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Hyperbolic geometry: The first 150 years
Author:
John W. Milnor
Journal:
Bull. Amer. Math. Soc. 6 (1982), 9-24
MSC (1980):
Primary 01A55, 01A60, 51M10; Secondary 57R15, 20H10
MathSciNet review:
634431
Full-text PDF
References |
Similar Articles |
Additional Information
-
[1827] C. F. Gauss, Disquisitiones generales circa superficies curvas, Commentationes Gottingensis 6 (Werke 4, 218-258; German summary, 341-347).
-
[1829-30] N. I. Lobachevsky, On the foundations of geometry, Kazan Messenger 25-28; German transl., 'Zwei geometrische Abhandlungen', Leipzig, 1898.
-
[1832] T. Clausen, Ueber die Function sin φ + (1/2., J. Reine Angew. Math. 8, 298-300.
-
[1832] J. Bolyai, The absolute science of space, independent of the truth or falsity of Euclid’s Axiom 11 (which can never be proved a priori), Maros-Vásárhelyini.
-
[1836] N. I. Lobachevsky, Imaginary geometry and its application to integration, Kazan (German transl. Abh. Gesch. Math. 19, 1904).
-
[1837] N. I. Lobachevsky, Géometrie imaginaire, J. Reine Angew. Math. 17, 295-320.
-
[1859] A. Cayley, Sixth memoir upon quantics, Phil. Trans. 149, 61-91.
-
[1868] B. Riemann, Ueber die Hypothesen welche der Geometrie zu Grunde liegen, Abh. K. G. Wiss. Göttingen 13 (from his Inaugural Address of 1854).
-
[1868] E. Beltrami, Saggio di interpetrazione della geometria non-euclidea, Gior. Mat. 6, 248-312 (Also Op. Mat. 1, 374-405; Ann. École Norm. Sup. 6 (1869), 251-288).
-
[1868] E. Beltrami, Teoria fondamentale degli spazii di curvatura costante, Annali di mat. ser. II 2, 232-255 (Op. Mat. 1, 406-429; Ann. École Norm. Sup. 6 (1869), 345-375).
-
[1871] F. Klein, Über die sogenannte Nicht-Euklidische Geometrie, Math. Ann. 4, 573-625 (cf. Ges. Math. Abh. 1, 244-350).
-
[1871] E. Betti, Sopra gli spazi di un numero qualunque di dimensioni, Annali di mat. ser. II 4, 140-158.
-
[1873] H. A. Schwarz, Ueber diejenigen Fälle in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt, J. Reine Angew. Math. 75, 292-335 (Ges. Math. Abh. 2, 211-259).
-
H.
Poincaré, Théorie des groupes fuchsiens, Acta
Math. 1 (1882), no. 1, 1–76 (French). MR
1554574, http://dx.doi.org/10.1007/BF02391835
-
H.
Poincaré, Mémoire, Acta Math. 3
(1883), no. 1, 49–92 (French). Les groupes kleinéens. MR
1554613, http://dx.doi.org/10.1007/BF02422441
-
E.
Picard, Sur un groupe de transformations des Points de
l’espace situés du même côté d’un
plan, Bull. Soc. Math. France 12 (1884), 43–47
(French). MR
1503932
-
Walther
Dyck, Beiträge zur Analysis situs, Math. Ann.
37 (1890), no. 2, 273–316 (German). MR
1510647, http://dx.doi.org/10.1007/BF01200237
-
Felix
Klein, Zur Nicht-Euklidischen Geometrie, Math. Ann.
37 (1890), no. 4, 544–572 (German). MR
1510658, http://dx.doi.org/10.1007/BF01724772
-
Luigi
Bianchi, Geometrische Darstellung der Gruppen linearer
Substitutionen mit ganzen complexen Coefficienten nebst Anwendungen auf die
Zahlentheorie, Math. Ann. 38 (1891), no. 3,
313–333 (German). MR
1510677, http://dx.doi.org/10.1007/BF01199425
-
[1891] W. Killing, Ueber die Clifford-Klein'schen Raumformen, Math. Ann. 39, 258-278.
