The Geometry of Surfaces in Euclidean Spaces

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


The original version of this article was written more than five years ago with S.Z. Shefel’, a profound and original mathematician who died in 1984. Since then the geometry of surfaces has continued to be enriched with ideas and results. This has required changes and additions, but has not influenced the character of the article, the design of which originated with Shefel’. Without knowing to what extent Shefel’ would have approved the changes, I should nevertheless like to dedicate this article to his memory. (Yu.D. Burago)


Gaussian Curvature Spherical Mapping Convex Surface Extrinsic Curvature Isometric Immersion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ahlfors, L., Sario, L. (1960): Riemann Surfaces. Princeton Univ. Press, Princeton, N.J., Zb1.196,338zbMATHGoogle Scholar
  2. Aleksandrov, A.D. (1937–1938): On the theory of mixed volumes of convex bodies. I. Extension of some concepts of the theory of convex bodies. Mat. Sb. 2, 947–972, Zb1.17,426. II. New inequalities between mixed volumes and their applications. ibid. 2, 1205–1238, Zb1.18,276. III. Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies. ibid. 3, 27–46, Zb1.18,424. IV. Mixed discriminants and mixed volumes. ibid. 3, 227–251 (Russian), Zb1.19,328Google Scholar
  3. Aleksandrov, A.D. (1939): Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it. Uch. Zap. Leningr. Univ. Ser. Mat. No. 37, part 6, 3–35 (Russian)Google Scholar
  4. Aleksandrov, A.D. (1942): Smoothness of a convex surface with bounded Gaussian curvature. Dokl. Akad. Nauk SSSR 36, 195–199 (Russian), Zb1.61,376zbMATHGoogle Scholar
  5. Aleksandrov, A.D. (1948): The Intrinsic Geometry of Convex Surfaces. Gostekhizdat, Moscow-Leningrad. German transl.: Die Innere Geometrie der Konvexen Flächen. Akademie-Verlag, Berlin, 1955, Zb1.38,352Google Scholar
  6. Aleksandrov, A.D. (1949): Surfaces representable as a difference of convex functions. Izv. Akad. Nauk KazSSR, Ser. Mat. Mekh. 3, 3–20 (Russian)Google Scholar
  7. Aleksandrov, A.D. (1950a): Surfaces representable as a difference of convex functions. Dokl. Akad. Nauk SSSR 72, 613–616 (Russian), Zb1.39,180zbMATHGoogle Scholar
  8. Aleksandrov, A.D. (1950b): Convex Polyhedra. Gostekhizdat, Moscow-Leningrad. German transl.: Konvexe Polyeder. Akademie-Verlag, Berlin, 1958, Zb1.41,509Google Scholar
  9. Aleksandrov, A.D. (1956–1958): Uniqueness theorems for surfaces in the large. I–IV. Vestn. Leningr. Univ. 1956, 11, No. 19, 5–17; 1957, 12, No. 7, 15–44; 1958, 13, No. 7, 14–26; 1958, 13, No. 13, 27–34. Engl. transl.: Am. Math. Soc. Transl., II. Ser. 21, 341–411, Zb1.101,138 and Zb1.101,139zbMATHGoogle Scholar
  10. Aleksandrov, A.D., Pogorelov, A.V. (1963): Theory of surfaces and partial differential equations. Proc. 4th All-Union Math. Congr. Leningr. 3–16 (Russian), Zb1.196,411zbMATHGoogle Scholar
  11. Aleksandrov, A.D., Zalgaller, V.A. (1962): Two-dimensional manifolds of bounded curvature. Tr. Mat. Inst. Steklova 63,1–262. Engl. transl.: Proc. Steklov Inst. Math. 76,183 pp. (1965), Zb1.122,170zbMATHMathSciNetGoogle Scholar
  12. Allard, W.K. (1972): On the first variation of a varifold. Ann. Math., II. Ser. 95, 417–491, Zb1.251.49028MathSciNetCrossRefGoogle Scholar
  13. Aminov, Yu.A. (1980): On the Grassmann image of a two-dimensional surface in a four-dimensional Euclidean space. Ukr. Geom. Sb. 23, 3–16 (Russian), Zb1.459.53003zbMATHMathSciNetGoogle Scholar
  14. Bakel’man, I.Ya. (1956): Differential geometry of smooth nonregular surfaces. Usp. Mat. Nauk 11, No. 2, 67–124 (Russian), Zb1.70,392zbMATHMathSciNetGoogle Scholar
  15. Bakel’man, I.Ya., Verner, A.L., Kantor, B.E. (1973): Introduction to Differential Geometry “in the Large”. Nauka, Moscow (Russian), Zb1.276.53039Google Scholar
