (v. 3)Subjects: LCSH: Geometry. | Geometry, Plane. | Non-Euclidean geometry. | AMS: Geometry –

Instructional exposition (textbooks, tutorial papers, etc.). mscClassification: LCC QA445 .C27 2017 | DDC 516–dc23LC record available at https://lccn.loc.gov/2017024690

Copying and reprinting. Individual readers of this publication, and nonprofit libraries actingfor them, are permitted to make fair use of the material, such as to copy select pages for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publicationis permitted only under license from the American Mathematical Society. Permissions to reuseportions of AMS publication content are handled by Copyright Clearance Center’s RightsLink�service. For more information, please visit: http://www.ams.org/rightslink.

Send requests for translation rights and licensed reprints to [email protected] from these provisions is material for which the author holds copyright. In such cases,

requests for permission to reuse or reprint material should be addressed directly to the author(s).Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of thefirst page of each article within proceedings volumes.

c© 2017 by the American Mathematical Society. All rights reserved.The American Mathematical Society retains all rightsexcept those granted to the United States Government.

Printed in the United States of America.

©∞ The paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 22 21 20 19 18 17

Contents

Preface to the Three Volume Set vii

Preface to Volume 3 xi

Chapter 1. A Graphical Introduction to Hyperbolic Geometry 11.1. The Creation of Hyperbolic Geometry 11.2. Graphical Approximations to Non-Euclidean Geometry 21.3. Counting Layer Size Combinatorially 41.4. Asymptotic Growth Rate 51.5. Another Graph Model of Hyperbolic Space 61.6. Exercises 8

Chapter 2. Hyperbolic Geometry 112.1. Introduction 112.2. The Origins of Hyperbolic Geometry 122.3. Why Call It Hyperbolic Geometry? 142.4. Understanding the One-dimensional Case 152.5. Generalizing to Higher Dimensions 182.6. Rudiments of Riemannian Geometry 182.7. Five Models of Hyperbolic Space 192.8. Stereographic Projection 222.9. Geodesics 262.10. Isometries and Distances in the Hyperboloid Model 302.11. The Space at Infinity 322.12. The Geometric Classification of Isometries 332.13. Curious Facts about Hyperbolic Space 342.14. The Sixth Model 432.15. Why Study Hyperbolic Geometry? 452.16. When Does a Manifold Have a Hyperbolic Structure? 492.17. How to get Analytic Coordinates at Infinity? 52

Chapter 3. Gravity As Curvature 553.1. Einstein Identified Gravity with the Curvature of Space-Time 55

Chapter 4. Curvature by Polyhedral Approximation 574.1. Approximating Smooth Surfaces by Polyhedra 574.2. The Curvature of a Polyhedral Disk 574.3. How Flat Is a Disk? 574.4. How Straight Is a Disk Boundary? 594.5. Duality Theorem: Angle Defect + Boundary Defect = 2π 614.6. The Curvature of a Polyhedral Disk 63

v

vi CONTENTS

4.7. Applications of the Duality Between Angle Defectand Boundary Defect 63

4.8. The Curvature of a Smooth Disk 66

Chapter 5. Curvature As a Length Derivative 695.1. Straight Lines and Boundary Defects on Smooth Surfaces 695.2. The Length Derivative in a Polyhedral Surface 715.3. The Length Derivative on a Cone 755.4. Evaluating the Double Limit 78

Chapter 6. Theorema Egregium 816.1. Umlauf Theorem: Smooth Version 816.2. The Theorem 816.3. Outline of the Proof 826.4. Flexibility in Calculating Boundary Defect 826.5. Completing the Proof of Gauss’s Theorema Egregium 85

Chapter 7. Curvature Appendix 877.1. The Generalized Umlauf Theorem 877.2. Two Technical Properties of a Smooth Surface 887.3. Specialized Polyhedral Approximations to D′. 907.4. Specialized Polyhedral Approximations in the Plane. 907.5. Specialized Polyhedral Approximations on Curved Surfaces. 927.6. Outline of the Proof of the Theorem. 947.7. Exercises 98

Bibliography 99

Preface to the Three Volume Set

Geometry measures space (geo = earth, metry = measurement). Einstein’stheory of relativity measures space-time and might be called geochronometry (geo= space, chrono= time, metry = measurement). The arc of mathematical historythat has led us from the geometry of the plane of Euclid and the Greeks after 2500years to the physics of space-time of Einstein is an attractive mathematical story.Geometrical reasoning has proved instrumental in our understanding of the real andcomplex numbers, algebra and number theory, the development of calculus with itselaboration in analysis and differential equations, our notions of length, area, andvolume, motion, symmetry, topology, and curvature.