-
[1895] H. Poincaré, Analysis situs, J. École Polyt. 1, 1-121 (Oeuv. 6, 193-288).
-
[1898] J. Hadamard, Les surfaces à courbures opposées et leur lignes géodésiques, J. Math. Pures Appl. Ser. 5 4, 27-73 (Oeuv. 2, 729-775).
-
[1900] C. F. Gauss, Werke 8, 175-239.
-
Heinrich
Tietze, Über die topologischen Invarianten mehrdimensionaler
Mannigfaltigkeiten, Monatsh. Math. Phys. 19 (1908),
no. 1, 1–118 (German). MR
1547755, http://dx.doi.org/10.1007/BF01736688
-
[1912] H. Gieseking, Analytische Untersuchungen ueber topologische Gruppen, Thesis, Muenster (Compare Magnus (1974), 153).
-
L.
E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten,
Math. Ann. 71 (1911), no. 1, 97–115 (German).
MR
1511644, http://dx.doi.org/10.1007/BF01456931
-
[1913] O. Veblen and J. W. Alexander, Manifolds of n dimensions, Ann. of Math. (2) 14, 163-178.
-
[1912] H. Weyl, Die Idee der riemannschen Fläche, Teubner, Leipzig.
-
[1914] F. Hausdorff, Grundzüge der Mengenlehre, von Veit, Leipzig.
-
[1919] G. Humbert, Sur la mesure des Classes d'Hermite de discriminant donné dans un corps quadratique imaginaire, et sur certains volumes non euclidiens, Comptes Rendus (Paris) 169, 448-454.
-
[1923] E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen, Akad. Verl., Leipzig.
-
Heinz
Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann.
95 (1926), no. 1, 313–339 (German). MR
1512281, http://dx.doi.org/10.1007/BF01206614
-
É.
Cartan, Leçons sur la Géométrie des Espaces de
Riemann, Gauthier-Villars, Paris, 1946 (French). 2d ed. MR 0020842
(8,602g)
-
H.
Hopf and W.
Rinow, Ueber den Begriff der vollständigen
differentialgeometrischen Fläche, Comment. Math. Helv.
3 (1931), no. 1, 209–225 (German). MR
1509435, http://dx.doi.org/10.1007/BF01601813
-
H.
S. M. Coxeter, Twelve geometric essays, Southern Illinois
University Press, Carbondale, Ill., 1968. MR 0310745
(46 #9843)
-
[1937] J. H. C. Whitehead, On doubled knots, J. London Math. Soc. 12, 63-71 (Works 2, 59-67).
-
H.
S. M. Coxeter, Non-Euclidean Geometry, Mathematical
Expositions, no. 2, University of Toronto Press, Toronto, Ont., 1942. MR 0006835
(4,50a)
-
L.
Lewin, Dilogarithms and associated functions, Foreword by J.
C. P. Miller, Macdonald, London, 1958. MR 0105524
(21 #4264)
-
Armand
Borel, Compact Clifford-Klein forms of symmetric spaces,
Topology 2 (1963), 111–122. MR 0146301
(26 #3823)
-
G.
A. Margulis, The isometry of closed manifolds of constant negative
curvature with the same fundamental group, Dokl. Akad. Nauk SSSR
192 (1970), 736–737 (Russian). MR 0266103
(42 #1012)
-
E.
M. Andreev, Convex polyhedra of finite volume in
Lobačevskiĭ space, Mat. Sb. (N.S.) 83
(125) (1970), 256–260 (Russian). MR 0273510
(42 #8388)
-
G.
D. Mostow, The rigidity of locally symmetric spaces, Actes du
Congrès International des Mathématiciens (Nice, 1970)
Gauthier-Villars, Paris, 1971, pp. 187–197. MR 0419683
(54 #7701)
-
G.
D. Mostow, Strong rigidity of locally symmetric spaces,
Princeton University Press, Princeton, N.J., 1973. Annals of Mathematics
Studies, No. 78. MR 0385004
(52 #5874)
-
Gopal
Prasad, Strong rigidity of 𝑄-rank 1 lattices, Invent.