  16. Borisov, A.B., Ogievetskij, V.I. (1974): Theory of dynamical affine and conformal symmetries as the theory of a gravitational field. Teor. Mat. Fiz. 21, 329–342. Engl. transl.: Theor. Math. Phys. 21, 1179–1188 (1974)CrossRefGoogle Scholar
  17. Borisov, Yu.F. (1958–1960): Parallel translation on a smooth surface. I–IV. Vestn. Leningr. Univ. 1958, 13, No. 7, 160–171; 1958,13, No. 19, 45–54; 1959,14, No. 1, 34–50; 1959,14, No. 13, 83–92. Corrections, 1960, No. 19, 127–129 (Russian), Zb1.80,151; Zb1.121,171; Zb1.128,163zbMATHGoogle Scholar
  18. Burago, D.Yu. (1984): Unboundedness in Euclidean space of a horn with a finite positive part of the curvature. Mat. Zametki 36, 229–237. Engl. transl.: Math. Notes 36, 607–612 (1984), Zb1.565.53034MathSciNetGoogle Scholar
  19. Burago, Yu.D. (1960): Realization of a two-dimensional metrized manifold by a surface in E 3. Dokl. Akad. Nauk SSSR 135, 1301–1302. Engl. transl.: Sov. Math. Dokl.1, 1364–1365 (1960), Zb1.100,365MathSciNetGoogle Scholar
  20. Burago, Yu.D. (1968a): Surfaces of bounded extrinsic curvature. Ukr. Geom. Sb. 5–6, 29–43 (Russian), Zb1.195,516MathSciNetGoogle Scholar
  21. Burago, Yu.D. (1968b): Isoperimetric inequalties in the theory of surfaces of bounded extrinsic curvature. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 10. Engl. transl.: Sem. Math., V.A. Steklov Math. Inst., Leningr. 10 (1968), Zb1.198,550Google Scholar
  22. Burago, Yu.D. (1970): Isometric embedding of a manifold of bounded curvature in a Euclidean space. Uch. Zap. LGPI 395, 48–86 (Russian)MathSciNetGoogle Scholar
  23. Burago, Yu.D., Zalgaller, V.A. (1960): Polyhedral embedding of a development Vestn. Leningr. Univ. 15, No. 7, 66–80 (Russian), Zb1.98,354zbMATHMathSciNetGoogle Scholar
  24. Burago, Yu.D., Zalgaller, V.A. (1980): Geometric Inequalities. Nauka, Leningrad, Zb1.436.52009. Engl. transl.: Springer-Verlag, Berlin-New York (1988), Zb1.633.53002zbMATHGoogle Scholar
  25. Busemann, H. (1958): Convex Surfaces. Interscience, New York-London, Zb1.196,551zbMATHGoogle Scholar
  26. Busemann, H., Feller, W. (1935): Krümmungseigenschaften konvexer Flächen. Acta Math. 66, 1–47, Zb1.12,274MathSciNetCrossRefGoogle Scholar
  27. Cecil, T.E., Ryan, P.J. (1985): Tight and Taut Immersions of Manifolds. Pitman, Boston, London, Zb1.596.53002zbMATHGoogle Scholar
  28. Cesari, L. (1956): Surface area. Ann. Math. Stud. 35, Zb1.73,41zbMATHGoogle Scholar
  29. Chen, B. (1973): Geometry of Submanifolds. Marcel Dekker, New York, Zb1.262.53036zbMATHGoogle Scholar
  30. Cheng, S.Y., Yau, S.T. (1976): On the regularity of the solution of the n-dimensional Minkowski problem. Commun. Pure Appl. Math. 29, 495–516, Zb1.363.53030zbMATHMathSciNetCrossRefGoogle Scholar
  31. Cheng, S.Y., Yau, S.T. (1977): On the regularity of the Monge-Ampère equation det(∂2u/∂xi∂xj) = F(x,u). Commun. Pure Appl. Math. 30, 41–68, Zb1.347.35019zbMATHMathSciNetCrossRefGoogle Scholar
  32. Chern, S.S., Lashof, R. (1957): On the total curvature of immersed manifolds. Am. J. Math. 79, 306–313, Zb1.78,139zbMATHMathSciNetCrossRefGoogle Scholar
  33. Cohn-Vossen, S.E. (1927): Zwei Sätze über die Starrheit der Eiflächen. Göttinger Nachrichten, 125–134, Zb1.53,712Google Scholar
  34. Cohn-Vossen, S.E. (1936): Bendability of surfaces in the large. Usp. Mat. Nauk 1, 33–76 (Russian), Zb1.16,225zbMATHGoogle Scholar
  35. Cohn-Vossen, S.E. (1959): Shortest paths and total curvature of a surface. In: Some Questions of Differential Geometry in the Large. Fizmatgiz, Moscow, pp. 174–244 (Russian), Zb1.91,341Google Scholar
  36. Connelly, R. (1978): A counterexample to the rigidity conjecture for polyhedra. Publ. Math. Inst. Hautes Etud. Sci. 47, 333–338, Zb1.375.53034Google Scholar