These three volumes form a very personal excursion through those parts of themathematics of 1- and 2-dimensional geometry that I have found magical. In allcases, this point of view is the one most meaningful to me. Every section is designedaround results that, as a student, I found interesting in themselves and not justas preparation for something to come later. Where is the magic? Why are thesethings true?

Where is the tension? Every good theorem should have tensionbetween hypothesis and conclusion. — Dennis Sullivan

Where is the Sullivan tension in the statement and proofs of the theorems?What are the key ideas? Why is the given proof natural? Are the theorems almostfalse? Is there a nice picture? I am not interested in quoting results without proof.I am not afraid of a little algebra, or calculus, or linear algebra. I do not careabout complete rigor. I want to understand. If every formula in a book cuts thereadership in half, my audience is a small, elite audience. This book is for thestudent who likes the magic and wants to understand.

A scientist is someone who is always a child, asking ‘Why?why? why?’. — Isidor Isaac Rabi, Nobel Prize in Physics1944

Wir mussen wissen, wir werden wissen. [We must know, wewill know.] — David Hilbert

The three volumes indicate three natural parts into which the material on 2-dimensional spaces may be divided:

Volume 1: The geometry of the plane, with various historical attempts tounderstand lengths and areas: areas by similarity, by cut and paste, by counting,by slicing. Applications to the understanding of the real numbers, algebra, numbertheory, and the development of calculus. Limitations imposed on the measurementof size given by nonmeasurable sets and the wonderful Hausdorff-Banach-Tarskiparadox.

vii

viii PREFACE TO THE THREE VOLUME SET

Volume 2: The topology of the plane, with all of the standard theorems of1- and 2-dimensional topology, the fundamental theorem of algebra, the Brouwerfixed-point theorem, space-filling curves, curves of positive area, the Jordan curvetheorem, the topological characterization of the plane, the Schoenflies theorem,the R. L. Moore decomposition theorem, the open mapping theorem, the trian-gulation of 2-manifolds, the classification of 2-manifolds via orientation and Eulercharacteristic, dimension theory.

Volume 3: An introduction to non-Euclidean geometry and curvature. Whatis the analogy between the standard trigonometric functions and the hyperbolictrig functions? Why is non-Euclidean geometry called hyperbolic? What are thegross intuitive differences between Euclidean and hyperbolic geometry?

The approach to curvature is backwards to that of Gauss, with definitionsthat are obviously invariant under bending, with the intent that curvature shouldobviously measure the degree to which a surface cannot be flattened into the plane.Gauss’s Theorema Egregium then comes at the end of the discussion.

Prerequisites: An undergraduate student with a reasonable memory of cal-culus and linear algebra, but with no fear of proofs, should be able to understandalmost all of the first volume. A student with the rudiments of topology—open andclosed sets, continuous functions, compact sets and uniform continuity—should beable to understand almost all of the second volume with the exeption of a littlebit of algebraic topology used to prove results that are intuitively reasonable andcan be assumed if necessary. The final volume should be well within the reach ofsomeone who is comfortable with integration and change of variables. We will makean attempt in many places to review the tools needed.

Comments on exercises: Most exercises are interlaced with the text in thoseplaces where the development suggests them. They are an essential part of thetext, and the reader should at least make note of their content. Exercise sectionswhich appear at the end of most chapters refer back to these exercises, sometimeswith hints, occasionally with solutions, and sometimes add additional exercises.Readers should try as many exercises as attract them, first without looking at hintsor solutions.

Comments on difficulty: Typically, sections and chapters become more diffi-cult toward the end. Don’t be afraid to quit a chapter when it becomes too difficult.Digest as much as interests you and move on to the next chapter or section.

