Skip to main content
SpringerLink
Log in

Surfaces of Negative Curvature

  • Chapter
Book cover Geometry III

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 48))

Abstract

This article is devoted to surfaces of negative Gaussian curvature K < 0 in three-dimensional Euclidean space E 3 and related problems. These surfaces constitute part of the class of saddle surfaces in E N. Hence the article serves as an extension of the third chapter of Part I of this book, written by Yu.D. Burago and S.Z. Shefel’. At the same time, this article is meant to be read independently, and so together with the references to Alekseevskij, Vinogradov and Lychagin (1988), Alekseevskij, Vinberg and Solodovnikov (1988), Burago and Shefel’ (1989), and Sabitov (1989b), we repeat certain facts in the text that are already reflected in these surveys. However, these repetitions are comparatively small.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Adel’son-Vel’skij, G.M. (1945): Generalization of a geometrical theorem of S.N. Bernstein. Dokl. Akad. Nauk SSSR 49, 399–401 (Russian), Zbl.61,373

    Google Scholar 

  • Aleksandrov, A.D. (1938): On a class of closed surfaces. Mat. Sb. 4, 69–77 (Russian), Zbl.20,261

    Google Scholar 

  • Aleksandrov, V.A. (1990): Efimov’s theorem on differential criteria for homeomorphism. Mat. Sb. 181, 183–188. Engl, transi.: Math. USSR. Sb. 69, No. 1, 197–202 (1991).

    Google Scholar 

  • Alekseevskij, D.V., Vinberg, E.B., Solodovnikov, A.G. (1988): The geometry of spaces of constant curvature. Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 29, 5–146, Zbl.699.53001. Engl, transi, in: Encycl. Math. Sc. 29, Springer-Verlag, Heidelberg (in preparation)

    Google Scholar 

  • Alekseevskij, D.V., Vinogradov, A.M., Lychagin, V.V. (1988): The basic ideas and concepts of differential geometry. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 28, 5–297, Zbl.675.53001. Engl, transi.: Encycl. Math. Sc. 28, Springer-Verlag, Heidelberg (1991)

    Google Scholar 

  • Alexander, S., Maltz, R. (1976): Isometric immersions of Riemannian products in Euclidean space. J. Differ. Geom. 11, 47–57, Zbl.334.53053

    MATH  MathSciNet  Google Scholar 

  • Aminov, Yu.A. (1968): n-dimensional analogues of S.N. Bernstein’s integral formula. Mat. Sb., Nov. Ser. 75, 375–399. Engl, transi.: Math. USSR, Sb. 4, 343–367 (1968), Zbl.176,187

    MathSciNet  Google Scholar 

  • Aminov, Yu.A. (1971): An energy condition for the existence of a vortex. Mat. Sb., Nov. Ser. 86, 325–334. Engl, transi.: Math. USSR, Sb. 15, 325–334 (1972), Zbl.221.52005

    Google Scholar 

  • Aminov, Yu.A. (1980): Isometric immersions of domains of n-dimensional Lobachevskij space in (2n — l)-dimensional Euclidean space. Mat. Sb., Nov. Ser. 111, 402–433. Engl, transi.: Math. USSR, Sb. 39, 359–386 (1981), Zbl.431.53023

    MathSciNet  Google Scholar 

  • Aminov, Yu.A. (1982): Embedding problems: geometrical and topological aspects. Itogi Nauki Tekh., Ser. Sovrem. Probl. Geom. 13, 119–156. Engl, transi.: J. Sov. Math. 25, 1308–1331 (1984), Zbl. 499.53018

    MathSciNet  Google Scholar 

  • Aminov, Yu.A. (1983): Isometric immersions of domain of three-dimensional Lobachevskij space in five-dimensional Euclidean space and the motion of a rigid body. Mat. Sb., Nov. Ser. 122, No. 1, 12–30. Engl, transi.: Math. USSR, Sb. 50,11–30 (1985), Zbl.539.53006

    MathSciNet  Google Scholar 

  • Aminov, Yu.A. (1988): Isometric immersions of domains of n-dimensional Lobachevskij space in Euclidean spaces with a flat normal connection. A model of a gauge field. Mat. Sb., Nov. Ser. 137, 275–299. Engl, transi.: Math. USSR, Sb. 65, No. 2, 279–303 (1990), Zbl.663.53018

    Google Scholar 

  • Amsler, M.H. (1955): Des surfaces à courbure négative constante dans l’espace à trois dimensions et de leurs singularités. Math. Ann. 130, 234–256, Zbl.68,351 Arnol’d, V.l. (1990): Catastrophe Theory (3rd edition). Nauka, Moscow. English transi, (of the 2nd. edition): Springer-Verlag, Berlin-Heidelberg-New York, 1984, Zbl.517.58002

    MATH  MathSciNet  Google Scholar 

  • Azov, D.G. (1983): On a class of hyperbolic Monge-Ampère equations. Usp. Mat. Nauk 38, No. 1, 153–154. Engl, transi.: Russ. Math. Surv. 38, No. 1,170–171 (1983), Zbl.524.35072

    MATH  MathSciNet  Google Scholar 

  • Azov, D.G. (1984): Some generalizations of a theorem of N.V. Efimov on hyperbolic Monge-Ampère equations. In: Differential equations and their applications, Collect. Artic, Moscow 1984, 60–64 (Russian), Zbl.595.35088

    Google Scholar 

  • Azov, D.G. (1985): Immersion by D. Blanusa’s method of some classes of complete n-dimensional Riemannian metrics in Euclidean spaces. Vestn. Mosk. Univ., Ser. I, No. 5, 72–74. English transi: Mosc. Univ. Math. Bull. 40, No. 5, 64–66 (1985), Zbl.581.53040

