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Geometry II pp 1–138Cite as

Geometry of Spaces of Constant Curvature

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Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 29))

Abstract

Spaces of constant curvature, i.e. Euclidean space, the sphere, and Lobachevskij space, occupy a special place in geometry. They are most accessible to our geometric intuition, making it possible to develop elementary geometry in a way very similar to that used to create the geometry we learned at school. However, since its basic notions can be interpreted in different ways, this geometry can be applied to objects other than the conventional physical space, the original source of our geometric intuition.

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References

  • Aleksandrov, A.D. (1950): Convex Polyhedra. Gostekhizdat, Moscow. German transi.: Akademie Verlag, Berlin 1958, Zb1. 41, 509

    Google Scholar 

  • Alekseevskij, D.V. (1973): S“ and En are the only Riemannian spaces admitting an essentially conformal transformation. Usp. Mat. Nauk 28, No.5, 225–226 (Russian), Zb1. 284. 53035

    Google Scholar 

  • Alekseevskij, D.V. (1975): Homogeneous Riemannian spaces of negative curvature. Mat. Sb., Nov. Ser. 96, 93–117. English transl.: Math. USSR, Sb. 25 (1975), 87109 (1976)

    Google Scholar 

  • Allendoerfer, C.B., Weil A. (1943): The Gauss-Bonnet theorem for Riemannian polyhedra. Trans. Am. Math. Soc. 53, 101–129, Zb1. 60, 381

    Google Scholar 

  • Andreev, E.M. (1970a): On convex polyhedra in Lobachevsky spaces. Mat. Sb., Nov. Ser 81(123), 445–478. English transi.: Math. USSR, Sb. 10, 413–440 (1970), Zb1. 194, 234

    Google Scholar 

  • Andreev, E.M. (1970b): On convex polyhedra of finite volume in Lobachevsky spaces. Mat. Sb., Nov. Ser. 83(125), 256–260. English transl.: Math. USSR, Sb. 12, 255–259 (1971), Zb1. 203, 549

    Google Scholar 

  • Andreev, E.M. (1970c): Intersection of plane boundaries of acute-angled polyhedra. Mat. Zametki 8, 521–527. English transi.: Math. Notes 8, 761–764 (1971), Zb1. 209, 265

    Google Scholar 

  • Aomoto, K. (1977): Analytic structure of Schläfli function. Nagoya Math.J. 68, 1–16, Zb1. 382. 33010

    Google Scholar 

  • Beardon, A.F. (1983): The Geometry of Discrete Groups. Springer, New York Berlin Heidelberg, Zb1. 528. 30001

    Google Scholar 

  • Berger, M. (1978): Géométrie, vols. 1–5. CEDIC/Fernand Nathan, Paris, Zb1.382. 51011, Zb1.382.51012, Zb1.423,51001, Zb1.423.51002, Zb1. 423. 51003

    Google Scholar 

  • Böhm, J., Hertel, E. (1981): Polyedergeometrie in n-dimensionalen Räumen konstanter Krümmung. Birkhäuser, Basel

    MATH  Google Scholar 

  • Borisenko, A.A. (1974): On complete surfaces in spaces of constant curvature. Ukr. Geom. Sb. 15, 8–15 (Russian), Zb1. 312, 53042

    Google Scholar 

  • Borisenko, A.A. (1981): On surfaces of non-positive exterior curvature in spaces of constant curvature. Mat. Sb. 114 (156), 339–354. English transi.: Math. USSR, Sb. 42, 297–310 (1982), Zb1. 473. 53049

    Google Scholar 

  • Bourbaki, N. (1969): Groupes et algèbres de Lie. Chap. 4–6. Hermann, Paris, Zb1. 186, 330

    Google Scholar 

  • Busemann, H., Kelly P.J (1953): Projective Geometry and Projective Metrics. Academic Press, New York, Zb1. 52, 373

    Google Scholar 

  • Cartan, E. (1928): Leçons sur la géometrie des espaces de Riemann. Gauthier-Villars, Paris, Jbuch 54, 755

    Google Scholar 

  • Cheeger, J., Ebin, D.G. (1975): Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam

    MATH  Google Scholar 

  • Chern, S. (1944): A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds. Ann. Math., II. Ser, 45, 747–752, Zb1. 60, 381

    Google Scholar 

  • Coxeter, H.S.M. (1934): Discrete groups generated by reflections. Ann. Math., II. Ser. 35, 588–621, Zb1. 10, 11

    Google Scholar 

  • Coxeter, H.S.M. (1935): The functions of Schläfli and Lobatchevsky. Q. J. Math., Oxf., Ser 6, 13–29, Zb1. 11, 170