Comments on the bibliography: The book was written with very littledirect reference to sources, and many of the proofs may therefore differ from thestandard ones. But there are many wonderful books and wonderful teachers thatwe can learn from. I have therefore collected an annotated bibliography that youmay want to explore. I particularly recommend [1, G. H. Hardy, A Mathematician’sApology ], [2, G. Polya, How to Solve It ], and [3, T. W. Korner, The Pleasure ofCounting ], just for fun, light reading. For a bit of hero worship, I also recommendthe biographical references [21, E. T. Bell, Men of Mathematics ], [22, C. Henrion,Women of Mathematics ], and [23, W. Dunham, Journey Through Genius ]. AndI have to thank my particular heroes: my brother Larry, who taught me aboutuncountable sets, space-filling curves, and mathematical induction; Georg Polya,who invited me into his home and showed me his mathematical notebooks; my ad-visor C. E. Burgess, who introduced me to the wonders of Texas-style mathematics;R. H. Bing, whose Sling, Dogbone Space, Hooked Rug, Baseball Move, epslums and

PREFACE TO THE THREE VOLUME SET ix

deltas, and Crumpled Cubes added color and wonder to the study of topology; andW. P. Thurston, who often made me feel like Gary Larson’s character of little brain(“Stop, professor, my brain is full.”) They were all kind and encouraging to me.And then there are those whom I only know from their writing: especially Euclid,Archimedes, Gauss, Hilbert, and Poincare.

Finally, I must thank Bill Floyd and Walter Parry for more than three decadesof mathematical fun. When we would get together, we would work hard everymorning, then talk mathematics for the rest of the day as we hiked the cities, coun-trysides, mountains, and woods of Utah, Virginia, Michigan, Minnesota, England,France, and any other place we could manage to get together. And special thanksto Bill for cleaning up and improving almost all of those figures in these bookswhich he had not himself originally drawn.

Preface to Volume 3

This is the last of a three volume set describing a very personal arc of thoughtthat begins with earth measurement (that is, geo-metry), passes through the topol-ogy of 2-dimensional surfaces, and ends with space-time measurement (that is, geo-chrono-metry, where Einstein identifies gravity with the curvature of space-time).The volumes are (1) The Geometry of the 2- Dimensional Spaces; (2) The Topologyof 2-Dimensional Spaces; and (3) An Introduction to Non Euclidean Geometry andCurvature.

Volume 3 gives a general introduction to hyperbolic non-Euclidean geometryin all dimensions, with an introduction to all of the standard models and theirrelationships to one another. It explains why the models are called hyperbolic.It discusses the ways in which this geometry differs from Euclidean geometry. Itcalculates the shortest paths in this geometry (the geodesics). It explains someof the reasons why this geometry is studied. Following the introduction to non-Euclidean geometry, Volume 3 approaches curvature in dimension 2 in a way thatdoes not begin with the Gauss map. Instead, it begins with a natural metric-invariant definition of curvature that measures in an obvious way the difficulty offlattening a surface into the plane without distorting lengths and areas. It endswith the Gauss map and a proof of Gauss’s Theorema Egregium. This approach,which is backward to the classical approach, is intuitive; but it entails difficultiesin proving that certain limits exist.

xi

Bibliography

Plain Fun

(top recommendations for easy, but rewarding, pleasure).

[1] Hardy, G. H., A Mathematician’s Apology, Cambridge University Press, 2004 (eighth print-ing).

[2] Polya, G., How to Solve It, Princeton Univerity Press, 2004.[3] Korner, T. W., The Pleasures of Counting, Cambridge University Press, 1996.

- More Fun

[4] Davis, P. J. and Hersh, R., The Mathematical Experience, Houghton Mifflin Company, 1981.[5] Rademacher, H., Higher Mathematics from an Elementary Point of View, Birkhauser, 1983.[6] Hilbert, D., and Cohn-Vossen, S., Geometry and the Imagination, (translated by P. Nemeyi),

Chelsea Publishing Company, New York, 1952. [College level exposition of rich ideas fromlow-dimensional geometry, with many figures.]