  37. Diskant, V.I. (1985): Stability in Aleksandrov’s problem for a convex body, one of whose projections is a ball. Ukr. Geom. Sb. 28, 50–62, Zb1.544.52004. Engl. transl.: J. Sov. Math. 48, No. 1, 41–49 (1990)zbMATHGoogle Scholar
  38. Diskant, V.I. (1988): Refinement of analogues of the generalized isoperimetric inequality. Ukr. Geom. Sb. 31, 56–59 (Russian), Zb1.672.53053zbMATHGoogle Scholar
  39. Dubrovin, A.A. (1974): On the regularity of a convex hypersurface in the neighbourhood of a shortest curve. Ukr. Geom. Sb. 15, 42–54 (Russian), Zb1.323.53047zbMATHMathSciNetGoogle Scholar
  40. Efimov, N.V. (1948): Qualitative questions in the theory of deformations of surfaces. Usp. Mat. Nauk 3, No. 2, 47–158. Engl. transl.: Am. Math. Soc. Transl. 6, 274–323, Zb1.30,69zbMATHMathSciNetGoogle Scholar
  41. Efimov, N.V. (1949): Qualitative questions in the theory of deformations of surfaces “in the small”. Tr. Mat. Inst. Steklova 30, 1–128 (Russian), Zb1.41,488MathSciNetGoogle Scholar
  42. Efimov, N.V. (1964): The appearance of singularities on a surface of negative curvature. Mat. Sb., Nov. Ser. 64, 286–320. Engl. transl.: Am. Math. Soc. Transl., II. Ser. 66, 154–190, Zb1.126,374MathSciNetGoogle Scholar
  43. Efimov, N.V. (1966): Surfaces with a slowly changing negative curvature. Usp. Mat. Nauk 21, No. 5, 3–58. Engl. transl.: Russ. Math. Surv. 21, No. 5, 1–55 (1966), Zb1.171,199MathSciNetGoogle Scholar
  44. Efimov, N.V. (1968): Differential criteria for homeomorphy of certain maps with application to the theory of surfaces. Mat. Sb., Nov. Ser. 76, 499–512. Engl. transl.: Math. USSR, Sb. 5, 475–488 (1968), Zb1.164,215MathSciNetGoogle Scholar
  45. Efimov, N.V. (1975): Nonimmersibility of the Lobachevskij half-plane. Vestn. Mosk. Univ., Ser. I 30, 83–86. Engl. transl.: Mosc. Univ.. Math. Bull. 30, 139–142 (1975), Zb1.297.53029zbMATHMathSciNetGoogle Scholar
  46. Federer, H. (1961): Currents and area. Trans. Am. Math. Soc. 98, 204–233, Zb1.187,313zbMATHMathSciNetCrossRefGoogle Scholar
  47. Fenchel, W., Jessen, B. (1938): Mengenfunktionen und konvexe Körper. Danske Vid. Selsk., Mat.Fyss. Medd. 16, No. 3, 1–31, Zb1.18,424Google Scholar
  48. Firey, W. (1968): Christoffel’s problem for general convex bodies. Mathematika 15, 7–21, Zb1.162,543zbMATHMathSciNetCrossRefGoogle Scholar
  49. Fomenko, V.T. (1964): Bendings and unique determination of surfaces of positive curvature with boundary. Mat. Sb., Nov. Ser. 63, 409–425 (Russian) Zb1.163,430MathSciNetGoogle Scholar
  50. Fomenko, V.T. (1965): Bending of surfaces that preserves congruence points. Mat. Sb., Nov. Ser. 66, 127–141 (Russian), Zb1.192,272MathSciNetGoogle Scholar
  51. Gromov, M.L. (1987): Partial Differential Relations. Ergeb. Math. Grenzgeb. (3) 9. Springer-Verlag, Berlin Heidelberg New York, Zb1.651.53001Google Scholar
  52. Gromov, M.L., Rokhlin, V.A. (1970): Embeddings and immersions of Riemannian manifolds. Usp. Mat. Nauk 25, No. 5, 3–62. Engl. transl.: Russ. Math. Surv. 25, No. 5, 1–57 (1970), Zb1.202,210zbMATHGoogle Scholar
  53. Hadamard, J. (1898): Les surfaces à courboures opposées et leurs lignes géodésiques. J. Math. Pures Appl. 4, 27–73Google Scholar
  54. Heinz, E. (1959): Über die Differentialungleichung 0< α ≤ rts 2β < ∞. Math. Z. 72, 107–126, Zb1.98,72zbMATHMathSciNetCrossRefGoogle Scholar
  55. Heinz, E. (1962): On Weyl’s embedding problem. J. Math. Mech. 11, 421–454, Zb1.119,166zbMATHMathSciNetGoogle Scholar
  56. Herglotz, G. (1943): Über die Starrheit der Eiflächen. Abh. Math. Semin. Hansische Univ. 15, 127–129, Zb1.28,94zbMATHMathSciNetCrossRefGoogle Scholar
  57. Hoffman, D., Osserman, R. (1982): The area of the generalized Gaussian image and the stability of minimal surfaces in S n and ℝn n. Math. Ann. 260, 437–452, Zb1.471.53037zbMATHMathSciNetCrossRefGoogle Scholar