    MATH  MathSciNet  Google Scholar 

  • Bäcklund, A.V. (1905): Concerning surfaces with constant negative curvature. New Era Printing Co., Lancaster

    Google Scholar 

  • Baikoussis, C, Koufogiorgos, T. (1980): Isometric immersions of complete Riemannian manifolds into Euclidean space. Proc. Amer. Math. Soc. 79, 87–88, Zbl.466.53003

    MATH  MathSciNet  Google Scholar 

  • Bakievich, N.I. (1960): Some boundary-value problems for equations of mixed type that arise in the study of infinitesimal bendings of surfaces of revolution. Usp. Mat. Nauk 15, No. 1, 171–176 (Russian), Zbl.91,341

    Google Scholar 

  • Barone, A., Esposito, F., Magee, C.J., Scott, A.C. (1971): Theory and application of the sine-Gordon equation. Riv. Nuovo Cimento 1, 227–267

    Google Scholar 

  • Barone, A., Paternò, G. (1982): Physics and Applications of the Josephson Effect. Wiley, Chichester-New York

    Google Scholar 

  • Bellman, R., Kalaba, R. (1965): Quasilinearization and Nonlinear Boundary-Value Problems. Elsevier, New York, Zbl.139,107

    MATH  Google Scholar 

  • Beltrami, E. (1868a): Saggio di interpretazione della geometria non-euclidea. Giorn. di Mat. Napoli 6, 285–315, Jbuch 1,275

    MATH  Google Scholar 

  • Beltrami, E. (1868b): Teoria fondamentale degli spazii di curvature constante. Ann. Mat. Pura Appl. 2, 232–255, Jbuch 1, 209

    Google Scholar 

  • Beltrami, E. (1872): Sulla superficie di rotazione che serve di tipo aile superficie pseudosferiche. Giorn. di Mat. Napoli 10,147–159

    MATH  Google Scholar 

  • Bernstein, S.N. (1960a): On a geometrical theorem and its applications to partial differential equations of elliptic type. Collected Works, Vol. 3, 251–258, Zbl. 178,447

    Google Scholar 

  • Bernstein, S.N. (1960b): Strengthening of a theorem on surfaces of negative curvature. Collected Works, Vol. 3, 361–370, Zbl.178,447

    Google Scholar 

  • Bianchi, L. (1927): Lezioni di geometria differenziale. Vol. 1, parte 1–2, 4. ed., Zanichelli, Bologna, Jbuch 48, 784

    MATH  Google Scholar 

  • Bieberbach, L. (1932): Eine singularitätenfreie Fläche konstanter negativer Krümmung im Hilbertschen Raum. Comment. Math. Helv. 4, 248–255, Zbl.5,82

    MathSciNet  Google Scholar 

  • Blanusa, D. (1953): Eine isometrisch singularitätenfreie Einbettung des n-dimensionalen hyperbolischer Räume im Hilbertschen Raum. Monatsh. Math. 57,102–108, Zbl.53,109

    MATH  MathSciNet  Google Scholar 

  • Blanusa, D. (1955): Über die Einbettung hyperbolischer Räume in Euklidische Räume. Monatsh. Math. 59, 217–229, Zbl.67,144

    MATH  MathSciNet  Google Scholar 

  • Blaschke, W. (1930): Vorlesungen über Differentialgeometrie und Geometrische Grundlagen Einsteins Relativitätstheorie. I. Elementare Differentialgeometrie. 3rd. ed., Springer-Verlag, Berlin, Jbuch 56, 588

    MATH  Google Scholar 

  • Bolin, B. (1956): An improved barotropic model and some aspects of using the balance equation on three-dimensional flows. Tellus 8, 61–75

    Google Scholar 

  • Brandt, I.S. (1970a): Surfaces of negative extrinsic curvature in a Riemannian space with non-positive Riemannian curvature. Dokl. Akad. Nauk SSSR 194, 747–749. Engl, transi.: Sov. Math. Dokl. 11, 1270–1272 (1970), Zbl.213,481

    MathSciNet  Google Scholar 

  • Brandt, I.S. (1970b): Some properties of surfaces with slowly changing negative extrinsic curvarture in a Riemannian space. Mat. Sb., Nov. Ser. 83,313–324. Engl, transi.: Math. USSR, Sb. 12,313–324 (1971), Zbl.203,243

    Google Scholar 

  • Brys’ev, A.B. (1985): An estimate of the domain of regularity of solutions of some non-linear differential inequalities. Ukr. Geom. Sb. 28, 19–21. Engl, transi.: J. Sov. Math. 48, No. 1, 15–17 (1990), Zbl.584.34006

    MATH  Google Scholar 

  • Burago, Yu.D. (1989), Shefel’, S.Z.: The geometry of surfaces in Euclidean spaces. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 48, 5–97. Engl, transi, in: Encycl. Math. Sc. 48, Springer-Verlag, Heidelberg, 1–85,1992 (Part I of this volume)

    MATH  Google Scholar 

  • Burago, Yu.D., Zalgaller, V.A. (1974): Sufficient criteria for convexity. Zap. Nauchn. Semin. Leningr., Otd. Mat. Inst. Steklova 45, 3–52. Engl, transi.: J. Sov. Math. 8, 395–435 (1978), Zbl. 348.52003

    MATH  MathSciNet  Google Scholar 

  • Cartan, E. (1919–1920): Sur les variétés de courbure constante d’un espace euclidien ou non-euclidien. Bull. Soc. Math. France 47, 125–160; 48,132–208, Zbuch.47,692

