Selected Titles in This Series - AMS

Selected Title s i n Thi s Serie s

17 Jacque s Hadamar d (Jerem y J . Gra y an d A b e Shenitzer , Editors) , Non-Euclidea n

geometry i n th e theor y o f automorphi c functions , 199 9

16 P . G . L . Dirichle t (wit h Supplement s b y R . Dedekind) , Lecture s o n numbe r theory ,

1999

15 Charle s W . Curtis , Pioneer s o f representatio n theory : Probenius , Burnside , Schur , an d

Brauer, 199 9

14 Vladimi r Maz'y a an d Tatyan a Shaposhnikova , Jacque s Hadamard , a universa l

mathematician, 199 8

13 Lar s Garding , Mathematic s an d mathematicians : Mathematic s i n Swede n befor e 1950 ,

1998

12 Walte r Rudin , Th e wa y I remembe r it , 199 7

11 Jun e Barrow-Green , Poincar e an d th e thre e bod y problem , 199 7

10 Joh n Stillwell , Source s o f hyperboli c geometry , 199 6

9 Bruc e C . Bernd t an d Rober t A . Rankin , Ramanujan : Letter s an d commentary , 199 5

8 Kare n Hunge r Parshal l an d Davi d E . Rowe , Th e emergenc e o f th e America n

mathematical researc h community , 1876-1900 : J . J . Sylvester , Feli x Klein , an d E . H . Moore ,

1994

7 Hen k J . M . Bos , Lecture s i n th e histor y o f mathematics , 199 3

6 Smilk a Zdravkovsk a an d Pete r L . Duren , Editors , Golde n year s o f Mosco w

mathematics, 199 3

5 Georg e W . Mackey , Th e scop e an d histor y o f commutativ e an d noncommutativ e

harmonic analysis , 199 2

4 Charle s W . McArthur , Operation s analysi s i n th e U.S . Arm y Eight h Ai r Forc e i n Worl d

War II , 199 0

3 Pete r L . Dure n e t al. , Editors , A centur y o f mathematic s i n America , par t III , 198 9

2 Pete r L . Dure n e t al. , Editors , A centur y o f mathematic s i n America , par t II , 198 9

1 Pete r L . Dure n e t al. , Editors , A centur y o f mathematic s i n America , par t I , 198 8

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Ill

the Theor y o f

c


e^wcvwc^ •

S O U R C E S

Non^Euclidean in

the Theor y o f AiULtonnorphic Functions

Jeremy J. Gra y and Abe Shenitzer , Editors

Translated by Abe Shenitze r

With Historical Introduction by Jeremy J . Gra y

7

American Mathematica l Societ y London Mathematica l Societ y

https://doi.org/10.1090/hmath/017

Editor ia l Boar d

American Mathematica l Societ y Londo n Mathematica l Societ y George E . Andrew s Davi d Fowler , Chai r Bruce Chandle r Jerem y J . Gra y Karen Parshall , Chai r S . J . Patterso n George B . Seligma n

Neevklidova geometriy a v teori i avtomorfnyk h funktsi i

by Jacque s Hadamar d

Originally publishe d i n Russia n b y Gostekhizdat , M-L , 1951 .

Original Russia n tex t translate d b y Ab e Shenitzer .

1991 Mathematics Subject Classification. Primar y 01-XX , 01A55 , 01A60 ; Secondary 30-03 , 30F35 , 34A20 , 51-03 .

Library o f Congres s Cataloging-in-Publicatio n Dat a

Hadamard, Jacques , 1865-1963 . Non-Euclidean geometr y i n th e theor y o f automorphi c function s / Jacque s Hadamar d ;

Jeremy J . Gray , Ab e Shenitzer, editor s ; translated b y Abe Shenitzer ; with historica l introductio n by Jerem y J . Gray .

p. cm . — (Histor y o f mathematics ; v. 17) Includes bibliographica l references . ISBN 0-8218-2030- 3 (softcove r : alk . paper ) 1. Automorphi c functions . 2 . Geometry , Non-Euclidean . I . Gray , Jeremy , 1947 - .