  58. Isanov, T.G. (1979a): The extension of infinitesimal bendings of a surface of positive curvature. Sib. Mat. Zh. 20, 1261–1268. Engl. transl.: Sib. Math. J. 20, 894–899 (1979), Zb1.429.53040zbMATHMathSciNetGoogle Scholar
  59. Isanov, T.G. (1979b): The extension of infinitesimal bendings of surfaces of class C m,λ. Sib. Mat. Zh. 20, 1306–1307. Engl. transl.: Sib. Math. J. 20, 929–930 (1979), Zb1.426.53003zbMATHMathSciNetGoogle Scholar
  60. Jacobowitz, H. (1972): Implicit function theorems and isometric embeddings. Ann. Math., II. Ser. 95, 191–225, Zb1.214,129zbMATHMathSciNetCrossRefGoogle Scholar
  61. Kagan, V.F. (1947–1948): Foundations of the Theory of Surfaces. I, II. Gostekhizdat, Moscow-Leningrad (Russian), Zb1.41,487Google Scholar
  62. Kerekjarto, B. (1923): Vorlesungen über Topologie. Berlin, Jbuch 49, 396zbMATHCrossRefGoogle Scholar
  63. Klimentov, S.B. (1982): The structure of the set of solutions of the basic equations of the theory of surfaces. Ukr. Geom. Sb. 25, 69–82 (Russian), Zb1.509.53021zbMATHMathSciNetGoogle Scholar
  64. Klimentov, S.B. (1984): Extension of inflnitesimal high-order bendings of a simply-connected surface of positive curvature. Mat. Zametki 36, 393–403. Engl. transl.: Math. Notes 36, 695–700 (1984), Zb1.581.53002MathSciNetGoogle Scholar
  65. Klimentov, S.B. (1986): On a way of constructing solutions of boundary-value problems of the theory of bendings of surfaces of positive curvature. Ukr. Geom. Sb. 29, 56–82 (Russian), Zb1.615.35014zbMATHGoogle Scholar
  66. Klimentov, S.B. (1987): On the extension of infinitesimal higher-order bendings of a simply-connected surface of positive curvature under boundary conditions. Ukr. Geom. Sb. 30, 41–49 (Russian), Zb1.631.53049zbMATHGoogle Scholar
  67. Klotz-Milnor, T. (1972): Efimov’s theorem about complete immersed surfaces of negative curvature. Adv. Math. 8, 474–542, Zb1.236.53055zbMATHCrossRefGoogle Scholar
  68. Kozlov, S.E. (1989): Estimates of the area of spherical images of two-dimensional surfaces in Riemannian manifolds. Mat. Zametki 46, No. 3, 120–122, Zb1.687.53018zbMATHMathSciNetGoogle Scholar
  69. Kreinovich, V.Ya. (1986): Query No. 369. Notices Amer. Math. Soc. 33, 945Google Scholar
  70. Kuiper, N. (1955): On C 1-isometric imbedding. Nederl. Akad. Wet. Proc., Ser. A 58 (= Indagationes Math. 17), 545–556, 683–689, Zb1.67,396zbMATHMathSciNetGoogle Scholar
  71. Kuiper, N. (1970): Minimal total absolute curvature for immersions. Invent. Math. 10, 209–238, Zb1.195,511MathSciNetCrossRefGoogle Scholar
  72. Levi, E. (1908): Sur l’application des équations integrales au problème de Riemann. Nachr. König. Gesell. Wiss. Göttingen Mat. 249–252Google Scholar
  73. Lewy, H. (1935–1937): A priori limitations for solutions of Monge-Ampère equations. I, II. Trans. Am. Math. Soc. 37, 417–434, Zb1.11,350; 41, 365–374, Zb1.17,211MathSciNetGoogle Scholar
  74. Lewy, H. (1936): On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull. Am. Math. Soc. 42, 689–692, Zb1.15,159MathSciNetCrossRefGoogle Scholar
  75. Lewy, H. (1938): On the existence of a closed convex surface realising a given Riemannian metric. Proc. Natl. Acad. Sci. USA 24, 104–106, Zb1.18,88CrossRefGoogle Scholar
  76. Liberman, I.M. (1941): Geodesic curves on convex surfaces. Dokl. Akad. Nauk SSSR 32, 310–313 (Russian), Zb1.61,376zbMATHMathSciNetGoogle Scholar
  77. Lin, C.S. (1985): The local isometric embedding in 3 of two-dimensional Riemannian manifolds with nonnegative curvature. J. Differ. Geom. 21, 213–230, Zb1.584.53002zbMATHGoogle Scholar
  78. Michael, J.H., Simon, L.M. (1973): Sobolev and mean-value inequalities on generalized submanifolds in ℝn. Commun. Pure Appl. Math. 26, 361–379, Zb1.252.53006zbMATHMathSciNetCrossRefGoogle Scholar
  79. Milka, A.D. (1974a): A theorem on a smooth point of a shortest curve. Ukr. Geom. Sb. 15, 62–70 (Russian), Zb1.321.53045zbMATHMathSciNetGoogle Scholar