    MathSciNet  Google Scholar 

  • Chebyshëv, P.L. (1878): Sur la coupe des habits. Lecture to the Association française pour l’avancement des sciences, 28 August 1878

    Google Scholar 

  • Chern, S.S., Kuiper, N.H. (1952): Some theorems on the isometric imbedding of compact Riemannian manifolds in Euclidean space. Ann. Math., II. Ser. 56,422–430, Zbl.52,176

    MathSciNet  Google Scholar 

  • Cohn-Vossen, S.E. (1928): Die parabolische Kurve. Math. Ann. 99, 273–308, Jbuch 54, 498

    MATH  MathSciNet  Google Scholar 

  • Dini, V. (1865): Sulle superficie nelle quali la somma due raggi di curvature principale è constante. Ann. Mat. Pura Appl. 7, 5–18.

    Google Scholar 

  • Dubrovin, B.A., Novikov, S.P., Fomenko, A.T. (1979): Modern Geometry. Methods and Applications. Nauka, Moscow, Zbl.433.53001. Engl, transi.: Grad. Texts Math. 93, Part I (1984) and 104, Part II (1985)

    Google Scholar 

  • Eberlein, P. (1979): Surfaces of nonpositive curvature. Mem. Amer. Math. Soc. 218, 90 pp., Zbl.497.53043

    MathSciNet  Google Scholar 

  • Eberlein, P. (1985): Structure of manifolds of nonpositive curvature. Lect. Notes Math. 1156, 86–153, Zbl.569.53020

    MathSciNet  Google Scholar 

  • Efimov, N.V. (1953): Investigation of a one-to-one projection of a surface of negative curvature. Dokl. Akad. Nauk SSSR 93, 609–611, Zbl.52,172

    MATH  MathSciNet  Google Scholar 

  • Efimov, N.V. (1963): The impossibility in three-dimensional Euclidean space of a complete regular surface with a negative upper bound of the Gaussian curvature. Dokl. Akad. Nauk SSSR 150, 1206–1209. Engl, transi.: Sov. Math. Dokl. 4, 843–846 (1963), Zbl.135,400

    MathSciNet  Google Scholar 

  • Efimov, N.V. (1964): The origin of singularities on surfaces of negative curvature. Mat. Sb., Nov. Ser. 64, 286–320 (Russian), Zbl.126,374

    MathSciNet  Google Scholar 

  • Efimov, N.V. (1966a): Hyperbolic problems in the theory of surfaces. Proc. Int. Congr. Math., Moscow 1966,177–188. Engl, transi.: Transi., II. Ser., Am. Math. Sec. 70, 26–38 (1968), Zbl.188,536

    Google Scholar 

  • Efimov, N.V. (1966b): Surfaces with slowly changing negative curvature. Usp. Mat. Nauk 21, No. 5, 3–58. Engl, transi: Russ. Math. Surv. 21, No. 5,1–55 (1966), Zbl.171,199

    MathSciNet  Google Scholar 

  • Efimov, N.V. (1968): Differential criteria for homeomorphism of certain mappings with application to the theory of surfaces. Mat. Sb., Nov. Ser. 76, 499–512. Engl, transi.: Math. USSR, Sb. 5,475–488 (1968), Zbl. 164,215

    MathSciNet  Google Scholar 

  • Efimov, N.V. (1976): Estimates of the dimensions of the domain of regularity of certain Monge-Ampère equations. Mat. Sb. Nov. Ser. 100, 356–363. Engl, transi: Math. USSR, Sb. 29, 319–326 (1978),Zbl.332.35018

    MathSciNet  Google Scholar 

  • Efimov, N.V. (1984): Surfaces of negative curvature. Math. Encyclopaedia, vol 4,163–173

    Google Scholar 

  • Efimov, N.V., Poznyak, E.G. (1961): Some transformations of the basic equations of the theory of surfaces. Dokl. Akad. Nauk SSSR 137, 25–27. Engl, transi: Sov. Math. Dokl. 2, 225–227 (1961), Zbl 108,339

    MathSciNet  Google Scholar 

  • Enz, U. (1964): Die Dynamik der Blochschen Wand. Helv. Phys. Acta 37, 245–251

    Google Scholar 

  • Finikov, S.P. (1950): Theory of Congruences. GTTI, Moscow-Leningrad. German transi.: Akademie-Verlag, Berlin (1959), Zbl.85,367

    Google Scholar 

  • Finikov, S.P. (1952): A Course of Differential Geometry. GTTI, Moscow (Russian) Zbl.48,149

    Google Scholar 

  • Fomichëva, Yu.G. (1978): On the existence of a surface of negative curvature in E 3 with a given product of the principal curvatures defined as a function of the normal. In: Modern geometry. Repub. Collect. Sei. Works, Leningrad 1978,136–139 (Russian), Zbl.423.53004

    Google Scholar 

  • Fomichëva, Yu.G. (1979a): Stability of the solution of the problem of constructing in E 3 a surface with a given spherical mapping from its negative curvature defined as a function of the normal. In: Questions of global geometry. Collect. Sci. Works, Leningrad 1979 (Russian), Zbl.477.53004

    Google Scholar 

  • Fomichëva, Yu.G. (1979b): On the existence in E 3 of an asymptotic quadrangle with a univalent spherical mapping and given negative Gaussian curvature. In: Studies in the geometry of immersed manifolds and in projective geometry. Collect. Sci. Works, Leningrad 1979, 109–121 (Russian), Zbl.532.53003