Math. 21 (1973), 255–286. MR 0385005
(52 #5875)
-
Wilhelm
Magnus, Noneuclidean tesselations and their groups, Academic
Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New
York-London, 1974. Pure and Applied Mathematics, Vol. 61. MR 0352287
(50 #4774)
-
Robert
Riley, A quadratic parabolic group, Math. Proc. Cambridge
Philos. Soc. 77 (1975), 281–288. MR 0412416
(54 #542)
-
Avner
Ash, Deformation retracts with lowest possible dimension of
arithmetic quotients of self-adjoint homogeneous cones, Math. Ann.
225 (1977), no. 1, 69–76. MR 0427490
(55 #522)
-
[1978] W. Thurston, Geometry and 3-manifolds, Lecture Notes, Princeton University.
-
Norbert
Wielenberg, The structure of certain subgroups of the Picard
group, Math. Proc. Cambridge Philos. Soc. 84 (1978),
no. 3, 427–436. MR 503003
(80b:57010), http://dx.doi.org/10.1017/S0305004100055250
-
Uffe
Haagerup and Hans
J. Munkholm, Simplices of maximal volume in hyperbolic
𝑛-space, Acta Math. 147 (1981), no. 1-2,
1–11. MR
631085 (82j:53116), http://dx.doi.org/10.1007/BF02392865
-
[1980] H. J. Munkholm, Simplices of maximal volume in hyperbolic space, Gromov’s norm, and Gromov's proof of Mostow's rigidity theory (following Thurston), pp. 109-124 of Topology Symposium—Siegen 1979, ed. Koschorke and Neumann, Springer Lecture Notes in Math., no. 788.
-
[1980] R. Rieley, Applications of a computer implementation of Poincaré's theorem on fundamental polyhedra, preprint, I.A.S.
-
[1980] R. Rieley, Seven excellent knots, preprint, I.A.S.
-
Marie-France
Vignéras, Variétés riemanniennes isospectrales
et non isométriques, Ann. of Math. (2) 112
(1980), no. 1, 21–32 (French). MR 584073
(82b:58102), http://dx.doi.org/10.2307/1971319
-
A.
Borel, Commensurability classes and volumes of hyperbolic
3-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)
8 (1981), no. 1, 1–33. MR 616899
(82j:22008)
-
Michael
Gromov, Hyperbolic manifolds (according to Thurston and
Jørgensen), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in
Math., vol. 842, Springer, Berlin, 1981, pp. 40–53. MR 636516
(84b:53046)
-
Dennis
Sullivan, Travaux de Thurston sur les groupes quasi-fuchsiens et
les variétés hyperboliques de dimension 3 fibrées sur
𝑆¹, Bourbaki Seminar, Vol. 1979/80, Lecture Notes in
Math., vol. 842, Springer, Berlin, 1981, pp. 196–214
(French). MR
636524 (83h:58079)
-
William
P. Thurston, Three-dimensional manifolds, Kleinian
groups and hyperbolic geometry, Bull. Amer.
Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524
(83h:57019), http://dx.doi.org/10.1090/S0273-0979-1982-15003-0
-
William
P. Thurston, Hyperbolic structures on 3-manifolds. I. Deformation
of acylindrical manifolds, Ann. of Math. (2) 124
(1986), no. 2, 203–246. MR 855294
(88g:57014), http://dx.doi.org/10.2307/1971277
- [1827] C. F. Gauss, Disquisitiones generales circa superficies curvas, Commentationes Gottingensis 6 (Werke 4, 218-258; German summary, 341-347).
- [1829-30] N. I. Lobachevsky, On the foundations of geometry, Kazan Messenger 25-28; German transl., 'Zwei geometrische Abhandlungen', Leipzig, 1898.
- [1832] T. Clausen, Ueber die Function sin φ + (1/2., J. Reine Angew. Math. 8, 298-300.
- [1832] J. Bolyai, The absolute science of space, independent of the truth or falsity of Euclid’s Axiom 11 (which can never be proved a priori), Maros-Vásárhelyini.
- [1836] N. I. Lobachevsky, Imaginary geometry and its application to integration, Kazan (German transl. Abh. Gesch. Math. 19, 1904).