  80. Milka, A.D. (1974b): An estimate of the curvature of a set adjacent to a shortest curve. Ukr. Geom. Sb. 15, 70–80 (Russian), Zb1.343.53037zbMATHMathSciNetGoogle Scholar
  81. Milka, A.D. (1975): A shortest curve with a nonrectifiable spherical image. Ukr. Geom. Sb. 16, 35–52 (Russian), Zb1.321.53046Google Scholar
  82. Milka, A.D. (1977): A shortest curve, all of whose points are singular. Ukr. Geom. Sb. 20, 95–98 (Russian), Zb1.429.53041zbMATHMathSciNetGoogle Scholar
  83. Milka, A.D. (1980): The unique determination of general closed convex surfaces in Lobachevskij space. Ukr. Geom. Sb. 23, 99–107 (Russian), Zb1.459.51012zbMATHMathSciNetGoogle Scholar
  84. Minkowski, H. (1903): Volumen und Oberfläche. Math. Ann. 57, Jbuch 34, 649MathSciNetCrossRefGoogle Scholar
  85. Nash, J. (1954): C1-isometric imbeddings. Ann. Math., II. Ser. 60, 383–396, Zb1.58,377zbMATHMathSciNetCrossRefGoogle Scholar
  86. Nash, J. (1956): The imbedding problem for Riemannian manifolds. Ann. Math., II. Ser. 63, 20–63, Zb1.70,386zbMATHMathSciNetCrossRefGoogle Scholar
  87. Nikolaev, I.G., Shefel’, S.Z. (1982): Smoothness of convex surfaces on the basis of differential properties of quasiconformal maps. Dokl. Akad. Nauk SSSR 267, 296–300. Engl. transl.: Sov. Math. Dokl. 26, 599–602 (1982), Zb1.527.53036MathSciNetGoogle Scholar
  88. Nikolaev, I.G., Shefel’, S.Z. (1985): Smoothness of convex surfaces and generalized solutions of the Monge-Ampère equation on the basis of differential properties of quasiconformal maps. Sib. Mat. Zh. 26, 77–89. Engl. transl.: Sib. Math. J. 26, 841–851 (1985), Zb1.585.53053zbMATHMathSciNetGoogle Scholar
  89. Nirenberg, L. (1953): The Weyl and Minkowski problems in differential geometry in the large. Commun. Pure Appl. Math. 6, 337–394, Zb1.51,124zbMATHMathSciNetCrossRefGoogle Scholar
  90. Ogievetskij, V.I. (1973): Infinite-dimensional algebra of general covariance group as the closure of finite-dimensional algebras of conformal and linear groups. Lett. Nuovo Cimento (2) 8, 988–990MathSciNetCrossRefGoogle Scholar
  91. Olovyanishnikov, S.P. (1946): On the bending of infinite convex surfaces. Mat. Sb., Nov. Ser. 18, 429–440 (Russian), Zb1.61,376Google Scholar
  92. Perel’man, G.Ya. (1988a): Polyhedral saddle surfaces. Ukr. Geom. Sb. 31, 100–108 (Russian), Zb1.681.53032zbMATHGoogle Scholar
  93. Perel’man, G.Ya. (1988b): Metric obstructions to the existence of certain saddle surfaces. Preprint P-1–88, LOMI, Leningrad (Russian)Google Scholar
  94. Perel’man, G.Ya. (1989): An example of a complete saddle surface in ℝ4 with Gaussian curvature different from zero. Ukr. Geom. Sb. 32, 99–102 (Russian), Zb1.714.53035zbMATHGoogle Scholar
  95. Perel’man, G.Ya. (1990a): A new statement of a theorem of N. V. Efimov. Lectures at the All-Union conference of young scientists on differential geometry, dedicated to the 80th anniversary of the birth of N.V. Efimov, Rostov-on-Don, 1990, p. 89. (Russian)Google Scholar
  96. Perel’man, G.Ya. (1990b): Saddle surfaces in Euclidean spaces. Dissertation, Leningrad State Univ., Leningrad, 1990 (Russian)Google Scholar
  97. Pogorelov, A.V. (1949a): Unique determination of convex surfaces. Tr. Mat. Inst. Steklova 29, 3–99 (Russian), Zb1.41,508zbMATHMathSciNetGoogle Scholar
  98. Pogorelov, A.V. (1949b): Regularity of convex surfaces with a regular metric. Dokl. Akad. Nauk SSSR 66, 1051–1053 (Russian), Zb1.33,214Google Scholar
  99. Pogorelov, A.V. (1949c): Convex surfaces with a regular metric. Dokl. Akad. Nauk SSSR 67, 791–794. Engl. transl.: Am. Math. Soc. Transl., I. Ser. 6, 424–429, Zb1.33,214zbMATHMathSciNetGoogle Scholar
  100. Pogorelov, A.V. (1951): Bending of Convex Surfaces. Nauka, Moscow-Leningrad. German transl.: Die Verbiegung Konvexer Flächen. Akademie-Verlag, Berlin, 1957, Zb1.45,425Google Scholar