    Google Scholar 

  • Galchenkova, R.I., Lumiste, Yu.G., Ozhigova, E.P., Pogrebysskij, LB. (1970): Ferdinand Minding. Nauka, Leningrad (Russian), Zbl.225.01007

    MATH  Google Scholar 

  • Galeeva, R.F., Sokolov, D.D. (1984a): On the geometrical interpretation of solutions of some non-linear equations of mathematical physics. In: Investigations on the theory of surfaces in Riemannian spaces. Interuniv. Collect. Sci. Works, Leningrad 1984,8–22 (Russian), Zbl.616.35074

    Google Scholar 

  • Galeeva, R.F., Sokolov, D.D. (1984b): Infinitesimal deformations of a hyperboloid of one sheet, ibid., 41–44 (Russian), Zbl.601.53004

    Google Scholar 

  • Gauss, D.F. (1823–1827): Disquisitiones générales circa superficies curvas. Comm. Soc. Reg. Scient. Göttingensis Recentiores 6, 99–146. Also in: Gauss, CF., Werke, Band 4, Gesellschaft der Wissen­schaft, Göttingen, 1873, pp. 217–258

    Google Scholar 

  • Geisberg, S.P. (1970): On the properties of the normal mapping generated by the equation rt — s 2 = -f 2 (x, y) . Mat. Sb., Nov. Ser. 82, 224–232. Engl, transi.: Math. USSR, Sb. 11, 201–208 (1970), Zbl.194,525

    MathSciNet  Google Scholar 

  • Gisina, F.A. et al. (1976): Dynamic Meteorology. Gidrometeoizdat, Leningrad (Russian)

    Google Scholar 

  • Gol’denveizer, A.L. (1976): The Theory of Thin Elastic Shells. 2nd ed., Nauka, Moscow (Russian), Zbl.461.73052

    Google Scholar 

  • Gol’denveizer, A.L. (1979): Mathematical rigidity of surfaces and physical rigidity of shells. Izv. Akad. Nauk SSSR Mekh. Tverd. Tela 1979, No. 6, 66–67. Engl, transi.: Mech. Solids 14, No. 6, 53–63 (1979)

    MathSciNet  Google Scholar 

  • Gol’denveizer, A.L., Lidskij, B.V., Tovstik, P.E. (1979): Free Oscillations of Thin Elastic Shells. Nauka, Moscow (Russian)

    Google Scholar 

  • Gribkov, I.V. (1977): The construction of some regular solutions of the “sine-Gordon” equation by means of surfaces of constant curvature. Vestn. Mosk. Univ., Ser. I., No. 4, 78–83. Engl, transi.: Mosc. Univ. Math. Bull. 32, No. 4, 63–67 (1977), Zbl.359.53003

    MathSciNet  Google Scholar 

  • Hartman, P., Wintner, A. (1951): On the asymptotic curves of a surface. Am. J. Math. 73, 149–172, Zbl.42,157

    MATH  MathSciNet  Google Scholar 

  • Hartman, P., Wintner, A. (1952): On hyperbolic partial differential equations. Am. J. Math. 74, 834–864, Zbl.48,333

    MATH  MathSciNet  Google Scholar 

  • Hartman, P., Wintner, A. (1953): On asymptotic parametrisation. Am. J. Math. 75, 488–496, Zbl.50,378

    MATH  MathSciNet  Google Scholar 

  • Hazzidakis, J.N. (1879): Über einige Eigenschaften der Flächen mit konstante Krümmungsmass. J. Reine Angew. Math. 88, 68–73, Jbuch 11, 527

    MATH  Google Scholar 

  • Heinz, E. (1955): Über Flächen mit eineindeutiger Projektion auf der Ebene, deren Krümmungen durch Ungleichungen eingeschränkt sind. Math. Ann. 129, 451–454, Zbl.65,372

    MATH  MathSciNet  Google Scholar 

  • Henke, W. (1981): Isometrische Immersionen des n-dimensional hyperbolischen Raumes H n im E4n-3. Manuscr. Math. 34, 265–278, Zbl.458.53035

    MATH  MathSciNet  Google Scholar 

  • Hubert, D. (1901): Flächen von konstanter Gauss’schen Krümmung. Trans. Amer. Math. Soc. No. 2, 87–99, Jbuch 32, 608

    MathSciNet  Google Scholar 

  • Hubert, D. (1903): Über Flächen von konstanter Gauss’sche Krümmung. In: Grundlagen der Geo­metrie. 2nd. ed., Teubner, Leipzig, pp. 162–175, Jbuch 34, 223

    Google Scholar 

  • Hubert, D., Cohn-Vossen, S.E. (1932): Anschauliche Geometrie. Springer Verlag, Berlin. Engl, transi.: Geometry and the Imagination. Chelsea, New York, 1952, Zbl.5,112

    Google Scholar 

  • Holmgren, E. (1902): Sur les surfaces à courbure constante négative. C. R. Acad. Sci. Paris 134, 740–743, Jbuch 33, 643

    MATH  Google Scholar 

  • Ivanova-Karatopraklieva, I. (1983): Sufficient conditions for rigidity of some classes of surfaces that project one-to-one on a plane. Banach Cent. Publ. 12, 83–93 (Russian), Zbl.561.53006

    MathSciNet  Google Scholar 

  • Ivanova-Karatopraklieva, I. (1984a): Non-rigidity of some classes of surfaces of revolution of mixed curvature. C. R. Akad. Bulg. Sci. 37, 569–572 (Russian), Zbl.546.53003

    MATH  MathSciNet  Google Scholar 

  • Ivanova-Karatopraklieva, I. (1984b): Conditions for rigidity of some classes of surfaces of mixed type. I. Serdica 10, 287–302 (Russian), Zbl.567.53002