[7] Dorrie, H., 100 Great Problems of Elementary Mathematics: Their History and Solution,Dover Publications, Inc., 1965, pp. 108-112. [We learned our first proof of the fundamentaltheorem of algebra here.]

[8] Courant, R. and Robbins, H., What is Mathematics?, Oxford University Press, 1941.

Classics

(a chance to see the thinking of the very best, in chronological order).

[9] Euclid, The Thirteen Books of Euclid’s Elements, Vol. 1-3, 2nd Ed., (edited by T. L. Heath)Cambridge University Press, Cambridge, 1926. [Reprinted by Dover, New York, 1956.]

[10] Archimedes, The Works of Archimedes, edited by T. L. Heath, Dover Publications, In.,Mineola, New York, 2002. See also the exposition in Polya, G., Mathematics and PlausibleReasoning, Vol. 1. Induction and Analogy in Mathematics, Chapter IX. Physical Mathemat-ics, pp. 155-158, Princeton University Press, 1954. [How Archimedes discovered the integralcalculus.]

[11] Wallis, J., in A Source Book in Mathematics, 1200-1800, edited by D. J. Struik., HarvardUniversity Press, 1969, pp. 244-253. [Wallis’s product formula for π.]

[12] Gauss, K. F., General Investigations of Curved Surfaces of 1827 and 1825, Princeton Uni-versity Library, 1902. [Available online. Difficult reading.]

[13] Fourier, J., The Analytical Theory of Heat, translated by Alexander Freeman, CambridgeUniversity Press, 1878. [Available online, 508 pages. The introduction explains Fourier’sthoughts in approaching the problem of the mathematical treatment of heat. Chapter 3 ex-plains his discovery of Fourier series.]

[14] Riemann, B., Collected Papers, edited by Roger Baker, Kendrick Press, Heber City, Utah,2004. [English translation of Riemann’s wonderful papers.]

99

100 BIBLIOGRAPHY

[15] Poincare, H., Science and Method, Dover Publications, Inc., 2003. [Discusses the role of thesubconscious in mathematical discovery.] Also, The Value of Science, translated by G. B.Halstead, Dover Publications, Inc., 1958.

[16] Klein, F., Vorlesungen uber Nicht-Euklidische Geometrie, Verlag von Julius Springer, Berlin,1928. [In German. An algebraic development of non-Euclidean geometry with respect to theKlein and projective models. Beautiful figures. Elegant exposition.]

[17] Hilbert, D., Gesammelte Abhandlungen (Collected Works), 3 volumes, Springer-Verlag,

1970. [In German. The transcendence of e and π appears in Volume 1, pp. 1-4. Hilbert’sspace-filling curve appears in Volume 3, pp. 1-2.]

[18] Einstein, A., The Meaning of Relativity, Princeton University Press, 1956.[19] Thurston, W. P., Three-Dimensional Geometry and Topology, edited by Silvio Levy, Prince-

ton University Press, 1997. [An intuitive introduction to dimension 3 by the foremost ge-ometer of our generation.]

[20] W. P. Thurston’s theorems on surface diffeomorphisms as exposited in Fathi, A., and Lau-denbach, F., and Poenaru, V., Travaux de Thurston sur les Surfaces, Seminaire Orsay,Societe Mathematique de France, 1991/1979. [In French.]

History

(concentrating on famous mathematicians).

[21] Bell, E. T., Men of Mathematics, Simon and Schuster, Inc., 1937. [The book that convincedme that mathematics is exciting and romantic.]

[22] Henrion, C., Women of Mathematics, Indiana University Press, 1997.[23] Dunham, W., Journey Through Genius, Penguin Books, 1991.

Supporting Textbooks

- Topology

[24] Munkres, J. R., Topology, a First Course, Prentice-Hall, Inc., 1975. [The early chaptersexplain the basics of topology that form the prerequisites for the latter half of this book. Thelater chapters contain rather different views of some of the later theorems in our book.]