  101. Pogorelov, A.V. (1952a): Unique Determination of General Convex Surfaces. Izdat. Akad. Nauk SSSR, Kiev. German transl.: Die Eindeutige Bestimmung Allgemeiner Konvexer Flächen, Akademie-Verlag, Berlin, 1956, Zb1.72,175Google Scholar
  102. Pogorelov, A.V. (1952b): Regularity of a convex surface with given Gaussian curvature. Mat. Sb. Nov. Ser. 31, 88–103, Zb1.48,405MathSciNetGoogle Scholar
  103. Pogorelov, A.V. (1953): Stability of isolated ridge points on a convex surface under bending. Usp. Mat. Nauk 8, No. 3, 131–134 (Russian), Zb1.51,384zbMATHMathSciNetGoogle Scholar
  104. Pogorelov, A.V. (1954): Unique determination of infinite convex surfaces. Dokl. Akad. Nauk 94, 21–23 (Russian), Zb1.55,154MathSciNetGoogle Scholar
  105. Pogorelov, A.V. (1956a): Nonbendability of general infinite convex surfaces with total curvature 2π. Dokl. Akad. Nauk SSSR 106, 19–20 (Russian), Zb1.70,168zbMATHMathSciNetGoogle Scholar
  106. Pogorelov, A.V. (1956b): Surfaces of Bounded Extrinsic Curvature. Izdat. Gosud. Univ., Kharkov (Russian), Zb1.74,176Google Scholar
  107. Pogorelov, A.V. (1969): Extrinsic Geometry of Convex Surfaces. Nauka, Moscow. Engl. transl.: Am. Math. Soc., Providence, RI, 1973, Zb1.311.53067Google Scholar
  108. Pogorelov, A.V. (1971): An example of a two-dimensional Riemannian metric that does not locally admit a realization in E 3. Dokl. Akad. Nauk SSSR 198, 42–43. Engl. transl.: Sov. Math. Dokl. 12, 729–730 (1971), Zb1.232.53013MathSciNetGoogle Scholar
  109. Pogorelov, A.V. (1975): The Multidimensional Minkowski Problem. Nauka, Moscow. Engl. transl.: J. Wiley & Sons, New York etc., 1978, Zb1.387.53023Google Scholar
  110. Poznyak, E.G. (1966): Regular realization in the large of two-dimensional metrics of negative curvature. Ukr. Geom. Sb. 3, 78–92 (Russian), Zb1.205,514zbMATHGoogle Scholar
  111. Poznyak, E.G. (1973): Isometric immersions of two-dimensional Riemannian metrics in Euclidean space. Usp. Mat. Nauk 28, No. 4, 47–76. Engl. transl.: Russ. Math. Surv. 28, No. 4, 47–77 (1973), Zb1.283.53001MathSciNetGoogle Scholar
  112. Poznyak, E.G., Shikin, E.V. (1974): Surfaces of negative curvature. Itogi Nauki Tekh., Ser. Algebra, Topologiya, Geometriya 12,171–208. Engl. transl.: J. Sov. Math. 5, 865–887 (1976), Zb1.318.53050Google Scholar
  113. Poznyak, E.G., Shikin, E.V. (1980): Isometric immersions of domains of the Lobachevskij plane in Euclidean spaces. Tr. Tbilis. Mat. Inst. Razmadze 64, 82–93 (Russian), Zb1.497.53006zbMATHMathSciNetGoogle Scholar
  114. Reshetnyak, Yu.G. (1956): A generalization of convex surfaces. Mat. Sb., Nov. Ser. 40, 381–398 (Russian), Zb1.72,176Google Scholar
  115. Reshetnyak, Yu.G. (1959): Investigation of manifolds of bounded curvature by means of isothermal coordinates. Izv. Sib. Otd. Akad. Nauk SSSR 10, 15–28 (Russian), Zb1.115,164Google Scholar
  116. Reshetnyak, Yu.G. (1960a): On the theory of spaces of curvature not greater than K. Mat. Sb., Nov. Ser. 52, 789–798 (Russian), Zb1.101,402Google Scholar
  117. Reshetnyak, Yu.G. (1960b): Isothermal coordinates in a manifold of bounded curvature. I, II. Sib. Mat. Zh. 1, 88–116, 248–276 (Russian), Zb1.108,338Google Scholar
  118. Reshetnyak, Yu.G. (1962): A special map of a cone into a manifold of bounded curvature. Sib. Mat. Zh. 3, 256–272 (Russian), Zb1.124,153zbMATHGoogle Scholar
  119. Reshetnyak, Yu.G. (1967): Isothermal coordinates on surfaces of bounded integral mean curvature. Dokl. Akad. Nauk SSSR 174, 1024–1025. Engl. transl.: Sov. Math. Dokl. 8, 715–717 (1967), Zb1.155,303MathSciNetGoogle Scholar
  120. Rozendorn, E.R. (1961): Construction of a bounded complete surface of nonpositive curvature. Usp. Mat. Nauk 16, No. 2, 149–156 (Russian), Zb1.103,154zbMATHMathSciNetGoogle Scholar