    MATH  MathSciNet  Google Scholar 

  • Ivanova-Karatopraklieva, I. (1985): Rigidity of some classes of surfaces of revolution of mixed curvature whose boundary is not a parallel. Serdica 11, 330–340 (Russian), Zbl.611.53006

    MATH  MathSciNet  Google Scholar 

  • John, F. (1968): On quasi-isometric mappings. I. Commun. Pure Appl. Math. 21, 77–110, Zbl. 157,458

    MATH  Google Scholar 

  • Kadomtsev, S.B. (1978): The impossibility of some special immersions of Lobachevskij space. Mat. Sb., Nov. Ser. 107, 175–198. Engl, transi: Math. USSR, Sb. 35,461–480 (1979), Zbl.394.53032

    MathSciNet  Google Scholar 

  • Kaidasov, Zh., Shikin, E.V. (1986): An isometric immersion in E 3 of a convex domain of the Lobachevskij plane containing two horocycles. Mat. Zametki 39, 612–617. Engl, transi.: Math. Notes 39, 335–338 (1986), Zbl.609.53005

    MathSciNet  Google Scholar 

  • Kantor, B.E. (1970): On the question of the normal image of a complete surface of negative curvature. Mat. Sb., Nov. Ser. 82, 220–223. Engl, transi.: Math. USSR, Sb. 11, 197–200 (1970) Zbl. 194,525

    MathSciNet  Google Scholar 

  • Kantor, B.E. (1976): Rigidity of surfaces of negative curvature. Sib. Mat. Zh. 17, 1052–1057. Engl. transi.: Sib. Math. J. 17, 777–781 (1977), Zbl.358.53003

    MATH  MathSciNet  Google Scholar 

  • Kantor, B.E. (1978a): On the rigidity of some surfaces of negative curvature. In: Questions of global geometry. 31st Herzen lecture. Leningrad State Ped. Inst., Leningrad, 16–18 (Russian)

    Google Scholar 

  • Kantor, B.E. (1978b): On the unique determination of some classes of surfaces of negative curvature. In: Modern geometry. Republ. Collect. Sci. Works, Leningrad 1978, 67–73 (Russian), Zbl.421.53002

    Google Scholar 

  • Kantor, B.E. (1980): The absence of closed asymptotic lines on a class of tubes of negative curvature. Sib. Mat. Zh. 21, No. 6, 21–27. Engl, transi.: Sib. Math. J. 21, 768–773 (1980), Zbl.467.53002

    MATH  MathSciNet  Google Scholar 

  • Kantor, B.E. (1981): On the uniqueness of the solution of a Monge-Ampère equation of hyperbolic type with two fixed characteristics. In: Modern geometry (Investigations on differential geometry). Interuniv. Collect. Sci. Works, Leningrad 1981, 78–81 (Russian), Zbl.548.53004

    Google Scholar 

  • Khineva, S. (1977): Infinitesimal bendings of surfaces of negative Gaussian curvature. God. Sofij. Univ., Fak. Mat. Mekh. 68 (1973/1974), 295–309 (Russian), Zbl.393.53001

    MATH  Google Scholar 

  • Klabukova, L.S. (1983): On the differential operator of problems of the theory of moment-free elastic shells with negative Gaussian curvature. Zh. Vychisl. Mat. Mat. Fiz. 23, 1477–1486. Engl, transi: USSR Comput. Math. Math. Phys. 23, No. 6, 120–126 (1983), Zbl.563.73081

    MathSciNet  Google Scholar 

  • Klotz-Milnor, T. (1972): Efimov’s theorem about complete immersed surfaces of negative curvature. Adv. Math. 8, 474–543, Zbl.236.53055

    MATH  Google Scholar 

  • Kosevich, A.M. (1972): The Foundations of the Mechanics of a Crystal Lattice. Nauka, Moscow (Russian)

    Google Scholar 

  • Kovaleva, G. A. (1968): An example of a homotopic tube of a surface with closed asymptotic lines. Mat. Zametki 3, 403–413. Engl, transi.: Math. Notes 3, 257–263 (1968), Zbl.169,237

    MathSciNet  Google Scholar 

  • Kuiper, N.H. (1955): On CMsometric imbeddings. I, II. Nederl. Acad. Wetensch. Proc. Ser. A 58 (Indagationes Math. 17), 545–556, 683–689, Zbl.67,396

    MATH  MathSciNet  Google Scholar 

  • Lamb, G.L. (1971): Analytical description of ultrashort optical pulse propagation in the resonant medium. Rev. Mod. Phys. 43,99–124

    MathSciNet  Google Scholar 

  • Lavrant’ev M.A., Shabat, B.V. (1973): Problems of Hydrodynamics and Their Mathematical Models. Nauka, Moscow (Russian)

    Google Scholar 

  • Liber, A.E. (1938): On a class of Riemannian spaces of constant negative curvature. Uch. Zap. Saratov. Univ. Ser. Fiz.-Mat. 1, No. 2, 105–122 (Russian)

    Google Scholar 

  • Mikhailovskij, V.I. (1962a): Infinitesimal bendings of surfaces of revolution of negative curvature with conical sleeve-like constraints. Dopov. Akad. Nauk Ukr. SSR 1962, No. 8, 990–993 (Russian), Zbl.139,149

    Google Scholar 

  • Mikhailovskij, V.I. (1962b): Infinitesimal “gliding” deformations of surfaces of revolution of negative curvature. Ukr. Mat. Zh. 14, 18–29 (Russian), Zbl.141,188