    Google Scholar 

  • Coxeter, H.S.M. (1957): Non-Euclidean Geometry, 3rd edition. University of Toronto Press, Toronto, Zb1.60,328

    Google Scholar 

  • Dubrovin, B.A., Novikov, S.P, Fomenko, A.T. (1979): Modern Geometry. Nauka, Moscow. English transl.: Springer, New York Berlin Heidelberg 1984, Zb1.433. 53001

    Google Scholar 

  • Efimov, N.V. (1978): Higher Geometry. Nauka, Moscow. English transi.: Mir, Moscow 1990, Zb1. 457. 51001

    Google Scholar 

  • Fenchel, W. (1989): Elementary Geometry in Hyperbolic Spaces. Walter de Gruyter, Berlin New York

    Book  Google Scholar 

  • Gantmakher, F.R. (1967): Theory of Matrices, 2nd. ed. Nauka, Moscow. French transi.: Dunod, Paris 1966, Zb1.50,248; Zb1.145,36

    Google Scholar 

  • Gorbatsevich, V.V., Onishchik, A.L. (1988): Lie groups of transformations. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 20, 103–240. English transi. in: Encycl. Math. Sci. 20. Springer, Berlin Heidelberg New York Tokyo (in preparation), Zb1.65622003

    Google Scholar 

  • Gromoll, D., Klingenberg, W., Meyer, W. (1968): Riemannsche Geometrie im Grossen. Springer, Berlin Heidelberg New York, Zb1. 155, 307

    Google Scholar 

  • Günther, P. (1960): Einige Sätze über das Volumenelement eines Riemannschen Raumes. Publ. Math., Debrecen 7, 78–93, Zb1. 97, 373

    Google Scholar 

  • Haagerup, U., Munkholm, H.D. (1981): Simplices of maximal volume in hyperbolic n-space. Acta Math. 147, 1–11, Zb1. 493. 51016

    Google Scholar 

  • Hodgson, C.D., Rivin, I., Smith, W.D. (1992): A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere. Bull. Am. Math. Soc. 27, 246–251

    Article  MATH  MathSciNet  Google Scholar 

  • Kagan, V.F. (1949): Foundations of Geometry, part 1. Gostekhizdat, Moscow Leningrad (Russian), Zb1. 49, 380

    Google Scholar 

  • Kagan, V.F. (1956): Foundations of Geometry, part 2. Gostekhizdat, Moscow Leningrad (Russian), Zb1. 49, 380

    Google Scholar 

  • Karpelevich, F.I. (1953): Surfaces of transitivity of a semi-simple subgroup of the group of motions of a symmetric space. Dokl. Akad. Nauk SSSR 93, 401–404 (Russian), Zb1. 52, 389

    Google Scholar 

  • Kellerhals, R. (1989): On the volume of hyperbolic polyhedra. Math. Ann. 285, 541–569, Zb1. 664. 51012

    Google Scholar 

  • Kellerhals, R. (1991): On the volumes of hyperbolic 5-orthoschemes and the trilogarithm. Preprint. Max-Planck-Institut für Mathematik. Bonn

    Google Scholar 

  • Klein, F. (1928): Vorlesungen über Nicht-Euklidische Geometrie. Verlag von J.Springer, Berlin,Jbuch 54, 593

    Google Scholar 

  • Kobayashi, Sh., Nomizu, K. (1963): Foundations of Differential Geometry. Vol.l. Interscience Publishers, New York London Sydney, Zb1. 119, 375

    Google Scholar 

  • Kobayashi, Sh., Nomizu, K. (1969): Foundations of Differential Geometry. Vol.2. Interscience Publishers, New York London Sydney, Zb1. 175, 485

    Google Scholar 

  • Kuiper, N.H. (1949): On conformally flat spaces in the large. Ann. Math., II. Ser. 50, 916–924, Zb1. 41, 93

    Google Scholar 

  • Lobachevskij, N.I. (1949a): Imaginary Geometry. Complete Works. Vol.l. Moscow Leningrad, 16–70 (Russian)

    Google Scholar 

  • Lobachevskij, N.I. (1949b): Applications of Imaginary Geometry to Certain Integrals. Complete Works. Vol.3. Moscow Leningrad, 181–294 (Russian)

    Google Scholar 

  • Maier, W. (1954): Inhaltsmessung im R3 fester Krümmung. Ein Beitrag zur imaginären Geometrie. Arch. Math. 5, 266–273, Zb1. 55, 381