II. Shenitzer , Abe . III . Title. IV . Series. QA353.A9H33 199 9 515'.9—dc21 99-3170 9

CIP

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o make fai r us e of the material, suc h a s to copy a chapte r fo r use in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the source i s given.

Republication, systemati c copying , o r multiple reproductio n o f any materia l i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Assistant t o the Publisher, America n Mathematica l Society , P. O. Box 6248, Providence , Rhod e Islan d 02940-6248 . Request s ca n also b e mad e b y e-mai l t o reprint-permissionOams.org.

© 199 9 b y the American Mathematica l Society . Al l rights reserved . Printed i n the United State s o f America .

The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the United State s Government .

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10 9 8 7 6 5 4 3 2 1 0 4 03 02 01 00 9 9

Contents

Acknowledgments i x

Introduction b y th e publisher s o f the Russia n translatio n x i

Historical introductio n JEREMY GRA Y 1

A brief histor y o f automorphi c functio n theory , 1880-193 0 JEREMY GRA Y 3

Chapter I . Th e grou p o f motion s o f th e hyperboli c plan e an d it s properl y

discontinuous subgroup s 1 7

Chapter II . Discontinuou s group s i n thre e geometries . Fuchsia n function s 3 7

Chapter III . Fuchsia n function s 5 7

Chapter IV . Kleinia n group s an d function s 7 1

Chapter V . Algebrai c function s an d linea r algebrai c differentia l equation s 7 9

Chapter VI . Fuchsia n group s an d geodesie s 8 7

References 9 3

Additional reference s 9 5


Acknowledgments

The editor s wis h to thank Sara h Shenitze r fo r carefull y readin g th e translatio n and eliminatin g a number o f infelicities an d outrigh t errors . W e also wish to than k Deb Smit h fo r he r car e i n th e preparatio n o f th e final versio n o f th e manuscrip t and fo r th e patienc e wit h whic h sh e handle d score s o f last-minute corrections .


Introduction b y th e Publisher s of th e Russia n Translatio n

In the Editors ' Introductio n t o B.A . Fuks ' book Non-euclidean geometry in the theory of conformal and pseudoconformal mappings, volum e V in thi s series , i t wa s mentioned tha t th e eminent Frenc h mathematician J . Hadamar d wrot e in the 1920 s a monograph i n connection with the preparation o f an edition of the collected work s of N.I . Lobachevski . Th e author' s manuscrip t wa s translated b y A.V . Vasil'ev an d edited b y B.A . Fuks .

Hadamard's monograp h i s a survey , an d mos t o f it s proposition s ar e state d without proof . Thi s calle d fo r a numbe r o f notes . Th e note s wer e writte n b y B.A . Fuks.

Fuks' boo k ca n serv e a s a n introductio n t o th e Hadamar d monograph .

The possibilit y o f establishin g a Lobachevskia n metri c i n a simpl y connecte d region o f the comple x plan e provide d i n th e pas t th e stimulu s fo r th e discover y o f automorphic functions . Th e metri c i n questio n playe d a n essential , an d a t time s crucial, rol e i n al l stage s o f the constructio n o f the gran d edific e o f thes e function s which have such important application s to many problems of mathematical analysis .

The basi c ai m o f Hadamard' s smal l boo k i s t o demonstrat e th e fundamenta l importance o f the Lobachevskia n metri c fo r th e theor y o f automorphi c functions .

While th e autho r usuall y omit s proofs , i n mos t case s h e provide s thei r under -lying idea s o r outlines . Whe n h e doe s this , h e tend s t o emphasiz e th e significanc e of the relevan t proposition s o r fact s o f Lobachevskian geometr y fo r eac h argument .