    Google Scholar 

  • Mikhailovskij, V.I. (1962c): Infinitesimal bendings of the lateral surface of a glued tube of revolution of negative curvature. Vestn. Kiev. Univ. Ser. Mat. Mekh., No. 5, Part 1, 58–64 (Russian)

    Google Scholar 

  • Mikhailovskij, V.I. (1962d): Infinitesimal bendings of piecewise regular surfaces of revolution of negative curvature. Ukr. Mat. Zh. 14, 422–426 (Russian), Zbl.146,432

    Google Scholar 

  • Mikhailovskij, V.I. (1988): On some boundary-value problems of the theory of infinitesimal bendings of surfaces of negative curvature. 9th All-Union Geometry Conference, Kishinev, pp. 218–219

    Google Scholar 

  • Miller, J.D. (1984): Some global inverse function theorems. J. Math. Anal. Appl. 100, 375–384, Zbl.549.58006

    MATH  MathSciNet  Google Scholar 

  • Minagawa, T., Rado, T. (1952): On the infinitesimal rigidity of surfaces. Osaka Math. J. 4, 241–285, Zbl.48,153

    MATH  MathSciNet  Google Scholar 

  • Minding, F. (1839): Wie sich entscheiden lässt, ob zwei gegebene krumme Flächen auf einander abwickelbar sind oder nicht; nebst Bemerkungen über die Flächen von unveränderlichen Krümmungsmasse. J. Reine Angew. Math. 19, 370–387

    MATH  Google Scholar 

  • Minding, F. (1840): Beiträge zur Theorie der kürzesten Linien und krummen Flächen. J. Reine Angew. Math. 20, 323–327

    MATH  Google Scholar 

  • Moore, J.D. (1971): Isometric immersions of Riemannian products. J. Differ. Geom. 5, 159–168, Zbl.213,238

    Google Scholar 

  • Nirenberg, L. (1963): Rigidity of a class of closed surfaces. In: Nonlinear problems. Proc. Symp. Madison 1962, 177–193, Zbl. 111,344

    Google Scholar 

  • Norden, A.P. (ed.) (1956): On the foundations of geometry. In: Classic Works on Lobachevskij Geometry and the Development of its Ideas. GITTL, Moscow (Russian), Zbl.72,156

    Google Scholar 

  • Novikov, S.P. (1986): Topology. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 12, 5–552, Zbl.668.55001. Engl, transi, in: Encycl. Math. Sc. 12, Springer-Verlag, Heidelberg (1992).

    MATH  Google Scholar 

  • Novikov, S.P., Fomenko, A.T. (1987): Elements of Differential Geometry and Topology. Nauka, Moscow (Russian), Zbl.628.53002

    MATH  Google Scholar 

  • Pogorelov, A.V. (1967): Geometrical Methods in the Nonlinear Theory of Elastic Shells. Nauka, Moscow (Russian), Zbl.168,452

    Google Scholar 

  • Popov, A.G. (1989): An analogue of the phase space for the sine-Gordon equation. Vestn. Mosk. Univ., Ser. Fiz. Astron. 30, No. 4, 19–22. Engl, transi.: Mose. Univ. Phys. Bull. 44, No. 4, 20–23 (1989)

    Google Scholar 

  • Pourciau, B. (1988): Global invertibility of nonsmooth mappings. J. Math. Anal. Appl. 131, 170–179, Zbl.666.49004

    MATH  MathSciNet  Google Scholar 

  • Poznyak, E.G. (1966): On the regular realization in the large of two-dimensional metrics of negative curvature. Dokl. Akad. Nauk SSSR 170, 786–789. Engl, transi.: Sov. Math. Dokl. 7, 1288–1291 (1966), Zbl.168,195

    MathSciNet  Google Scholar 

  • Poznyak, E.G. (1977a): Geometrical investigations connected with the equation Z xy = sin Z. Itogi Nauki Tekh., Ser. Sovrem. Probl. Geom. 8, 225–241. English transi.: J. Sov. Math. 13, 677–686 (1980), Zbl.432.53003

    Google Scholar 

  • Poznyak, E.G. (1977b): Isometric immersion in E 3 of some non-compact domains of the Lobachevskij plane. Mat. Sb., Nov. Ser. 102, 3–12. Engl, transi.: Math. USSR, Sb. 31,1–8 (1977), Zbl.344.53008

    Google Scholar 

  • Poznyak, E.G. (1979): Geometrical interpretation of regular solutions of the equation Z xy = sin Z. Differ. Uravn. 15, 1332–1336. Engl, transi.: Differ. Equations 15, 948–951 (1980), Zbl.413.35078

    MATH  MathSciNet  Google Scholar 

  • Poznyak, E.G. (1991): Isometric immersions of Riemannian spaces. Differential geometry structures on manifolds and their applications. Collection of materials of the Ail-Union geometry school, Chernovtsy Univ., pp. 177–182. (Deposited at VINITI, No. 562-B91)

    Google Scholar 

  • Poznyak, E.G., Popov, A.G. (1991): The geometry of the sine-Gordon equation, Itogi Nauki Tekh., Ser. Sovrem. Problem. Geom. 23,99–130

    MATH  MathSciNet  Google Scholar 

  • Poznyak, E.G., Shikin, E.V. (1974): Surfaces of negative curvature. Itogi Nauki Tekh., Ser. Algebra, Topologija, Geom. 12, 171–207. English transi.: J. Sov. Math. 5, 865–887 (1976), Zbl.318.53050

    Google Scholar 

  • Poznyak, E.G., Shikin, E.V. (1980): Isometric immersions of a domain of the Lobachevskij plane in Euclidean spaces. Tr. Tbilis Mat. Inst. Razmadze 64, 82–93 (Russian), Zbl.497.53006