  121. Rozendorn, E.R. (1966): Weakly irregular surfaces of negative curvature. Usp. Mat. Nauk 21, No. 5, 59–116. Engl. transl.: Russ. Math. Surv. 21, No. 5, 57–112 (1966), Zb1.173,232zbMATHMathSciNetGoogle Scholar
  122. Rozendorn, E.R. (1967): The influence of the intrinsic metric on the regularity of a surface of negative curvature. Mat. Sb., Nov. Ser. 73, 236–254. Engl. transl.: Math. USSR, Sb. 2, 207–223 (1967), Zb1.154,212MathSciNetGoogle Scholar
  123. Rozendorn, E.R. (1981): Bounded complete weakly nonregular surfaces with negative curvature bounded away from zero. Mat. Sb., Nov. Ser. 116, 558–567. Engl. transl.: Math. USSR, Sb. 44, 501–509 (1983), Zb1.477.53003MathSciNetGoogle Scholar
  124. Rozendorn, E.R. (1989): Surfaces of negative curvature. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 48, 98–195. Engl. transl. in: Encycl. Math. Sc. 48, Springer-Verlag, Heidelberg, 87–178, 1992 (Part II of this volume)zbMATHMathSciNetGoogle Scholar
  125. Rozenson, N.A. (1940–1943): Riemannian spaces of class I. Izv. Akad. Nauk SSSR Ser. Mat. 4, 181–192; 5, 325–352; 7, 253–284 (Russian), Zb1.24,282; Zb1.60,383zbMATHGoogle Scholar
  126. Sabitov, I.Kh. (1976): Regularity of convex surfaces with a metric that is regular in Hölder classes. Sib. Mat. Zh. 17, 907–915. Engl. transl.: Sib. Math. J. 17, 681–687 (1977), Zb1.356.53017zbMATHMathSciNetGoogle Scholar
  127. Sabitov, I.Kh. (1989): Local theory of bendings of surfaces. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 48, 196–270. Engl. transl. in: Encycl. Math. Sc. 48, Springer-Verlag. Heidelberg, 179–250, 1992 (Part III of this volume)zbMATHMathSciNetGoogle Scholar
  128. Sabitov, I.Kh., Shefel’, S.Z. (1976): Connections between the orders of smoothness of a surface and its metric. Sib. Mat. Zh. 17, 916–925. Engl. transl.: Sib. Math. J. 17, 687–694 (1977), Zb1.358.53015zbMATHGoogle Scholar
  129. Shefel’, G.S. (1984): Geometric properties of transformation groups of Euclidean space. Dokl. Akad. Nauk SSSR 277, 803–806. Engl. transl.: Sov. Math. Dokl. 30, 178–181 (1984), Zb1.597.20033MathSciNetGoogle Scholar
  130. Shefel’, G.S. (1985): Transformation groups of Euclidean space. Sib. Mat. Zh. 26, No. 3, 197–215. Engl. transl.: Sib. Math. J. 26, 464–478 (1985), Zb1.574.53006MathSciNetGoogle Scholar
  131. Shefel’, S.Z. (1963): Research into the geometry of saddle surfaces. Preprint, Inst. Mat. Sib. Otd. Akad. Nauk SSSR (Russian)Google Scholar
  132. Shefel’, S.Z. (1964): The intrinsic geometry of saddle surfaces. Sib. Mat. Zh. 5, 1382–1396 (Russian), Zb1.142,189Google Scholar
  133. Shefel’, S.Z. (1967): Compactness conditions for a family of saddle surfaces. Sib. Mat. Zh. 8, 705–714. Engl. transl.: Sib. Math. J. 8, 528–535 (1967), Zb1.161,418CrossRefGoogle Scholar
  134. Shefel’, S.Z. (1969): Two classes of k-dimensional surfaces in n-dimensional space. Sib. Mat. Zh. 10, 459–466. Engl. transl.: Sib. Math. J. 10, 328–333 (1969), Zb1.174,531Google Scholar
  135. Shefel’, S.Z. (1970): Completely regular isometric immersions in Euclidean space. Sib. Mat. Zh. 11, 442–460. Engl. transl.: Sib. Math. J. 11, 337–350 (1970), Zb1.201,242Google Scholar
  136. Shefel’, S.Z. (1974): C1-smooth isometric immersions. Sib. Mat. Zh. 15, 1372–1393. Engl. transl.: Sib. Math. J. 15, 972–987 (1974), Zb1.301.53033Google Scholar
  137. Shefel’, S.Z. (1975): C 1-smooth surfaces of bounded positive extrinsic curvature. Sib. Mat. Zh. 16, 1122–1123. Engl. transl. Sib. Math. J. 16, 863–864 (1976), Zb1.323.53043Google Scholar
  138. Shefel’, S.Z. (1977): Smoothness of the solution of the Minkowski problem. Sib. Mat. Zh. 18, 472–475. Engl. transl. Sib. Math. J. 18, 338–340 (1977), Zb1.357.52004Google Scholar
  139. Shefel’, S.Z. (1978): Surfaces in Euclidean space. In: Mathematical Analysis and Mixed Questions of Mathematics. Nauka, Novosibirsk, 297–318 (Russian)Google Scholar