    Google Scholar 

  • Milnor, J. (1982): Hyperbolic geometry: the first 150 years. Bull. Am. Math. Soc., New Ser. 6, 9–24, Zb1. 486. 01006

    Google Scholar 

  • Rashevskij, P.K. (1967): Riemannian Geometry and Tensor Calculus, 2nd ed. Nauka, Moscow. German transl.: VEB Deutscher Verlag der Wissenschaften, Berlin 1959, Zb1.52,388, Zb1.92,392, Zb1.152,390

    Google Scholar 

  • Rivin, I. (1992): Some applications of Lobachevsky’s volume formula. Preprint. NEC

    Google Scholar 

  • Research Institute and Mathematics Department, Princeton University Rozenfel’d, B.A. (1966): Multi-Dimensional Spaces. Nauka, Moscow (Russian), Zb1. 143, 438

    Google Scholar 

  • Rozenfel’d, B.A. (1969): Non-Euclidean Spaces. Nauka, Moscow (Russian), Zb1. 176, 189

    Google Scholar 

  • Schläfli, L. (1858): On the multiple integral f f… f dx dy… dz whose limits are pi = a l + biy +… + hiz @ 0, p2 @ 0,…,p„ 0 and x2 + y2 +… + z2 * 1. Q. J. Math. 2, 269–300

    Google Scholar 

  • Schläfli, L. (1860): On the multiple integral f f… f dx dy… dz whose limits are pi = al + biy +… + hlz @ 0, p2 0,…,pß 0 and x2 + y2 +… + z2 * 1. Q. J. Math. 3, 54–68; 97–108

    Google Scholar 

  • Shabat, B.V. (1985): Introduction to Complex Analysis, Vol. 1–2. Nauka, Moscow. French transl.: Mir, Moscow 1990, Zb1.574.30001, Zb1. 578. 32001

    Google Scholar 

  • Shirokov, P.A. (1983): A Short Outline of the Foundations of Lobachevskij Geometry, 2nd ed. Nauka, Moscow. English transl.: P. Nordhoff Ltd., Groningen 1964, Zb1.68,142

    Google Scholar 

  • Solodovnikov, A.S. (1969): The group of projective transformations in a complete analytic Riemannian space. Dokl. Akad. Nauk SSSR 186, 1262–1265. English transl.: Sov. Math., Dokl. 10, 750–753 (1969), Zb1. 188, 540

    Google Scholar 

  • Vinberg, E.B. (1985): Hyperbolic reflection groups. Usp. Mat. Nauk. 40, No. 1, 29–66. English transl.: Russ. Math. Surv. 40, No. 1, 31–75 (1985), Zb1. 579. 51015

    Google Scholar 

  • Vinberg, E.B. (1991): The volume of polyhedra on a sphere and in Lobachevsky space. Proc. First Siberian Winter School “Algebra and Analysis” (Kemerovo, 1987). Amer. Math. Soc. Transl. 148, 15–18

    MathSciNet  Google Scholar 

  • Vinberg, E.B., Onishchik A.L. (1988): Foundations of the theory of Lie groups. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. Fundam. Napravleniya 20, 5–102. English transi. in: Encycl. Math. Sci. 20. Springer, Berlin Heidelberg New York Tokyo (in preparation), Zb1. 656. 22002

    Google Scholar 

  • Volkov, Yu.A., Vladimirova, S.M. (1971): Isometric immersions of a Euclidean plane in Lobachevsky space. Mat. Zametki 10, 327–332. English transi.: Math. Notes 10, 619–622, Zb1. 235. 53041

    Google Scholar 

  • Wolf, J.A. (1967): Spaces of Constant Curvature. McGraw-Hill, New York, Zb1. 162, 533

    Google Scholar 

  • Zagier, D. (1986): Hyperbolic manifolds and special values of Dedekind zeta-functions. Invent. Math. 83, 285–301, Zb1. 591. 12014

    Google Scholar 

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Authors
  1. D. V. Alekseevskij
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  2. E. B. Vinberg
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  3. A. S. Solodovnikov
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Moscow University, 119899, Moscow, Russia

    E. B. Vinberg

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Alekseevskij, D.V., Vinberg, E.B., Solodovnikov, A.S. (1993). Geometry of Spaces of Constant Curvature. In: Vinberg, E.B. (eds) Geometry II. Encyclopaedia of Mathematical Sciences, vol 29. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02901-5_1

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  • DOI: https://doi.org/10.1007/978-3-662-02901-5_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08086-9

  • Online ISBN: 978-3-662-02901-5

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