    MATH  MathSciNet  Google Scholar 

  • Poznyak, E.G., Shikin, E.V. (1986): Analytic tools of the theory of imbeddings of negative curvature two-dimensional manifolds-. Izv. Vyssh. Uchebn. Zaved., Mat. 1986, No. 1,56–60. Engl. transi.: Soviet Math. 30, 72–79 (1986), Zbl.607.53033

    MATH  Google Scholar 

  • Poznyak, E.G., Sokolov, D.D. (1977): Isometric immersions of Riemannian spaces in Euclidean spaces. Itogi Nauki Tekh., Ser. Algebra, Topologija, Geom. 15, 173–211. English transi.: J. Sov. Math. 14, 1407–1428 (1980), Zbl.448.53040

    Google Scholar 

  • Rozendorn, E.R. (1962): On complete surfaces of negative curvature K ≤ - 1 in the Euclidean spaces E 3 and E 4. Mat. Sb., Nov. Ser. 58, 453–478 (Russian), Zbl.192,272

    MathSciNet  Google Scholar 

  • Rozendorn, E.R. (1966): Investigation of the main equations of the theory of surfaces in asymptotic coordinates. Mat. Sb., Nov. Ser. 70, 490–507 (Russian), Zbl.145,415

    MathSciNet  Google Scholar 

  • Rozendorn, E.R. (1972): Non-immersibility of complete q-metrics of negative curvature in the class of weakly non-regular surfaces. Mat. Sb., Nov. Ser. 89, 83–92. Engl, transi.: Math. USSR, Sb. 18, 83–92 (1973), Zbl.241.53035

    MathSciNet  Google Scholar 

  • Rozendorn, E.R. (1980): Reduction of a problem of meteorology to a geometrical problem. Usp. Mat. Nauk 35, No. 6, 167–168. Engl, transi.: Russ. Math. Surv. 35, No. 1, 101–102 (1980), Zbl.458.53013

    MATH  MathSciNet  Google Scholar 

  • Rozendorn, E.R. (1985): A class of surfaces of negative curvature with singularities on curves. Vestn. Mosk. Univ. Ser. I, No. 1, 50–52. Engl, transi.: Mosc. Univ. Math. Bull. 40, No. 1, 75–79 (1985) Zbl.576.53004

    MathSciNet  Google Scholar 

  • Rozendorn, E.R. (1987): The integral form of writing the equations of infinitesimal bendings for surfaces of negative curvature. Ail-Union conference on geometry “in the large”. Novosibirsk, p. 105 (Russian)

    Google Scholar 

  • Rozendorn, E.R. (1988): Investigation of the regularity of a surface with a regular metric of negative curvature. All-Union School on optimal control, geometry and analysis. Kemerovo, p. 43 (Russian)

    Google Scholar 

  • Rozhdestvenskij, B.L. (1962): A system of quasilinear equations of the theory of surfaces. Dokl. Akad. Nauk SSSR 143, 50–52. Engl, transi.: Sov. Math. Dokl. 3, 351–353 (1962), Zbl.137,407

    MathSciNet  Google Scholar 

  • Rozhdestvenskij, B.L., Yanenko, N.N. (1978): Systems of quasilinear equations and their applications to gas dynamics. 2nd. ed., Nauka, Moscow. Engl, transi.: Transi. Math. Monographs, Vol. 55, Providence (1983), Zbl.513.35002, Zbl.177,140

    MATH  Google Scholar 

  • Sabitov, I.Kh. (1989a): On isometric immersions of the Lobachevskij plane in E 4. Sib. Mat. Zh. 30, No. 5, 179–186. Engl, transi: Sib. Math. J. 30, No. 5, 805–811 (1989), Zbl.697.53014

    MathSciNet  Google Scholar 

  • Sabitov, I.Kh. (1989b): Local theory of bendings of surfaces. Itogi Nauki Tekh., Ser. Sovem. Probl. Mat., Fundam. Napravleniya 48, 196–270. Engl, transi, in: Encycl. Math. Sc. 48, Springer-Verlag, Heidelberg, 179–250, 1992 (Part III of this volume)

    MATH  MathSciNet  Google Scholar 

  • Schur, F. (1886): Über die Deformation der Räume konstanten Riemannschen Krümmungsmasses. Math. Ann. 27, 163–176, Jbuch 18,444

    MathSciNet  Google Scholar 

  • Seifert, H., Threlfall, W. (1934): Lehrbuch der Topologie. Teubner, Leipzig-Berlin. Reprint: Chelsea Publ. Co., New York, 1947, Zbl.9,86 Seminar on geometry in the large (1986): Scientific seminar of the department of mathematical analysis. Joint enlarged meeetings devoted to the 75th birthday of N.V. Efimov (26–30 September 1985). Vestn. Mosk. Univ. Ser. Mat. Mekh., No. 5, 92–98

    Google Scholar 

  • Shiga, K. (1984): Hadamard manifolds. In: Geometry of geodesies and related topics. Proc. Symp., Tokyo 1982, Adv. Stud. Pure Math. 3, 239–281, Zbl.565.53025

    Google Scholar 

  • Shikin, E.V. (1975): Isometric imbeddings in E 3 of noncompact domains of nonpositive curvature. Itogi Nauki Tekh., Ser. Probl. Geom. 7, 249–266 (Russian), Zbl.551.53006

    Google Scholar 

  • Shikin, E.V. (1980): Isometric immersion of two-dimensional manifolds of negative curvature by the method of Darboux. Mat. Zametki 27, 779–794. Engl, transi.: Math. Notes 27, 373–382 (1980), Zbl.438.53004