By no w i t i s clea r tha t Hadamard' s boo k canno t serv e a s a textboo k o n th e theory o f automorphi c function s an d tha t i t ca n onl y b e recommende d t o reader s who hav e a certai n amoun t o f knowledge o f thi s theory . A goo d sourc e fo r th e re -quired background knowledg e is chapters I I and II I of the recently published secon d edition o f V.V. Golubev' s Lectures on the analytic theory of differential equations, Gostekhizdat, M.-L. , 1950 . Chapter s I I and II I of B.A. Fuks ' Non-euclidean geome-try in the theory of conformal and pseudoconformal mappings (Gostekhizda t 1951) , volume V i n thi s series , ar e a goo d sourc e o f indispensabl e informatio n abou t th e Lobachevskian metri c i n a simpl y connecte d domai n o f th e comple x plane , abou t the group of motions generated b y this metric, and about it s properly discontinuou s subgroups.

Chapter I o f Hadamard' s book , "Th e grou p o f motion s o f th e Lobachevskia n plane an d it s properl y discontinuou s subgroups" , i s o f a n introductor y nature . I t describes th e subsequentl y use d realization s o f th e Lobachevskia n plane , bring s together th e mos t importan t propertie s o f the properl y discontinuou s subgroup s o f its group of motions, and partl y explains the characteristi c singula r feature s o f their fundamental domains .

xi

xii INTRODUCTIO N

Chapter II , "Properl y discontinuou s group s o f 3 geometries. Fuchsia n groups" , is devote d t o th e stud y o f th e properl y discontinuou s subgroup s o f th e group s o f motions o f th e geometrie s o f Riemann , Euclid , an d Lobachevski . (I n particular , i t deals wit h th e condition s unde r whic h a polygo n i n th e Lobachevskia n plan e ca n serve a s a fundamenta l regio n o f a properly discontinuou s subgrou p o f it s grou p o f motions.)

Chapter II I deal s with Fuchsia n function s an d Chapte r I V with Kleinia n func -tions; i n particular , thes e chapter s includ e th e theor y o f Poincare' s thet a series .

Chapter V is devoted t o application s o f the automorphi c function s constructe d in th e earlie r chapter s t o th e proble m o f uniformizatio n o f algebrai c curve s an d t o the solutio n o f ordinary linea r differentia l equation s wit h algebrai c coefficients .

The las t chapter , Chapte r VI , is titled "Fuchsia n group s an d geodesi c lines". I t is relatively shor t an d i s in the nature o f an appendix . Th e book includes reference s to work s tha t contai n comprehensiv e account s o f touched-o n issues . Additiona l references ar e foun d i n the editor' s notes .

REMARK. I n th e Englis h translation , th e Russia n footnote s appea r a s Note s at th e en d o f eac h o f the si x chapters . (Eds. )


References

Bernhard Riemann. Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachtrdge. Collected papers. Nac h de r Ausgab e vo n H . Webe r un d R . Dedekin d ne u heraus -gegeben vo n R . Narasimhan . Springer/Teubner , Berlin , etc./Leipzig , 1990 .

B. A . Fuks , Non-Euclidea n geometr y i n th e theor y o f conforma l an d pseudoconforma l map -pings. Gostekhizdat , 1951 . (Russian )

L. R . Ford , Automorphic functions (2n d edition) . Chelsea , Ne w York , 1951 .

L. P . Eisenhart , Continuous groups of transformations. Princeton , 1933 ; repr . Ne w York , 1961.

A. K . Sushkevich , Foundations of higher algebra. Gostekhizdat , 1941 . (Russian )

H. Poincare , Su r le s forme s cubique s ternaires . C . R . Acad . Sci . Paris , 9 0 (Jun e 1880) , 1336 -1339; Oeuvres, 5 , 1950 , 25-27 .

G. Giraud , Lecons sur les fonctions automorphes. Gauthier-Villars , Paris , 1920 .

J. Lagrange , Nouvau x memoire s d e l'Acad . Royal e de s Science s e t Belle s Lettres , Paris , 1779 ; repr. i n Oeuvres de Lagrange, J . A . Serret , ed. , 1 4 vols. , Gauthier-Villars , Paris , 1867-1892 .