[25] Massey, W. S., Algebraic Topology: An Introduction. Springer-Verlag, New York -Heidelberg-Berlin, 1967 (Sixth printing: 1984), Chapter I, pp. 1-54. [A particularly nice

introduction to covering spaces.][26] Hatcher, A., Algebraic Topology , Cambridge University Press, 2001. [A very nice introduc-

tion to algebraic topology, a bit of which we need in Volume 2.][27] Munkres, J. R., Elements of Algebraic Topology, Addison-Wesley, 1984. [Another nice in-

troduction.][28] Alexandroff, P., Elementary Concepts of Topology, translated by Alan E. Farley, Dover

Publications, Inc., 1932. [A wonderful small book.][29] Alexandrov, P. S., Combinatorial Topology, 3 volumes, translated by Horace Komm, Gray-

lock Press, Rochester, NY, 1956.[30] Seifert, H., and Threlfall, W., A Textbook of Topology, translated by Michael A. Goldman;

and Seifert, H., Topology of 3-Dimensional Fibered Spaces, translated by Wolfgang Heil,Academic Press, 1980. [Available online.]

[31] Hurewicz, W., and Wallman, H., Dimension Theory, Princeton University Press, 1941. [SeeChapters 4, 5, and 6 of Volume 2.]

BIBLIOGRAPHY 101

- Algebra

[32] Herstein, I. N., Abstract Algebra, third edition, John Wiley & Sons, Inc., 1999. [See ourChapter 6 of Volume 1.]

[33] Dummit, D. S., and Foote, R. M., Abstract Algebra, third edition, John Wiley & Sons, Inc.,2004. [See our Chapter 6 of Volume 1.]

[34] Hardy, G. H., and Wright, E. M., An Introduction to the Theory of Numbers, fourthedition, Oxford University Press, 1960. [See our Chapter 5 of Volume 1.]

- Analysis

[35] Apostol, T. M., Mathematical Analysis , Addison-Wesley, 1957. [Good background for Rie-mannian metrics in Chapter 1 of Volume 1, and also the chapters of Volume 3.]

[36] Lang, S., Real and Functional Analysis, third edition, Springer, 1993. [Chapter XIV gives thedifferentiable version of the open mapping theorem. The proof uses the contraction mappingprinciple. See our Chapter 12 of Volume 2 for the topological version of the open mappingtheorem.]

[37] Spivak, M., Calculus on Manifolds, W. A. Benjamin, Inc., New York, N. Y., 1965. [Goodbackground for Riemannian metrics in Chapter 1 of Volume 1 and for Volume 3.]

[38] Janich, K., Vector Analysis, translated by Leslie Kay, Springer, 2001. [Good background forRiemannian metrics in Chapter 1 of Volume 1 and for Volume 3.]

[39] Saks, S., Theory of the Integral, second revised edition, translated by L. C. Young, DoverPublications, Inc., New York, 1964. [Wonderfully readable.]

[40] H. L. Royden, H. L., Real Analysis, third edition, Macmillan, 1988. [The place where wefirst learned about nonmeasurable sets.]

References from our Paper on Hyperbolic Geometry in Flavors ofGeometry (reprinted here as our Volume 3, Chapter 2)

[41] Flavors of Geometry, edited by Silvio Levy, Cambridge University Press, 1997.[42] Alonso, J. M., Brady, T., Cooper, D., Ferlini, V., Lustig, M., Mihalik, M., Shapiro, M.,

Short, H., Notes on word hyperbolic groups, Group Theory from a Geometrical Viewpoint:21 March — 6 April 1990, ICTP, Trieste, Italy, (E. Ghys, A. Haefliger, and A. Verjovsky,eds.), World Scientific, Singapore, 1991, pp. 3–63.

[43] Benedetti, R., and Petronio, C., Lectures on Hyperbolic Geometry, Universitext, Springer-Verlag, Berlin, 1992. [Expounds many of the facts about hyperbolic geometry outlined inThurston’s influential notes.]

[44] Bolyai, W., and Bolyai, J., Geometrische Untersuchungen, B. G. Teubner, Leipzig andBerlin, 1913. (reprinted by Johnson Reprint Corp., New York and London, 1972) [Historicaland biographical materials.]

[45] Cannon, J. W., The combinatorial structure of cocompact discrete hyperbolic groups, Geom.Dedicata 16 (1984), 123–148.