  140. Shefel’, S.Z. (1979): Conformal correspondence of metrics and smoothness of isometric immersions. Sib. Mat. Zh. 20, 397–401. Engl. transl.: Sib. Math. J. 20, 284–287 (1979), Zb1.414.53004MathSciNetGoogle Scholar
  141. Shefel’, S.Z. (1982): Smoothness of a conformal map of Riemannian spaces. Sib. Mat. Zh. 23,153–159. Engl. transl.: Sib. Math. J. 23, 119–124 (1982), Zb1.494.53020MathSciNetGoogle Scholar
  142. Shefel’, S.Z. (1985): Geometric properties of immersed manifolds. Sib. Mat. Zh. 26, No. 1, 170–188. Engl. transl.: Sib. Math. J. 26, 133–147 (1985), Zb1.567.53005MathSciNetGoogle Scholar
  143. Shor, L.A. (1967): Isometric nondeformable convex surfaces. Mat. Zametki 1, 209–216. Engl. transl.: Math. Notes 1, 140–144 (1968), Zb1.163,441zbMATHMathSciNetGoogle Scholar
  144. Usov, V.V. (1976a): The length of the spherical image of a geodesic on a convex surface. Sib. Mat. Zh. 17, 233–236. Engl. transl.: Sib. Math. J. 17, 185–188 (1976), Zb1.332.53036MathSciNetCrossRefGoogle Scholar
  145. Usov, V.V. (1976b): Spatial rotation of curves on convex surfaces. Sib. Mat. Zh. 17, 1427–1430. Engl. transl.: Sib. Math. J. 17, 1043–1045 (1976), Zb1.404.53050zbMATHMathSciNetCrossRefGoogle Scholar
  146. Usov, V.V. (1977): The indicator of a shortest curve on a convex surface. Sib. Mat. Zh. 18, 899–907. Engl. transl.: Sib. Math. J. 18, 637–644 (1977), Zb1.378.53034zbMATHMathSciNetGoogle Scholar
  147. Vekua, I.N. (1959): Generalized Analytic Functions. Fizmatgiz, Moscow. Engl. transl.: Pergamon Press, Oxford etc. (1962), Zb1.92,297zbMATHGoogle Scholar
  148. Vekua, I.N. (1982): Some General Methods of Constructing Various Versions of Shell Theory. Nauka, Moscow, Zb1.598.73100. Engl. transl.: Pitman, Boston etc. (1985)Google Scholar
  149. Verner, A.L. (1968): Cohn-Vossen’s theorem on the integral curvature of complete surfaces. Sib. Mat. Zh. 9, 199–203. Engl. transl.: Sib. Math. J. 9, 150–153 (1968), Zb1.159,230zbMATHMathSciNetCrossRefGoogle Scholar
  150. Verner, A.L. (1967–1968): The extrinsic geometry of the simplest complete surfaces of nonpositive curvature. I, II. Mat. Sb., Nov. Ser 74, 218–240; 75, 112–139. Corrections: 77, 136. Engl. transl. Math. USSR, Sb. 3, 205–224; 4, 99–123, Zb1.164,216; Zb1.167,194MathSciNetGoogle Scholar
  151. Verner, A’.L. (1970a): Unboundedness of a hyperbolic horn in Euclidean space. Sib. Mat. Zh. 11, 20–29. Engl. transl.: Sib. Math. J. 11, 15–21 (1970), Zb1.212,263MathSciNetCrossRefGoogle Scholar
  152. Verner, A.L. (1970b): Constricting saddle surfaces. Sib. Mat. Zh. 11, 750–769. Engl. transl.: Sib. Math. J. 11, 567–581 (1970), Zb1.219.53051zbMATHMathSciNetCrossRefGoogle Scholar
  153. Volkov, Yu.A. (1963): Stability of the solution of the Minkowski problem. Vestn. Leningr. Univ., Ser. Mat. Mekh. Astronom. 18, No. 1, 33–43 (Russian), Zb1.158,197zbMATHGoogle Scholar
  154. Volkov, Yu.A. (1968): Estimate of the deformation of a convex surface depending on a change of its intrinsic metric. Ukr. Geom. Sb. 5–6, 44–69 (Russian), Zb1.207,208Google Scholar
  155. Weyl, H. (1916): Über die Bestimmung einer geschlossen konvexen Fläche durch ihr Linielement. Zürich Naturf. Ges. 61, 40–72, Jbuch 46, 1115Google Scholar
  156. Willmore, T.J. (1982): Total Curvature in Riemannian Geometry. Ellis Horwood, Chichester; Halstead Press, New York, Zb1.501.53038zbMATHGoogle Scholar
  157. Zalgaller, V.A. (1950): Curves with bounded variation of rotation on convex surfaces. Mat. Sb., Nov. Ser. 26, 205–214 (Russian), Zb1.39,181MathSciNetGoogle Scholar
  158. Zalgaller, V.A. (1958): Isometric embedding of polyhedra. Dokl. Akad. Nauk SSSR 123, 599–601 (Russian), Zb1.94,360zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

There are no affiliations available

Personalised recommendations