Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

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
Similar Articles
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