[46] Cannon, J. W., The theory of negatively curved spaces and groups, Ergodic Theory, SymbolicDynamics and Hyperbolic Spaces, (T. Bedford, M. Keane, and C. Series, eds.) OxfordUniversity Press, Oxford and New York, 1991, pp. 315–369.

[47] Cannon, J. W., The combinatorial Riemann mapping theorem, Acta Mathematica 173(1994), 155–234.

[48] Cannon, J. W., Floyd, W. J., Parry, W. R., Squaring rectangles: the finite Riemann mappingtheorem, The Mathematical Heritage of Wilhelm Magnus — Groups, Geometry & Special

102 BIBLIOGRAPHY

Functions, Contemporary Mathematics 169, American Mathematics Society, Providence,1994, pp. 133–212.

[49] Cannon, J. W., Floyd, W. J., Parry, W. R., Sufficiently rich families of planar rings,preprint.

[50] Cannon, J. W., Swenson, E. L., Recognizing constant curvature groups in dimension 3,preprint.

[51] Coornaert, M., Delzant, T., Papadopoulos, A., Geometrie et theorie des groupes: les groupes

hyperboliques de Gromov, Lecture Notes 1441, Springer-Verlag, Berlin-Heidelberg-NewYork,1990.

[52] Euclid, The Thirteen Books of Euclid’s Elements, Vol. 1-3, 2nd Ed., (T. L. Heath, ed.)Cambridge University Press, Cambridge, 1926 (reprinted by Dover, New York, 1956).

[53] Gabai, D., Homotopy hyperbolic 3-manifolds are virtually hyperbolic, J. Amer. Math. Soc.7 (1994), 193–198.

[54] Gabai, D., On the geometric and topological rigidity of hyperbolic 3-manifolds, Bull. Amer.Math. Soc. 31 (1994), 228–232.

[55] Ghys, E., de la Harpe, P., Sur les groupes hyperboliques d’apres Mikhael Gromov, Progressin Mathematics 83, Birkhauser, Boston, 1990.

[56] Gromov, M., Hyperbolic groups, Essays in Group Theory, (S. Gersten, ed.), MSRI Publi-cation 8, Springer-Verlag, New York, 1987. [Perhaps the most influential recent paper ingeometric group theory.]

[57] Hilbert, D., Cohn-Vossen, S., Geometry and the Imagination, Chelsea Publishing Company,New York, 1952. [College level exposition of rich ideas from low-dimensional geometry withmany figures.]

[58] Iversen, B., Hyperbolic Geometry, London Mathematical Society Student Texts 25, Cam-bridge University Press, Cambridge, 1993. [Very clean algebraic approach to hyperbolic ge-ometry.]

[59] Klein, F., Vorlesungen uber Nicht-Euklidische Geometrie, Verlag von Julius Springer, Berlin,1928. [Mostly algebraic development of non-Euclidean geometry with respect to Klein andprojective models. Beautiful figures. Elegant exposition.]

[60] Kline, M. Mathematical Thought from Ancient to Modern Times, Oxford University Press,

New York, 1972. [A 3-volume history of mathematics. Full of interesting material.][61] Lobatschefskij, N. I., Zwei Geometrische Abhandlungen, B. G. Teubner, Leipzig and Berlin,

1898. (reprinted by Johnson Reprint Corp., New York and London, 1972) [Original papers.][62] Mosher, L., Geometry of cubulated 3-manifolds, Topology 34 (1995), 789–814.[63] Mosher, L., Oertel, U., Spaces which are not negatively curved, preprint.[64] Mostow, G. D., Strong Rigidity of Locally Symmetric Spaces, Annals of Mathematics Studies

78, Princeton University Press, Princeton, 1973.[65] Poincare, H., Science and Method, Dover Publications, New York, 1952. [One of Poincare’s

several popular expositions of science. Still worth reading after almost 100 years.][66] Ratcliffe, J. G., Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149,

Springer-Verlag, New York, 1994. [Fantastic bibliography, careful and unified exposition.][67] Riemann, B., Collected Papers, Kendrick Press, Heber City, Utah, 2004. [English translation

of Riemann’s wonderful papers][68] Swenson, E. L., Negatively curved groups and related topics, Ph.D. dissertation, Brigham

Young University, 1993.[69] Thurston, W. P., The Geometry and Topology of 3-Manifolds, lecture notes, Princeton Uni-

versity, Princeton, 1979. [Reintroduced hyperbolic geometry to the topologist. Very excitingand difficult.]

[70] Weyl, H., Space—Time—Matter, Dover, New York, 1922. [Weyl’s exposition and develop-ment of relativity and gauge theory which begins at the beginning with motivation, philoso-phy, and elementary developments as well as advanced theory.]

BIBLIOGRAPHY 103

Further Technical References (arranged by chapter)

For the entirety of Volume 2

[71] Newman, M. H. A., Elements of the Topology of Plane Sets of Points, Cambridge UniversityPress, 1939.[A good alternative introduction to the topology of the plane.]

- Volume 1, Chapter 1

[72] Feynman, R., The Character of Physical Law, The M.I.T. Press, 1989, p. 47.[All ofFeynman’s writing is fun and thought provoking.]


[73] Gilbert, W. J., and Vanstone, S. A., An Introduction to Mathematical Thinking, PearsonPrentice Hall, 2005. [The place where I learned the algorithmic calculations about theEuclidean algorithm. See our Chapter 2.]



[74] Reid, C., Hilbert, Springer Verlag, 1970. [A wonderful biography of Hilbert, with an extendeddiscussion of the Hilbert address in which he stated the Hilbert problems. See our Chapter4.]


[75] Apostol , T. M., Calculus , Volume 1, Blaisdell Publishing Company, New York, 1961. [Theplace where I first learned areas by counting. See our Chapter 5.]


[76] Hilton, P., and Pedersen, J., Approximating any regular polygon by folding paper, Math.Mag. 56 (1983), 141-155. [Method for approximating many angles algorithmically by paper-folding.]

[77] Hilton, P., and Pedersen, J., Folding regular star polygons and number theory Math. Intel-ligencer 7 (1985), 15-26. [More paper-folding.]

[78] Burkard Polster, Variations on a Theme in Paper Folding, Amer. Math. Monthly 111 (2004),39-47. [More paper-folding approximations to angles. See Chapter 6 and the impossibility oftrisecting an angle.]

104 BIBLIOGRAPHY


[79] Wagon, S., The Banach-Tarski Paradox, Cambridge University Press, 1994.[A wonderfulexposition of the Hausdorff-Banach-Tarski paradox, without the emphasis on the graph ofthe free group. See our Chapter 7.]

- Volume 1, Chapter 7; Volume 3, Chapter 1.

[80] Coxeter, H. S. M., and Moser, W. O., Generators and Relations for Discrete Groups, secondedition, Springer-Verlag, 1964. [The place where I learned that groups can be viewed as graphs(the Cayley graph or the Dehn Gruppenbild). See our Chapter 7 where we use the graph ofthe free group on two generators and Chapter 25 where we use graphs as approximationsto non Euclidean geometry.]


[81] Peano, G. , Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen 36(1), 1890, pp. 157-160. [The first space-filling curve, described algebraically. See our Chapter12.]

[82] Peano, G., Selected works of Giuseppe Peano, edited by Kennedy, Hubert C., and translated.With a biographical sketch and bibliography, Allen & Unwin, London, 1973.

[83] Hilbert, D., Uber die stetige Abbildung einer Line auf ein Flachenstuck, MathematischeAnnalen 38 (3), 1891, pp. 459-460. [Hilbert gave the first pictures of a space-filling curve.See our Chapter 12.]

[84] G. Polya, Uber eine Peanosche Kurve, Bull. Acad. Sci. Cracovie, A, 1913, pp. 305-313.[Polya’s triangle-filling curve. See our Chapter 12.]

[85] Lax, P. D., The differentiability of Polya’s function, Adv. Math., 10, 1973, pp. 456-464.[Lax recommends the non-isosceles triangle in Polya’s construction since it simplifies thedescription of the path followed to the point represented by a binary expansion. See ourChapter 12.]


[86] Mandelbrot, B., The Fractal Geometry of Nature, W. H . Freeman & Co, 1982. [Mandelbrotsuggests the use of Hausdorff dimension as a means of recognizing sets that are locallycomplicated or chaotic. He defines these to be fractals. See our Chapter 13.]