N. V . Efimov , Higher geometry. Mir , Moscow , 1980 , distribute d b y Importe d Publications , Chicago, IL .

J. Hadamard , Lecons sur le calcul des variations. Hermann , Paris , 1910 .

F. Klein , Vorlesungen iiber das Ikosaeder. Teubner , Leipzig , 1884 .

C. Jordan , Cours d'Analyse, Vol III. Gauthier-Villars , Paris , 1893 .

C. Jordan , Memoir e su r le s equation s differen t ielles lineaire s a integral e algebrique . J . Rein e Angew. Math. , 8 4 (1878) , 89-216 .

C. Jordan , Su r l a determinatio n de s groupe s d'ordr e fini contenu s dan s l e group e lineaire , Atti Acad . Nap . 8 , no . 1 1 (1880) ; Oeuvres , II , 177-218 .

D. Hilber t an d S . Cohn-Vossen , Geometry and the imagination. Chelsea , Ne w York , 1952 .

L. Kronecker , Uebe r System e vo n Funktionen mehrere r Variabeln . Monatsber . Berline r Akad . Wissenschaften (1869) , 689 .

H. Poincare , Su r le s courbe s definie s pa r le s equation s differen t ielles. J . Math . Pure s Appl. , ser. 4 , vol. 1 , 1885, 167-244 , and vol . 2 , 1886 , 151-217 ; in Oeuvres 2 , 1916 , 90-161 and 167-221 .

W. Blaschke , Vorlesungen iiber Differentialeometrie. Dover , Ne w York , 1945 .

P. Appel l an d E . Goursat , Theorie des fonctions algebriques d'une variable, Vol. II, "Fonc -tions automorphes" , P . Fatou . Gauthier-Villars , Paris , 1930 .

V. V . Golubev , Lectures on the analytic theory of differential equations. Gostekhizdat , 1950 . (Russian)

R. Frick e an d F . Klein , Vorlesungen iiber die Theorie der automorphen Funktionen, I, II Springer, Berlin , 1926 .

R. Fricke , Ueber die Poincare'schen Reihen der (-l)-ten Dimension. Dedekin d Festschrift , 1901.

93

94 REFERENCES

[23] F . Klein , Ueber lineare Differentialleichungen 2. Ordnung. Teubner , Leipzig , 1906 .

[24] X . Vaksely , C . R . Acad . Sci . Paris , 17 8 (1924) , 1135 .

[25] H . Poincare , Su r le s fonction s fuchsiennes . C . R . Acad . Sci . Paris , 9 2 (Jun e 1888) , 859 .

[26] H . Poincare , Su r le s fonction s fuchsiennes . Bull . Soc . Math . France , X I (1883) , 112 .

[27] R . Courant , Theory of functions of a complex variable. Note s b y A . A . Blank , Couran t Institute o f Mathematica l Sciences , Ne w York , 1948 .

[28] A . Schoenflies , Uebe r di e Kreisbogenpolygone . Math . Annalen , 4 2 (1893) , 377 ; 4 4 (1895) , 105.

[29] H . Poincare , Su r le s groupe s de s equation s lineaires . Act a Math. , 4 (1883) , 201-302 .

[30] V . F . Kagan , Foundations of the theory of surfaces. Gostekhizdat , 1947 . (Russian )

[31] E . Picard , (1 ) Memoir e su r l a theori e de s equation s au x derivee s partielle s e t l a method e des approximation s successives . J . Math . Pure s AppL , serie s 4 , v . 6 (1890) , 145-211 ; sam e journal, serie s 4 , v . 9 (1893) , 217-273 ; (2 ) Su r l'equatio n Au — ke u; sam e journal , serie s 4 , v . 9 (1893) , 273-293 ; sam e journal , serie s 5 , v . 4 (1898) , 313-317 ; J . Rein e Angew . Math. , 13 0 (1905), 243-258 .

[32] E . Picard , Traite d'Analyse, Vol. 3. Gauthier-Villars , Paris , 1926 .