- [1837] N. I. Lobachevsky, Géometrie imaginaire, J. Reine Angew. Math. 17, 295-320.
- [1859] A. Cayley, Sixth memoir upon quantics, Phil. Trans. 149, 61-91.
- [1868] B. Riemann, Ueber die Hypothesen welche der Geometrie zu Grunde liegen, Abh. K. G. Wiss. Göttingen 13 (from his Inaugural Address of 1854).
- [1868] E. Beltrami, Saggio di interpetrazione della geometria non-euclidea, Gior. Mat. 6, 248-312 (Also Op. Mat. 1, 374-405; Ann. École Norm. Sup. 6 (1869), 251-288).
- [1868] E. Beltrami, Teoria fondamentale degli spazii di curvatura costante, Annali di mat. ser. II 2, 232-255 (Op. Mat. 1, 406-429; Ann. École Norm. Sup. 6 (1869), 345-375).
- [1871] F. Klein, Über die sogenannte Nicht-Euklidische Geometrie, Math. Ann. 4, 573-625 (cf. Ges. Math. Abh. 1, 244-350).
- [1871] E. Betti, Sopra gli spazi di un numero qualunque di dimensioni, Annali di mat. ser. II 4, 140-158.
- [1873] H. A. Schwarz, Ueber diejenigen Fälle in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt, J. Reine Angew. Math. 75, 292-335 (Ges. Math. Abh. 2, 211-259).
- [1882] H. Poincaré, Theorie des groupes fuchsiens, Acta Math. 1, 1-62; (Oeuv. 2, 108-168). MR 1554574
- [1883] H. Poincaré, Mémoire sur les groupes kleinéens, Acta Math. 3, 49-92; (Oeuv. 2, 258-299). MR 1554613
- [1884] E. Picard, Sur un groupe de transformations des points de l'espace situés du même côté d'un plan, Bull. Soc. Math. France 12, 43-47. MR 1503932
- [1890] W. Dyck, Beiträge zur Analysis situs. II, Math. Ann. 37 273-316. MR 1510647
- [1890] F. Klein, Zur Nicht-Euklidischen Geometrie, Math. Ann. 37, 544-572 (Ges. Abh. 1, 353-383). MR 1510658
- [1891] L. Bianchi, Geometrische Darstellung der Gruppen Linearer Substitutionen mit ganzen complexen Coefficienten nebst Anwendungen auf die Zahlentheorie, Math. Ann. 38, 313-333. MR 1510677
- [1891] W. Killing, Ueber die Clifford-Klein'schen Raumformen, Math. Ann. 39, 258-278.
- [1895] H. Poincaré, Analysis situs, J. École Polyt. 1, 1-121 (Oeuv. 6, 193-288).
- [1898] J. Hadamard, Les surfaces à courbures opposées et leur lignes géodésiques, J. Math. Pures Appl. Ser. 5 4, 27-73 (Oeuv. 2, 729-775).
- [1900] C. F. Gauss, Werke 8, 175-239.
- [1908] H. Tietze, Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatshefte für Math. Phys. 19, 1-118. MR 1547755
- [1912] H. Gieseking, Analytische Untersuchungen ueber topologische Gruppen, Thesis, Muenster (Compare Magnus (1974), 153).
- [1912] L. E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann. 71, 97-115. MR 1511644
- [1913] O. Veblen and J. W. Alexander, Manifolds of n dimensions, Ann. of Math. (2) 14, 163-178.
- [1912] H. Weyl, Die Idee der riemannschen Fläche, Teubner, Leipzig.
- [1914] F. Hausdorff, Grundzüge der Mengenlehre, von Veit, Leipzig.
- [1919] G. Humbert, Sur la mesure des Classes d'Hermite de discriminant donné dans un corps quadratique imaginaire, et sur certains volumes non euclidiens, Comptes Rendus (Paris) 169, 448-454.
- [1923] E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen, Akad. Verl., Leipzig.
- [1925] H. Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95, 313-339. MR 1512281
- [1928] E. Cartan, Leçons sur la géométrie des espaces de Riemann, Gauthier-Villars, Paris. MR 20842
- [1931] H. Hopf and W. Rinow, Über den Begriff der vollständigen differentialgeometrischen Fläche, Comment. Math. Helv. 3, 209-225. MR 1509435
- [1935] H. S. M. Coxeter, The functions of Schläfli and Lobatschefsky, Quart. J. Math. 6, 13-29 (pp. 3-20 of twelve geometric essays, S. Ill. Univ. Press, Carbondale, 1968). MR 310745
- [1937] J. H. C. Whitehead, On doubled knots, J. London Math. Soc. 12, 63-71 (Works 2, 59-67).
- [1942] H. S. M. Coxeter, Non-Euclidean geometry, Univ. Toronto Press, Toronto. MR 6835
- [1958] L. Lewin, Dilogarithms and associated functions, MacDonald, London. MR 105524
- [1963] A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2, 111-112. MR 146301
- [1970] G. A. Margulis, Isometry of closed manifolds of constant negative curvature with the same fundamental group, Soviet Math. Dokl. 11, 722-723. MR 266103
- [1970] E. M. Andreev, On convex polyhedra [of finite volume] in Lobačevskiĭ space, Mat. Sbornik 81, 445-478 and 83, 256-260 (transl. 10, 413-440 and 12, 255-259). MR 273510
- [1971] G. D. Mostow, The rigidity of locally symmetric spaces, Proc. (1970) Internat. Congr. Math., vol. 2, pp. 187-197, Gauthier-Villars, Paris. MR 419683
- [1973] G. D. Mostow, Strong rigidity of locally symmetric spaces, Ann. Math. Studies, No. 78, Princeton Univ. Press, Princeton, N. J. MR 385004
- [1973] G. Prasad, Strong rigidity of Q-rank 1 lattices, Invent. Math. 21, 255-286. MR 385005
- [1974] W. Magnus, Noneuclidean tesselations and their groups, Academic Press, New York. MR 352287
- [1975] R. Riley, A quadratic parabolic group, Math. Proc. Cambridge Philos. Soc. 77, 281-288. MR 412416
- [1977] A. Ash, Deformation retracts with lowest possible dimension of arithmetic quotients of self-adjoint homogeneous cones, Math. Ann. 225, 69-76. MR 427490
- [1978] W. Thurston, Geometry and 3-manifolds, Lecture Notes, Princeton University.
- [1978] N. Wielenberg, The structure of certain subgroups of the Picard group, Math. Proc. Cambridge Philos. Soc. 84, 427-436. MR 503003
- [1979] U. Haagerup and H. Munkholm, Simplices of maximal volume in hyperbolic n-space, preprint, Odense University. MR 631085
- [1980] H. J. Munkholm, Simplices of maximal volume in hyperbolic space, Gromov’s norm, and Gromov's proof of Mostow's rigidity theory (following Thurston), pp. 109-124 of Topology Symposium—Siegen 1979, ed. Koschorke and Neumann, Springer Lecture Notes in Math., no. 788.
- [1980] R. Rieley, Applications of a computer implementation of Poincaré's theorem on fundamental polyhedra, preprint, I.A.S.
- [1980] R. Rieley, Seven excellent knots, preprint, I.A.S.
- [1980] M.-F. Vignéras, Variétés riemanniennes isospectrales et non isométriques, Ann. of Math. (2) 112, 21-32. MR 584073
- [1981] A. Borel, Commensurability classes and volumes of hyperbolic 3-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8, 1-33. MR 616899
- [1981] M. Gromov, Hyperbolic manifolds according to Thurston and Jórgensen, Sémin. Bourbaki 546, Springer Lecture Notes in Math., no. 842, pp. 40-53. MR 636516
- [1981] D. Sullivan, Travaux de Thurston sur les groupes quasi-fuchsiens et les variétés hyperboliques de dimension 3 fibrées sur S, Sémin. Bourbaki 554, Springer Lecture Notes in Math., no. 842, pp. 196-214. MR 636524
- [1981] W. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry (to appear). MR 648524
- [1981] W. Thurston, Hyperbolic structures on 3-manifolds, I: Deformation of acylindrical manifolds (to appear). MR 855294
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0273-0979-1982-14958-8
PII:
S 0273-0979(1982)14958-8
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