Hyperbolic geometry: The first 150 years
HTML articles powered by AMS MathViewer
- by John W. Milnor PDF
- Bull. Amer. Math. Soc. 6 (1982), 9-24
References
-
[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, DOI 10.1007/BF02391835
- H. Poincaré, Mémoire, Acta Math. 3 (1883), no. 1, 49–92 (French). Les groupes kleinéens. MR 1554613, DOI 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, DOI 10.24033/bsmf.272
- Walther Dyck, Beiträge zur Analysis situs, Math. Ann. 37 (1890), no. 2, 273–316 (German). MR 1510647, DOI 10.1007/BF01200237
- Felix Klein, Zur Nicht-Euklidischen Geometrie, Math. Ann. 37 (1890), no. 4, 544–572 (German). MR 1510658, DOI 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, DOI 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, DOI 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, DOI 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, DOI 10.1007/BF01206614
- É. Cartan, Leçons sur la Géométrie des Espaces de Riemann, Gauthier-Villars, Paris, 1946 (French). 2d ed. MR 0020842
- H. Hopf and W. Rinow, Ueber den Begriff der vollständigen differentialgeometrischen Fläche, Comment. Math. Helv. 3 (1931), no. 1, 209–225 (German). MR 1509435, DOI 10.1007/BF01601813
- H. S. M. Coxeter, Twelve geometric essays, Southern Illinois University Press, Carbondale, Ill.; Feffer & Simons, Inc., London-Amsterdam, 1968. MR 0310745 [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
- L. Lewin, Dilogarithms and associated functions, Macdonald, London, 1958. Foreword by J. C. P. Miller. MR 0105524
- Armand Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111–122. MR 146301, DOI 10.1016/0040-9383(63)90026-0
- 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
- E. M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (125) (1970), 256–260 (Russian). MR 0273510
- 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
- G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR 0385004
- Gopal Prasad, Strong rigidity of $\textbf {Q}$-rank $1$ lattices, Invent. Math. 21 (1973), 255–286. MR 385005, DOI 10.1007/BF01418789
- Wilhelm Magnus, Noneuclidean tesselations and their groups, Pure and Applied Mathematics, Vol. 61, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0352287
- Robert Riley, A quadratic parabolic group, Math. Proc. Cambridge Philos. Soc. 77 (1975), 281–288. MR 412416, DOI 10.1017/S0305004100051094
- 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 427490, DOI 10.1007/BF01364892 [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, DOI 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, DOI 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, DOI 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
- Michael Gromov, Hyperbolic manifolds (according to Thurston and Jørgensen), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin-New York, 1981, pp. 40–53. MR 636516
- Dennis Sullivan, Travaux de Thurston sur les groupes quasi-fuchsiens et les variétés hyperboliques de dimension $3$ fibrées sur $S^{1}$, Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin-New York, 1981, pp. 196–214 (French). MR 636524
- 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, DOI 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, DOI 10.2307/1971277
Additional Information
- Journal: Bull. Amer. Math. Soc. 6 (1982), 9-24
- MSC (1980): Primary 01A55, 01A60, 51M10; Secondary 57R15, 20H10
- DOI: https://doi.org/10.1090/S0273-0979-1982-14958-8
- MathSciNet review: 634431