[33] L . Lichtenstein , Integratio n de r Differentialgleichunge n Au = ke u au f geschlossene n Flachen . Acta. Math. , 4 0 (1916) , 1-34 .

[34] H . Poincare , Memoir e su r le s fonction s zeta-fuchsiennes . Act a Math. , 5 (1884) , 209-278 , Oeuvres 2 , 1916 , 402-462 .

[35] I . M . Gel'fand , Lectures on linear algebra, trans . A . Shenitzer . Interscienc e Publishers , Ne w York, 1961 ; repr. Dover , Ne w York , 1989 .

[36] J . Hadamard , Le s surface s a courbur e opposee s e t leur s ligne s geodesiques . J . Math . Pure s Appl., serie s 5 , v . I V (1898) , 27-75 .

[37] J . Hadamard , Su r l a courbur e dan s le s espaces a plus d e deux dimensions . Proc . Soc . Science s Physiques e t Naturelle s d e Bordeau x (1898) .

Additional Reference s

A. F . Beardon , The geometry of discrete groups, Springer-Verlag , Ne w York , etc. , 1983 .

B. A . Dubrovin , A . T . Fomenko , S . P . Novikov , Modern geometry - Methods and applications, Part 1 , transl . R . G . Burns , Springer-Verlag , Ne w York , etc. , 1984 .

J. J . Gray , 'O n th e histor y o f the Rieman n mappin g theorem' , Rend. Circ. Mat. Palermo Suppl, Studies in the history of modern mathematics, 1 , 1994 , 47-94 .

C. Hermite , Oeuvres, 4 vols. , Paris , 1905-1917 .

C. F . Klein , Vorlesungen uber das Ikosaeder und die Auflosung der Gleichungen vom funften Grade. Teubner , Leipzig , 1884 ; secon d edition , 1993 , wit h a n introductor y essa y b y P . Slodowy , Birkhauser-Verlag, Basel , etc. ; Englis h translation , Lectures on the icosahedron, trans , b y G . G . Morrice, 1888 ; Dove r reprint , 1956 .

I. Kra , Automorphic functions and Kleinian groups, Benjamin , Reading , MA , 1972 .

D. Mumford , Curves and their Jacobians, Universit y o f Michiga n Press , An n Arbor , MI , 1976 .

H. Poincare , 'Memoir e su r le s groupe s Kleineens' , Act a Math. , 3 (1883) , 49-92 ; i n Oeuvres, II , 258-299; 2n d edition , Gauthier-Villars , Paris , 1952 .

H. Poincare , 'Su r l'uniformisatio n de s fonction s analytiques' , Act a Math. , 3 1 (1907) , 1-63 ; i n Oeuvres, IV , 70-139 , Gauthier-Villars , Paris , 1950 .

H. Poincare , Papers on Fuchsian functions, trans . J . Stillwell , Springer-Verlag , Ne w York , etc. , 1985.

H. Poincare , Three supplementary essays on the discovery of Fuchsian functions. A critica l edition o f th e origina l manuscript s wit h a n introductor y essa y b y Jerem y J . Gra y an d Scot t A . Walter, Blanchard , Paris , an d Akademie-Verlag , Berlin , 1997 .

F. Schottky , 'Uebe r di e conform e Abbildun g mehrfac h zusammenhangende r ebene r Flachen' , Journal fu r Mathematik , 8 3 (1877) , 300-351 .

H. A . Schwarz , 'Uebe r diejenige n Falle , i n welche n di e Gaussisch e hypergeometrisch e Reih e ein e algebraische Functio n ihre s vierte n Elemente s darstellt' , Journa l fu r Mathematik , 7 5 (1872) , 292-335; i n Gesammelte mathematische Abhandlungen, 2, 211-259 , Berlin , 1890 .

J.-P. Serre , A course in arithmetic, Springer-Verlag , Ne w York , etc. , 1973 .

J. J . Stoker , Differential geometry, Wiley , Ne w York , etc. , 1989 .

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