[87] Falconer, K. J., The Geometry of Fractal Sets, Cambridge University Press, 1985. [Seereference [86] and our Chapter 13.]

[88] Devaney, R. L., Differential Equations, Dynamical Systems, and an Introduction to Chaoswith Morris Hirsch and Stephen Smale, 2nd edition, Academic Press, 2004; 3rd edition,

Academic Press, 2013. [See reference [84] and our Chapter 13.]

- Volume 2, Chapter 8 and 11

[89] Moore, R. L., Concerning upper semi-continuous collections of continua, Trans. Amer.Math. Sc. 27 (1925), pp. 416-428. [Moore shows that his topological characterization of theplane or 2-sphere allows him to prove his theorem about decompositions of the 2-sphere. Seeour Volume 2, Chapters 8 and 11.]

BIBLIOGRAPHY 105

[90] Wilder, R. L., Topology of Manifolds, American Mathematical Society, 1949 . [Our proof ofthe topological characterization of the sphere is primarily modelled on Wilder’s proof, withwhat we consider to be conceptual simplifications. See our Chapter 8.]


[91] Rado, T., Uber den Begriff der Riemannschen Flache, Acts. Litt. Sci. Szeged 2 (1925), pp.101-121. [The first proof that 2-manifolds can be triangulated. See our Chapter 20.]


[92] Andrews, Peter, The classification of surfaces, Amer. Math. Monthly 95 (1988), 861-867l[93] Armstrong, M. A., Basic Topology, McGraw-Hill, London, 1979.[94] Burgess, C. E., Classification of surfaces, Amer. Math. Monthly 92 (1985), 349-354.[95] Francis, George K., Weeks, Jeffrey R., Conway’s ZIP proof, Amer. Math. Monthly 106

(1999), 393-399.


[96] Rolfsen, D., Knots and Links, AMS Chelsea, vol 346, 2003. [See our Chapter 22.]

For the entirety of Volume 3, see the references above taken from our article inFlavors of Geometry, beginning with reference [41].


[97] Misner, C. W., and Thorne, K. S., and Wheeler, J. A., Gravitation, W. H. Freeman andCompany, 1973.

- Volume 3, Chapters 4 and 5

[98] Abelson, H., and diSessa, A., Turtle Geometry, MIT Press, 1986. [The authors use the pathsof a computer turtle to model straight paths on a curved surface.]

For additional informationand updates on this book, visit

www.ams.org/bookpages/mbk-110

AMS on the Web www.ams.org

This is the final volume of a three volume collection devoted to the geom-etry, topology, and curvature of 2-dimensional spaces. The collection provides a guided tour through a wide range of topics by one of the twen-tieth century’s masters of geometric topology. The books are accessible to college and graduate students and provide perspective and insight to mathematicians at all levels who are interested in geometry and topology.

Einstein showed how to interpret gravity as the dynamic response to the curvature of space-time. Bill Thurston showed us that non-Euclidean geometries and curvature are essential to the understanding of low-dimensional spaces. This third and final volume aims to give the reader a firm intuitive under-standing of these concepts in dimension 2. The volume first demonstrates a number of the most important properties of non-Euclidean geometry by means of simple infinite graphs that approximate that geometry. This is followed by a long chapter taken from lectures the author gave at MSRI, which explains a more classical view of hyperbolic non-Euclidean geometry in all dimensions. Finally, the author explains a natural intrinsic obstruction to flattening a triangulated polyhedral surface into the plane without distorting the constituent triangles. That obstruction extends intrinsically to smooth surfaces by approximation and is called curvature. Gauss’s original definition of curvature is extrinsic rather than intrinsic. The final two chapters show that the book’s intrinsic definition is equivalent to Gauss’s extrinsic definition (Gauss’s

“Theorema Egregium” (“Great Theorem”)).

MBK/110

Two-Dimensional Spaces, Volume 3 NON-EUCLIDEAN GEOMETRY ... · Two-Dimensional Spaces, Volume 3 NON-EUCLIDEAN GEOMETRY AND CURVATURE 10.1090/mbk/110

Documents