    MATH  MathSciNet  Google Scholar 

  • Shikin, E.V. (1982): Isometric imbeddings in three-dimensional Euclidean space of two-dimensional manifolds of negative curvature. Mat. Zametki 31, 601–612. Engl, transi.: Math. Notes 31,305–312 (1982), Zbl.491.53002

    MATH  MathSciNet  Google Scholar 

  • Shikin, E.V. (1990): On the parabolicity of immersible and the hyperbolicity of non-immersible two-dimensional manifolds of negative curvature. Vestn. Mosk. Univ., Ser. I Mat. Mekh. No. 5, 42–45 (Russian). Engl, transi.: Mose. Univ. Math. Bull. 45, No. 5, 40–42 (1991), Zbl.718.53004

    MATH  MathSciNet  Google Scholar 

  • Smyth, B. and Xavier, F. (1987): Efimov’s theorem in dimension greater than two. Invent Math. 90, 443–450, Zbl.642.53007

    MATH  MathSciNet  Google Scholar 

  • Sokolov, D.D. (1980): Surfaces in pseudo-Euclidean space. Itogi Nauki Tekh., Ser. Probl. Geom. 11, 177–201. Engl, transi.: J. Sov. Math. 77, 1676–1688 (1981), Zbl.469.53053

    Google Scholar 

  • Spivak, M. (1975): Some left-over problems from classical differential geometry. Proc. Symp. Pure Math. 27, Part 1, 245–252, Zbl.306.53004

    Google Scholar 

  • Steuerwald, R. (1936): Über Enneper’sche Flächen und Bäcklund’sche Transformation. Abh. Bayer. Akad. Wiss. 40, 1–106, Zbl.16,224

    Google Scholar 

  • Ten, L.V. (1980): Rigidity of complete surfaces of negative curvature that coincide with a hyperbolic paraboloid outside a compact domain. Usp. Mat. Nauk 35, No. 6, 175–176. Engl, transi.: Russ. Math. Surv. 35, No. 6,111–112 (1980), Zbl.458.53037

    MATH  MathSciNet  Google Scholar 

  • Tenenblat, K., Terng, C.L. (1980): Bäcklund’s theorem for n-dimensional submanifolds of R 2n-l. Ann. Math., II. Ser. 111, 477–490, Zbl.462.35079

    MATH  MathSciNet  Google Scholar 

  • Terng, C.L. (1980): A higher dimensional generalization of the sine-Gordon equation and its soliton theory. Ann. Math., II. Ser. 111, 491–510, Zbl.447.53001

    MATH  MathSciNet  Google Scholar 

  • Tunitskij, D.V. (1987): On a regular isometric immersion in E 3 of unbounded domains of negative curvature. Mat. Sb., Nov. Ser. 134, 119–134. Engl, transi.: Math. USSR, Sb. 62, No. 1, 121–138 (1989), Zbl.636.53003

    Google Scholar 

  • Vekua, I.N. (1982): Some General Methods of Constructing Different Versions of the Theory of Shells. Nauka, Moscow, Zbl.598.73100. Engl, transi.: Pitman, Boston etc. (1985)

    Google Scholar 

  • Vinberg, E.B., Shvartsman, O.V. (1988): Discrete groups of motions of spaces of constant curvature. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 29, 147–259, Zbl.699.20017. Engl, transi, in: Encycl. Math. Sc. 29, Springer-Verlag, Heidelberg (in preparation)

    Google Scholar 

  • Vinogradskij, A.S. (1970): Boundary properties of surfaces with slowly changing negative curvature. Mat. Sb., Nov. Ser. 82, 285–299. Engl, transi.: Math. USSR, Sb. 11, 257–271 (1970), Zbl.195,230

    Google Scholar 

  • Vorob’eva, L.I. (1976): The impossibility of a C2-isometric immersion in E 3 of a Lobachevskij half-plane. Vestn. Mosk. Univ. Ser. I, No. 5, 42–46. Engl, transi.: Mose. Univ. Math. Bull. 30, No. 5/6, 32–35 (1975), Zbl.327.53036

    MathSciNet  Google Scholar 

  • Wintner, A. (1945): The nonlocal existence problem of ordinary differential equations. Am. J. Math. 67, 277–284

    MATH  MathSciNet  Google Scholar 

  • Wissler, C. (1972): Global Tschebyscheff-Netze auf Riemannschen Mannigfaltigkeiten und Fortsetzung von Flächen konstanter negativer Krümmung. Comment. Math. Helv. 47, 348–372, Zbl.257.53003

    MATH  MathSciNet  Google Scholar 

  • Xavier, F. (1984): Convex hulls of complete minimal surfaces. Math. Ann. 269, 179–182, Zbl.528.53009

    MATH  MathSciNet  Google Scholar 

  • Xavier, F. (1985): A non-immersion theorem for hyperbolic manifolds. Comment. Math. Helv. 60, 280–283, Zbl.566.53046

    MATH  MathSciNet  Google Scholar 

Download references

Authors
  1. E. R. Rozendorn
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. LOMI, Fontanka 27, 191011, St. Petersburg, Russia

    Yu. D. Burago  & V. A. Zalgaller  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Rozendorn, E.R. (1992). Surfaces of Negative Curvature. In: Burago, Y.D., Zalgaller, V.A. (eds) Geometry III. Encyclopaedia of Mathematical Sciences, vol 48. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02751-6_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-02751-6_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08102-6

  • Online ISBN: 978-3-662-02751-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation