×

The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research. (English. Russian original) Zbl 1360.01027

Trans. Mosc. Math. Soc. 2016, 149-175 (2016); translation from Tr. Mosk. Mat. O.-va 77, No. 2, 184-218 (2016).
The paper is devoted to the presentation of researches on geometry in the Moscow Mathematical Society. Geometry is meant here in a broader sense. The material is organized into the following subjects: manifolds with a metric, in particular manifolds with a Riemannian metric or Riemannian spaces, isometric realizations of abstract metrics as metrics of surfaces in spaces, bendings and isometric transformations of a surface, various properties of surfaces. An extensive bibliography (182 items!) is provided.

MSC:

01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
01A61 History of mathematics in the 21st century
01A74 History of mathematics at institutions and academies (non-university)
53-03 History of differential geometry
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Per A. N. Kolmogorov (ed.) and S. P. Novikov (ed.), Investigations in the metric theory of surfaces. Collection of articles, Mathematics. New in Foreign Science, vol. 18, Transl. from the English and from the French by I. Kh. Sabitov, Mir, Moscow (1980). (Russian)
[2] VER A. M. Vasil’ev, N. V. Efimov, and P. K. Rashevski\i , Research into differential geometry at the Moscow University in the Soviet period (for the 50th anniversary of Soviet Power), Vestnik Moskov. Univ. Ser. Mat. Mekh. 1967 (1967), no. 5, 12-23. (Russian)
[3] M0 Mathematics in the USSR over 40 years (1917-1957), vol. 1: Survey talks, GIFML, Moscow, 1959. (Russian)
[4] R0 P. K. Rashevski\i , Tensor differential geometry, Mathematics in the USSR over 30 years, GITTL, Moscow-Leningrad, 1948, 283-918.
[5] ES L. E. Evtushik and I. Kh. Sabitov, Geometry in the department of mathematical analysis, Sovrem. Probl. Mat. Mekh. 3 (2009), no. 1, 42-45. (Russian)
[6] 4 Geometry and topology, Sovrem. Probl. Mat. Mekh. 3 (2009), no. 2. (Russian)
[7] 5 Soviet mathematics over 20 years, Uspekhi Mat. Nauk 1938 (1938), no. 4, 3-13. (Russian)
[8] 6 On the scientific work of some departments of the Institute of Mathematics of Moscow University in 1936. Department of probability theory and mathematical statistics. Department of tensor differential geometry, Uspekhi Mat. Nauk 1938 (1938), no. 4, 289-294. (Russian)
[9] 7 Mathematics. Science in the USSR over fifteen years (1917-1932), GTTI, Moscow, 1932. (Russian)
[10] ED D. F. Egorov, Successes in mathematics in the USSR, Science and technology in the USSR (1917-1927), Rabotnik Prosvesch. 1 (1928), 223-234.
[11] L0 L. A. Lyusternik, “Matematicheski\uSbornik”, Uspekhi Mat. Nauk 1 (1946), no. 1, 242-247. (Russian)
[12] LaL A. F. Lapko and L. A. Lyusternik, From the history of Soviet mathematics, Uspekhi Mat. Nauk 22 (1967), no. 6, 13-140. (Russian) · Zbl 0183.00503
[13] AD A. D. Aleksandrov, Geometry and topology in the Soviet Union, Uspekhi Mat. Nauk 2 (1947), no. 5, 9-92. (Russian)
[14] E L. Euler, Opera postuma, Mathematica et Physica, vol. I, Academiae Scientiarum Petropolitanae, St. Petersburg, 1862, 494-495.
[15] L A.-M. Legendre, El\'ements de g\'eom\'etrie, Paris, 1794.
[16] C A. Cauchy, Sur les polygones et les poly\`“edres: second m\'”emoire, J. d’Ecole Polytechnique, IX (1813), cahier XVI, 87-98.
[17] M1 A. L-v, On N. I. Lobachevski\u’s theory of parallel lines, Mat. Sb. 3 (1868), no. 2, 78-120. (Russian)
[18] M2 J. Bertrand, Sur la somme des angles du triangle, Mat. Sb. 4 (1870), no. 4, 198-207. (Russian)
[19] M3 V. Ya. Bunyakovski\i , A note concerning the question on parallel lines, Mat. Sb. 6 (1872), no. 1, 77-82. (Russian)
[20] M4 L. K. Lakhtin, On life and scientific works of Nikola\uIvanovich Lobachevski\i (on the occasion of the centenary of his birth), Mat. Sb. 17 (1894), no. 3, 474-493. (Russian)
[21] M5 L. K. Lakhtin, On one concrete interpretation of the Lobachevski\uplanimetry, Mat. Sb. 17 (1895), no. 4, 767-790. (Russian)
[22] KlF F. Klein, \"Uber die sogenannte Nicht-Euklidische Geometrie, Math. Ann. 4 (1871), 573-625. · JFM 03.0231.02
[23] Ri B. Riemann, \`“Uber die Hypothesen, welche der Geometrie zu Grunde liegen, G\'”ott. Abd. 13 (1868). · JFM 01.0022.02
[24] O A. P. Norden (ed.), On foundations of geometry. A collection of classical papers on Lobachevski\ugeometry and the development of its ideas, GITTL, Moscow, 1956. (Russian)
[25] Bog S. A. Bogomolov, Introduction to Riemann’s non-Euclidean geometry, Leningrad-Moscow: ONTI GTTI, 1934. (Russian)
[26] M6 K. M. Peterson, On relations and affinities between curved surfaces, Mat. Sb. 1 (1866), no. 1, 391-438. (Russian)
[27] B O. Bonnet, M\'emoire sur la th\'eorie des surfaces applicables sur une surface donn\'ee, J. d’Ecole Polytechnique. Premi\`ere Partie XXIV (1865), no. 41, 209-230; Deuxi\`eme Partie XXV (1867), no. 42, 1-151
[28] Bo A. I. Bobenko, Exploring Surfaces through Methods from the Theory of Integrable Systems: The Bonnet Problem, Surveys on Geometry and Integrable Systems, Adv. Stud. Pure Math. 51 (2008) 1-51. · Zbl 1165.53006
[29] S1 I. Kh. Sabitov, Isometric surfaces with common mean curvature and the problem of Bonnet pairs, Mat. Sb. 203 (2012), no. 1, 115-158; English transl., Sb. Math. 203 (2012), no. 1, 111-152. · Zbl 1266.53006
[30] M7 K. M. Peterson, On the bending of surfaces of the 2nd order, Mat. Sb. 10 (1883), no. 4, 476-523. (Russian)
[31] M8 B. K. Mlodzeevski\i , Karl Mikha\ulovich Peterson and his geometric works, Mat. Sb. 24 (1903), no. 1, 1-21. (Russian)
[32] Ca E. Cartan, Sur les couples de surfaces applicables avec conservation des courbures principales, Bull. des Sciences Mathem. 66 (1942), 55-85; \OE uvres Compl\`etes. Partie III, vol. 2, 1591-1620. · Zbl 0027.08903
[33] P K. Peterson, \`“Uber Kurven und Fl\'”achen, Moskau und Leipzig, 1868.
[34] R S. D. Rossinski\i , Karl Mikha\ulovich Peterson (1828-1881), Uspekhi Mat. Nauk 4 (1949), no. 5, 3-13. (Russian)
[35] M10a D. F. Egorov, Towards a general theory of the correspondence of surfaces, Mat. Sb. 18 (1896), no. 1, 86-107. (Russian)
[36] M8a B. K. Mlodzeevski\i , On surfaces related to Peterson surfaces, Mat. Sb. 21 (1900), no. 3, 450-460. (Russian)
[37] M8b B. K. Mlodzeevski\i , On bending of Peterson surfaces, Mat. Sb. 24 (1900), no. 3, 417-473. (Russian)
[38] M9 B. K. Mlodzeevski\i , On one transformation of infinitesimal bendings, Mat. Sb. 28 (1911), no. 1, 205-214. (Russian)
[39] M10 D. F. Egorov, On bending over a principal base for one family of planar or conical lines, Mat. Sb. 28 (1911), no. 1, 167-187. (Russian)
[40] M11 S. S. Byushgens, On bending of surfaces over a principal base, Mat. Sb. 28 (1912), no. 4, 507-528. (Russian)
[41] M12 S. P. Finikov, On bending of surfaces of the 2nd order over a principal base, Mat. Sb. 28 (1912), no. 4, 529-543. (Russian)
[42] M13 S. S. Byushgens, On cyclic congruences and Bianchi surfaces, Mat. Sb. 30 (1916), no. 2, 296-313. (Russian)
[43] F2a S. P. Finikov, General problem of bending over a principal base, Moscow, 1917. (Russian)
[44] B1 S. S. Byushgens, Bending over a principal base, Moscow, 1918. (Russian)
[45] M14  D. Th. Egoroff, Sur les surfaces, engendr\'ees par la distrubution des lignes d’une famille donn\'ee, Mat. Sb. 31 (1923), no. 3, 153-184. · JFM 49.0519.02
[46] M15 S. Finikoff, Sur les surfaces de M. Bianchi, Mat. Sb. 32 (1924), no. 1, 249-254.
[47] M16 S. P. Finikov, On one case of special bending of a congruence, Mat. Sb. 32 (1924), no. 1, 241-248. (Russian)
[48] M17 S. Bucheguennce, Sur certaines familles invariables de courbes, Mat. Sb. 32 (1925), no. 2, 348-352. · JFM 51.0552.01
[49] M18 A. F. Maslov, On Moutard’s transformation and quadratic solutions of an equation with equal invariants, Mat. Sb. 32 (1925), no. 3, 569-598. (Russian)
[50] M19 S. Bucheguennce, Sur une class des hypersurfaces, Mat. Sb. 32 (1925), no. 4, 625-631.
[51] M20 S. Bucheguennce, Sur les surfaces ayant une famille des parall\`“eles planes ou sph\'”eriques, Mat. Sb. 32 (1925), no. 4, 632-645.
[52] M21 S. Finikoff, Sur la d\'eformation des surfaces \`“a r\'”eseaux cin\'ematiquemant conjug\'es persistant, Mat. Sb. 33 (1926), no. 3, 129-160.
[53] M22 A. Th. Masloff, Sur la d\'eformation des surfaces avec conservation d’un syst\`“eme conjugu\'”e, Mat. Sb. 33 (1926), no. 1, 43-48.
[54] M23 A. Th. Masloff, Sur la d\'eformation continue d’une classe des surfaces, Mat. Sb. 33 (1926), no. 4, 367-370. · JFM 52.0708.02
[55] M24 S. Finikoff, Sur la congruence rectiligne de roulement d’une infinit\'e de mani\`eres, Mat. Sb. 34 (1927), no. 1, 49-54. · JFM 53.0639.06
[56] M25 L. N. Sretenski\i , On the bending of surfaces, Mat. Sb. 36 (1929), no. 2, 19-111. (Russian)
[57] M26 S. Finikoff, Sur les quadriques de Lie et les congruences de M. Demoulin, Mat. Sb. 38 (1931), nos. 1-2, 48-97.
[58] F1 S. P. Finikov, Theory of congruences, Gostekhizdat, Moscow, 1950. (Russian)
[59] F2 S. P. Finikov, Theory of pairs of congruences, Gostekhizdat, Moscow, 1956. (Russian)
[60] F3 S. P. Finikov, Bending over a principal base and related geometric problems, ONTI NKTP SSSR, Moscow-Leningrad, 1937. (Russian)
[61] De N. Delaunay, Sur les surfaces n’ayant qu’un cot\'e et sur les points singuliers des courbes planes, Bull. Soc. Math. France 26 (1898), 43-52. · JFM 29.0415.12
[62] LL L. K. Lakhtin, A note on one-sided surfaces, Mat. Sb. 24 (1904), no. 2, 178-193. (Russian)
[63] M27 B. K. Mlodzeevski\i , Investigation into the bending of surfaces, Uchen. Zapiski Moskov. Univ. Otdel Fiz.-Mat. Nauk 1886 (1886), no. 7, 1-141. (Russian)
[64] Ra P. K. Rashevski\i , Course of differential geometry, GITTL, Moscow, 1956. (Russian)
[65] M28 D. F. Egorov, Boleslav Kornelievich Mlodzeevski\u(Obituary), Mat. Sb. 32 (1925), no. 3, 449-452. (Russian)
[66] R1 S. D. Rossinski\i , Boleslav Kornelievich Mlodzeevski\u(1858-1923). Biographical outline, Moscow Univ., Moscow, 1950. (Russian)
[67] VB V. I. Bogachev, The history of the discovery of the theorems of Egorov and Luzin, Historico-mathematical studies. Second series, 2009 (2009), no. 13 (48), 54-67. (Russian) · Zbl 1198.01018
[68] Eg D. F. Egorov, On one class of orthogonal systems, Uchen. Zapiski Moskov. Univ. 1901 (1901), no. 18, 1-239. (Russian)
[69] Eg1 D. F. Egorov, Works on differential geometry, Nauka, Moscow, 1970. (Russian)
[70] D G. Darboux, Le\ccons sur les syst\`“emes orthogonaux et les coordonn\'”ees curvilignes, Paris, 1910.
[71] PK P. I. Kuznetsov, Dmitri\uF\"edorovich Egorov (on the centenary of his birth), Uspekhi Mat. Nauk 26 (1971), no. 5, 169-206; English transl., Russian Math. Surveys 26 (1972), no. 5, 125-164.
[72] NFS I. M. Nikonov, A. T. Fomenko, and A. I. Shafarevich, D. F. Egorov’s papers on differential geometry, Historico-mathematical studies. Second series, 2009 (2009), no. 13 (48), 49-53. (Russian)
[73] KS Yu. M. Kolyagin and O. A. Savvina, Dmitri\uF\"edorovich Egorov. The path of a scientist and a Christian, PSTGU, Moscow, 2010. (Russian)
[74] Eg2 D. Th. Egorov, Sur les syst\`emes orthogonaux admettant un groupe continu de transformations de Combescure, C. R. Acad. Sci. Paris 131 (1900), 668-671; ibid. 132 (1901), 74-77.
[75] DN1 B. A. Dubrovin and S. P. Novikov, Hamiltonian formalism of one-dimensional systems of hydrodynamical type and the Bogolyubov-Whitman averaging method, Dokl. Akad. Nauk SSSR 270 (1983), no. 4, 781-785; English transl., Soviet Math.–Dokl. 27 (1983), 665-669. · Zbl 0553.35011
[76] Ts1 S. P. Tsarev, On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamical type, Dokl. Akad. Nauk SSSR 282 (1985), no. 3, 534-537; English transl., Soviet Math.–Dokl. 31 (1985), 488-491. · Zbl 0605.35075
[77] DN2 B. A. Dubrovin and S. P. Novikov, Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Uspekhi Mat. Nauk 44 (1989), no. 6, 29-98. English transl., Russian Math. Surveys 44 (1989), no. 6, 35-124.
[78] Ts2 S. P. Tsarev, Geometry of Hamiltonian systems. Generalized hodograph method, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 5, 1048-1068; English transl., Math. USSR-Izv. 37 (1991), no. 2, 397-419. · Zbl 0796.76014
[79] MF O. I. Mokhov and E. V. Ferapontov, Nonlocal Hamiltonian operators of hydrodynamical type associated with metrics of constant curvature, Uspekhi Mat. Nauk 45 (1990), no. 3, 191-192; English transl., Russian Math. Surveys 45 (1990), no. 3, 218-219. · Zbl 0712.35080
[80] Fe1 E. V. Ferapontov, Differential geometry of nonlocal Hamiltonian operators of hydrodynamical type, Funk. Anal. Prilozh. 25 (1991), no. 3, 37-49; English transl., Funct. Anal. Appl. 25 (1991), no. 3, 195-204. · Zbl 0742.58018
[81] Fe2 E. V. Ferapontov, Hamiltonian systems of hydrodynamical type and their realizations on hypersurfaces of a pseudo-Euclidean space, Problems in geometry, vol. 22, Itogi Nauki Tekhn., VINITI, Moscow, 1990, 59-96; English transl., J. Soviet Math. 55 (1991), no. 5, 1970-1995.
[82] Kr I. M. Krichever, Algebraic-geometric \(n\)-orthogonal curvilinear systems of coordinates and solutions of equations of associativity, Funk. Anal. Prilozh. 31 (1997), no. 1, 32-50; English transl., Funct. Anal. Appl. 31 (1997), no. 1, 25-39. · Zbl 1004.37052
[83] Mo O. I. Mokhov, Compatible and almost compatible pseudo-Riemannian metrics, Funk. Anal. Prilozh. 35 (2001), no. 2, 24-36; English transl., Funct. Anal. Appl. 35 (2001), no. 2, 100-110. · Zbl 1005.53016
[84] PTs M. V. Pavlov and S. P. Tsarev, Tri-Hamiltonian structures of Egorov systems of hydrodynamical type, Funk. Anal. Prilozh. 37 (2003), no. 1, 38-54; English transl., Funct. Anal. Appl. 37 (2003), no. 1, 32-45. · Zbl 1019.37048
[85] BLP V. M. Bukhshtaber, D. V. Le\i kin, and M. V. Pavlov, Egorov hydrodynamical chains, Chazy equation, and the group \(SL(2,C)\), Funk. Anal. Prilozh. 37 (2003), no. 4, 13-26; English transl., Funct. Anal. Appl. 37 (2003), no. 4, 251-262. · Zbl 1075.37527
[86] Za V. E. Zakharov, Description of the \(n\)-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type. I: Integration of the Lam\'e equations, Duke Math. J. 94 (1998), no. 1, 103-139. · Zbl 0963.37068
[87] F5 S. P. Finikov, Serge\uSergeevich Byushgens (on the seventieth anniversary of his birth), Uspekhi Mat. Nauk 8 (1953), no. 4, 185-192. (Russian)
[88] VL A. M. Vasil’ev and G. F. Laptev, Serge\uPavlovich Finikov. Obituary, Uspekhi Mat. Nauk 19 (1964), no. 4, 155-162; English transl., Russian Math. Surveys 19 (1964), no. 4, 151-159.
[89] BT V. T. Bazylev, On the 90th anniversary of S. P. Finikov’s birth, Problems in geometry, vol. 6, Itogi Nauki Tekhn., VINITI, Moscow, 1974, 17-24. (Russian)
[90] CB1 C. Burstin, Ein Beitrag zum Problem der Einbettung der Riemannschen R\`“aume in euklidischen R\'”aumen, Mat. Sb. 38 (1931), no. 3-4, 74-85.
[91] CB2 C. Burstin, Beitr\`“age der Verbiegung von Hyperfl\'”achen in euklidischen R\"aumen, Mat. Sb. 38 (1931), no. 3-4, 86-93.
[92] S2 I. Kh. Sabitov, On the history of one interpretation of bendings over a principal base, Historico-mathematical studies. Second series, 2000 (2000), no. 5 (40), 164-166. (Russian) · Zbl 1074.01523
[93] K V. F. Kagan, Foundations of the theory of surfaces. Part 2, OGIZ, GITTL, Moscow-Leningrad, 1948. (Russian)
[94] Lu N. N. Luzin, Proof of one theorem in the theory of bendings, Izv. Akad. Nauk SSSR, Div. Techn. Sci. 1939 (1939), no. 2, 81-106; no. 7, 115-132; no. 10, 65-84. (Russian)
[95] A A. D. Aleksandrov, On infinitesimal bendings of surfaces, Mat. Sb. 1 (1936), no. 3, 307-322. (Russian)
[96] KF S. Cohn-Vossen, Bendability of surfaces in the large, Uspekhi Mat. Nauk 1936 (1936), no. 1, 33-76. (Russian)
[97] DA A. G. Dorfman, Solution of the equation of bending for some classes of surfaces, Uspekhi Mat. Nauk 12 (1957), no. 2, 147-150. (Russian)
[98] e1 N. V. Efimov, Bending of a neighbourhood of a parabolic point on a surface, Mat. Sb. 6 (1939), no. 3, 427-474. (Russian)
[99] e2 N. V. Efimov, Study of bending of a surface with a point of flattening, Mat. Sb. 19 (1946), no. 3, 461-488. (Russian)
[100] e3 N. V. Efimov, Study of deformations of a surface containing a point with zero Gaussian curvature, Mat. Sb. 23 (1948), no. 1, 89-125. (Russian)
[101] e4 N. V. Efimov, Qualitative questions of the theory of deformations of surfaces “in the small”, Trudy Mat. Inst. Steklova 30 (1949), 3-128. (Russian)
[102] e5 N. V. Efimov, Qualitative questions in the theory of deformations of surfaces, Uspekhi Mat. Nauk 3 (1948), no. 2, 47-158. (Russian)
[103] e6 N. V. Efimov, On rigidity “in the small”, Dokl. Akad. Nauk SSSR 60 (1948), no. 5, 761-764. (Russian)
[104] EM Encyclopedia of elementary mathematics. Book 5: Geometry, Nauka, Moscow, 1966; German transl., VEB Deutscher Verlag der Wissenschaften, Berlin, 1971.
[105] AVS D. V. Alekseevski\i , A. M. Vinogradov, and V. V. Lychagin, Basic ideas and concepts in differential geometry, Current problems in mathematics. Fundamental directions, vol. 28, Geometry-1, Itogi Nauki Tekhn., VINITI, Moscow, 1988, 5-289; English transl., Geometry I, Encycl. Math. Sci., vol. 28, Springer, Berlin, 1991.
[106] AVL D. V. Alekseevski\i , \`E. B. Vinberg, and A. S. Solodovnikov, Geometry of spaces of constant curvature, Current problems in mathematics. Fundamental directions, vol. 29, Geometry-2, Itogi Nauki Tekhn., VINITI, Moscow, 1988, 5-146; English transl., Geometry II: Spaces of constant curvature, Encycl. Math. Sci., vol. 29, Springer, Berlin, 1993, 1-138.
[107] VS \`E. B. Vinberg and O. V. Shvartsman, Discrete groups of motions in spaces of constant curvature, Current problems in mathematics. Fundamental directions, vol. 29, Geometry-2, Itogi Nauki Tekhn., VINITI, Moscow, 1988, 147-259; English transl., Geometry II: Spaces of constant curvature, Encycl. Math. Sci., vol. 29, Springer, Berlin, 1993, 139-248.
[108] Bur Yu. D. Burago, Geometry of surfaces in Euclidean spaces, Current problems in mathematics. Fundamental directions, vol. 48, Geometry-3, Itogi Nauki Tekhn., VINITI, Moscow, 1989, 5-97; English transl., Geometry III: Theory of surfaces, Encycl. Math. Sci., vol. 48, Springer, Berlin, 1992, 1-85. · Zbl 0711.53003
[109] YuR Yu. G. Reshetnyak, Two-dimensional manifolds of bounded curvature, Current problems in mathematics. Fundamental directions, vol. 70, Geometry-4, Itogi Nauki Tekhn., VINITI, Moscow, 1989, 7-189; English transl., Geometry IV: Non-regular Riemannian geometry, Encycl. Math. Sci., vol. 70, Springer, Berlin, 1993, 3-163.
[110] BN V. N. Berestovski\i and I. G. Nikolaev, Multidimensional generalized Riemannian spaces, Current problems in mathematics. Fundamental directions, vol. 70, Geometry-4, Itogi Nauki Tekhn., VINITI, Moscow, 1989, 190-272; English transl., Geometry IV: Non-regular Riemannian geometry, Encycl. Math. Sci., vol. 70, Springer, Berlin, 1993, 168-243.
[111] Ba1 I. K. Babenko, Closed geodesics, asymptotic volumes, and characteristics of group growth, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 4, 675-711; English transl., Math. USSR-Izv. 33 (1989), no. 1, 1-37.
[112] Ba2 I. K. Babenko, Asymptotic invariants of smooth manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 56 (1992), no. 4, 707-751; English transl., Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 1-38. · Zbl 0812.57022
[113] AF A. T. Fomenko, The multidimensional Plateau problem in Riemannian manifolds, Mat. Sb. 89 (1972), no. 3, 475-519; English transl., Math. USSR-Sbornik 18 (1972), no. 3, 487-527. · Zbl 0276.49032
[114] IT A. O. Ivanov and A. A. Tuzhilin, One-dimensional Gromov minimal filling problem, Mat. Sb. 203 (2012), no. 5, 65-118; English transl., Sb. Math. 203 (2012), nos. 5-6, 677-726. · Zbl 1248.05057
[115] Mo1 O. I. Mokhov, Realization of Frobenius manifolds as submanifolds in pseudo-Euclidean spaces, Trudy Mat. Inst. Steklova 267 (2009), 226-244; English transl., Proc. Steklov Inst. Math. 267 (2009), no. 1, 217-234. · Zbl 1242.53114
[116] SSH I. Kh. Sabitov and S. Z. Shefel’, The connections between the order of smoothness of a surface and its metric, Sibirsk. Mat. Zh. 17 (1976), no. 4, 916-925; English transl., Siberian Math. J. 17 (1976), 687-694. · Zbl 0358.53015
[117] GR M. L. Gromov and V. A. Rokhlin, Embeddings and immersions in Riemannian geometry, Uspekhi Mat. Nauk 25 (1970) no. 5, 3-62; English transl., Russian Math. Surveys 25 (1970), no. 5, 1-57. · Zbl 0222.53053
[118] S3 I. Kh. Sabitov, Regularity of convex regions with a metric that is regular in the H\"older classes, Sibirsk. Mat. Zh. 17 (1976), no. 4, 907-915; English transl., Siberian Math. J. 17 (1976), 681-687. · Zbl 0386.53041
[119] Roz \`E. R. Rozendorn, Some problems in mapping theory with an application to the study of surfaces of negative curvature under reduced conditions of regularity, Trudy Moskov. Mat. Ob-va 53 (1990), 171-191; English transl., Trans. Moscow Math. Soc. 1991 (1991), 177-197. · Zbl 0742.53003
[120] s4 I. Kh. Sabitov, The rigidity of “corrugated” surfaces of revolution, Mat. Zametki 14 (1973), 517-522; English transl., Math. Notes 14 (1973), 854-857. · Zbl 0288.53006
[121] eu N. V. Efimov and Z. D. Usmanov, Infinitesimal bending of a surface with a point of flatness, Dokl. Akad. Nauk SSSR 208 (1973), 28-31; English transl., Soviet Math.–Dokl. 14 (1973), 22-25. · Zbl 0289.53005
[122] s5 I. Kh. Sabitov, Local theory of the bendings of surfaces, Current problems in mathematics. Fundamental directions, vol. 48, Itogi Nauki Tekhn., VINITI, Moscow, 1989, 196-270; English transl., Geometry III: Theory of surfaces, Encycl. Math. Sci., vol. 48, Springer, Berlin, 1992, 179-250. · Zbl 0781.53008
[123] KSU S. B. Klimentov, I. Kh. Sabitov, and Z. D. Usmanov, Deformations of surfaces “in the small”: from N. V. Efimov to contemporary research, Current problems in mathematics and mechanics, vol. VI: Mathematics, no. 2. On the 100th anniversary of N. V. Efimov’s birth, Moscow Univ., Moscow, 2011, 34-48. (Russian)
[124] IS I. Ivanova-Karatopraklieva and I. Kh. Sabitov, Deformation of surfaces. I, Problems in geometry, vol. 23, Itogi Nauki Tekhn., VINITI, Moscow, 1991, 131-184; English transl., J. Math. Sci. 70 (1994), no. 2, 1685-1716. · Zbl 0835.53003
[125] s6 I. Kh. Sabitov, On relations between infinitesimal bendings of different orders, Ukrain. Geom. Sb. 35 (1992), 118-124; English transl., J. Math. Sci. 72 (1994), no. 4, 3237-3241. · Zbl 0835.53005
[126] EP1 \`E. G. Poznyak, An example of a closed surface with singular point, having a countable fundamental system of infinitesimal deformations, Uspekhi Mat. Nauk 12 (1957), no. 3, 363-367. (Russian)
[127] EP2 \`E. G. Poznyak, On nonrigidity of the second order, Uspekhi Mat. Nauk 16 (1961), no. 1, 157-161. (Russian)
[128] s7 I. Kh. Sabitov, Infinitesimal bendings of troughs of revolution. I, Mat. Sb. 98 (1975), no. 1, 113-129; English transl., Math. USSR-Sbornik 27 (1975), no. 1, 103-117; Infinitesimal bendings of troughs of revolution. II, Mat. Sb. 99 (1976), no. 1, 49-57; English transl., Math. USSR-Sbornik 28 (1976), 41-48. · Zbl 0371.53041
[129] s8 I. Kh. Sabitov, Investigation of the rigidity and inflexibility of analytic surfaces of revolution with flattening at the pole, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1986 (1986), no. 5, 29-36; English transl., Moscow Univ. Math. Bull. 41 (1986), no. 5, 33-41. · Zbl 0633.53006
[130] s9 I. Kh. Sabitov, Rigidity and inflexibility “in the small” and “in the large” of surfaces of revolution with flattenings at the poles, Mat. Sb. 204 (2013), no. 10, 127-160; English transl., Sb. Math. 204 (2013), nos. 9-10, 1516-1547. · Zbl 1292.53007
[131] s10 I. Kh. Sabitov, Second-order infinitesimal bendings of surfaces of revolution with flattenings at the poles, Mat. Sb. 205 (2014), no. 12, 111-140; English transl., Sb. Math. 205 (2014), nos. 11-12, 1787-1814. · Zbl 1315.53005
[132] s11 I. Kh. Sabitov, Quasiconformal mappings of a surface that are generated by its isometric transformations, and bendings of the surface onto itself, Fundam. Prikl. Mat. 1 (1995), no. 1, 281-288 (Russian) · Zbl 0871.53012
[133] s12 I. Kh. Sabitov, Possible generalizations of the Minagawa-Rado lemma on the rigidity of a surface of revolution with a fixed parallel, Mat. Zametki 19 (1976), no. 1, 123-132; English transl., Math. Notes 19 (1976), 74-79. · Zbl 0357.53003
[134] IS1 I. Ivanova-Karatopraklieva and I. Kh. Sabitov, Bending of surfaces. II, Current mathematics and applications. Thematic surveys, vol. 8, Itogi Nauki Tekhn., VINITI, Moscow, 1995, 108-167; English transl., Geometry. 1, J. Math. Sci. 74 (1995), no. 3, 997-1043. · Zbl 0861.53002
[135] IMS I. Ivanova-Karatopraklieva, P. E. Markov, and I. Kh. Sabitov, Bending of surfaces. III, Fundam. Prikl. Mat. 12 (2006), no. 1, 3-56; English transl., J. Math. Sci. (N. Y.) 149 (2008), no. 1, 861-895. · Zbl 1149.53301
[136] e7 N. V. Efimov, The impossibility in Euclidean \(3\)-space of a complete regular surface with a negative upper bound of the Gaussian curvature, Dokl. Akad. Nauk SSSR 150 (1963), 1206-1209; English transl., Soviet Math.–Dokl. 4 (1963), 843-846. · Zbl 0135.40001
[137] e8 N. V. Efimov, Emergence of singularities on surfaces of negative curvature, Mat. Sb. 64 (1964), no. 2, 286-320. (Russian)
[138] RSH \`“E. R. Rozendorn and E. V. Shikin, The papers of N. V. Efimov on surfaces of negative curvatures, Modern problems in mathematics and mechanics, vol. VI: Mathematics, no. 2. On the 100th anniversary of N. V. Efimov”s birth, Moscow Univ., Moscow, 2011, 49-56. (Russian)
[139] TS Remembering Nikola\uVladimirovich Efimov..., Moscow Centre for Contin. Math. Educ., Moscow, 2014. (Russian)
[140] Ro \`E. R. Rozendorn, The construction of a bounded, complete surface of nonpositive curvature, Uspekhi Mat. Nauk 16 (1961), no. 2, 149-156. (Russian)
[141] Ro1 \`E. R. Rozendorn, Surfaces of negative curvature, Current problems in mathematics. Fundamental directions, vol. 48, Itogi Nauki Tekhn., VINITI, Moscow, 1989, 98-195; English transl., Geometry. III, Encycl. Math. Sci., vol. 48, Springer, Berlin, 1992, 87-178.
[142] TC A. N. Tikhonov, A. A. Samarski\i , O. A. Ole\i nik, et al., \`Eduard Genrikhovich Poznyak (on the occasion of his seventieth birthday), Uspekhi Mat. Nauk 48 (1993), no. 4, 245-247; English transl., Russian Math. Surveys 48 (1993), no. 4, 267-269.
[143] EP3 \`E. G. Poznyak, On a regular global realization of two-dimensional Riemann metrics of negative curvature, Mat. Zametki 1 (1967), no. 2, 244-250; English transl., Math. Notes 1 (1967), 162-165.
[144] Sh1 E. V. Shikin, Isometric embeddings in \(\mathbbR^3\) of noncompact domains of nonpositive curvature, Problems in Geometry, vol. 7, Itogi Nauki Tekhn., VINITI, Moscow, 1975, 249-265. (Russian)
[145] Sh2 \`E. G. Poznyak, On isometric immersion in three-dimensional Euclidean space of two-dimensional manifolds of negative curvature, Mat. Zametki 31 (1982), no. 4, 601-612. (Russian)
[146] Tu D. V. Tunitski\i , Regular isometric immersion in \(E\sp 3\) of unbounded domains of negative curvature, Mat. Sb. 134 (1987), no. 1, 119-134; English transl., Math. USSR-Sbornik 62 (1989), no. 1, 121-138.
[147] EP4 \`E. G. Poznyak, Isometric immersions of two-dimensional Riemannian metrics in Euclidean space, Uspekhi Mat. Nauk 28 (1973), no. 4, 47-76; English transl., Russian Math. Surveys 28 (1973), no. 4, 47-77. · Zbl 0289.53004
[148] PSH \`E. G. Poznyak and E. V. Shikin, Surfaces of negative curvature, Algebra. Topology. Geometry, vol. 12, Itogi Nauki Tekhn., VINITI, Moscow, 1974, 171-207; English transl., J. Soviet Math. 5 (1976), 865-887.
[149] PS \`E. G. Poznyak and D. D. Sokolov, Isometric immersions of Riemannian spaces in Euclidean spaces, Algebra. Topology. Geometry, vol. 15, Itogi Nauki Tekhn., VINITI, Moscow, 1977, 173-211; English transl., J. Soviet Math. 14 (1980), 1407-1428. · Zbl 0449.53044
[150] PSH1 \`E. G. Poznyak and E. V. Shikin, Small parameter in the theory of isometric imbeddings for two-dimensional Riemannian manifolds into Euclidean spaces, Current mathematics and applications. Thematic surveys, vol. 8, Itogi Nauki Tekhn., VINITI, Moscow, 1995, 59-107; English transl., Geometry. 1, J. Math. Sci. 74 (1995), no. 3, 1078-1116. · Zbl 0849.53041
[151] PP \`E. G. Poznyak and A. G. Popov, Geometry of the sine-Gordon equation, Problems in geometry, vol. 23, Itogi Nauki Tekhn., VINITI, Moscow, 1991, 99-130; English transl., J. Math. Sci. 70 (1994), no. 2, 1666-1684. · Zbl 0835.35123
[152] Ya N. N. Yanenko, Some questions in the theory of the imbedding of Riemannian metrics in Euclidean spaces, Uspekhi Mat. Nauk 8 (1953), no. 1, 21-100. (Russian)
[153] Ya1 N. N. Yanenko, On the theory of the imbedding of surfaces in a multi-dimensional Euclidean space, Trudy Moskov. Mat. Ob-va 3 (1954), 89-180. (Russian)
[154] s13 I. Kh. Sabitov, Isometric immersions and embeddings of locally Euclidean metrics in \(\mathbbR\sp 2\), Izv. Ross. Akad. Nauk Ser. Mat. 63 (1999), no. 6, 147-166; English transl., Izv. Math. 63 (1999), no. 6, 1203-1220. · Zbl 0978.53031
[155] s14 I. Kh. Sabitov, Isometric embedding of locally Euclidean metrics in \(\mathbbR\sp 3\), Sibirsk. Mat. Zh. 26 (1985), no. 3, 156-167; English transl., Siberian Math. J. 26 (1985), 431-440. · Zbl 0584.53003
[156] MS1 S. N. Mikhalev and I. Kh. Sabitov, Isometric embeddings of locally Euclidean metrics in \(\mathbbR\sp 3\) as conical surfaces, Mat. Zametki 95 (2014), no. 2, 234-247; English transl., Math. Notes 95 (2014), nos. 1-2, 212-223. · Zbl 1315.53004
[157] MS2 S. N. Mikhalev and I. Kh. Sabitov, Isometric embeddings in \(\mathbbR\sp 3\) of an annulus with a locally Euclidean metric which are multivalued of cylindrical type, Mat. Zametki 98 (2015), no. 3, 378-385; English transl., Math. Notes 98 (2015), nos. 3-4, 441-447. · Zbl 1337.53006
[158] s15 I. Kh. Sabitov, On developable ruled surfaces of low smoothness, Sibirsk. Mat. Zh. 50 (2009), no. 5, 1163-1175; English transl., Siberian Math. J. 50 (2009), no. 5, 919-928. · Zbl 1224.53010
[159] s16 I. Kh. Sabitov, On the extrinsic curvature and the extrinsic structure of \(C\sp 1\)-smooth normal developable surfaces, Mat. Zametki 87 (2010), no. 6, 900-906; English transl., Math. Notes 87 (2010), nos. 5-6, 874-879. · Zbl 1275.53010
[160] s17 I. Kh. Sabitov, Isometric immersions and embeddings of locally Euclidean metrics, Cambridge Scientific Publishers, Cambridge, 2008. · Zbl 1218.53001
[161] SH1 M. I. Shtogrin, Piecewise-smooth developable surfaces, Trudy Mat. Inst. Steklova 263 (2008), 227-250; English transl., Proc. Steklov Inst. Math. 263 (2008), no. 1, 214-235. · Zbl 1180.53005
[162] SH2 M. I. Shtogrin, Bending of a developable surface with preservation of its edge and generators, Trudy Mat. Inst. Steklova 266 (2009), 263-271; English transl., Proc. Steklov Inst. Math. 266 (2009), no. 1, 251-259. · Zbl 1180.53006
[163] SH3 M. I. Shtogrin, Isometric embeddings of the surfaces of Platonic solids, Uspekhi Mat. Nauk 62 (2007), no. 2, 183-184; English transl., Russian Math. Surveys 62 (2007), no. 2, 395-397. · Zbl 1154.52006
[164] SH4 M. I. Shtogrin, On closed convex polyhedra admitting continuous bendings in the class of piecewise-smooth surfaces, Current problems of mathematics and mechanics, vol. VI: Mathematics, no. 3. On the 100th anniversary of N. V. Efimov’s birth, Moscow Univ., Moscow, 2011, 192-207. (Russian)
[165] SH5 M. I. Shtogrin, Bending of a piecewise developable surface, Trudy Mat. Inst. Steklova 275 (2011), 144-166; English transl., Proc. Steklov Inst. Math. 275 (2011), no. 1, 133-154. · Zbl 1297.53008
[166] DSS1 N. P. Dolbilin, M. A. Shtan’ko, and M. I. Shtogrin, Nonbendability of a division of a sphere into squares, Dokl. Akad. Nauk 354 (1997), no. 4, 443-445; English transl., Dokl. Math. 55 (1997), no. 3, 385-387.
[167] DSS2 N. P. Dolbilin, M. A. Shtan’ko, and M. I. Shtogrin, Rigidity of a quadrillage of a torus by squares, Uspekhi Mat. Nauk 54 (1999), no. 4, 167-168; English transl., Russian Math. Surveys 54 (1999), no. 4, 839-840.
[168] SHa M. I. Shtogrin, Rigidity of quadrillage of the pretzel, Uspekhi Mat. Nauk 54 (1999), no. 5, 183-184; English transl., Russian Math. Surveys 54 (1999), no. 5, 1044-1045. · Zbl 0969.52007
[169] s18 I. Kh. Sabitov, On a class of inflexible polyhedra, Sibirsk. Mat. Zh. 55 (2014), no. 5, 475-483; English transl., Siberian Math. J. 55 (2014), no. 5, 961-967. · Zbl 1321.52026
[170] ss18 I. Kh. Sabitov, Volumes of polyhedra, 2nd ed., Moscow Centre for Contin. Math. Educ., Moscow, 2009. (Russian)
[171] AV V. Alexandrov, An example of a flexible polyhedron with nonconstant volume in the spherical space, Beitr. Algebra Geometrie, 38 (1997), no. 1, 11-18. · Zbl 0881.52007
[172] SH6 M. I. Shtogrin, On flexible polyhedral surfaces, Trudy Mat. Inst. Steklova 288 (2015), 171-183; English transl., Proc. Steklov Inst. Math. 288 (2015), 153-164. · Zbl 1357.52021
[173] BE V. M. Bukhshtaber and N. Yu. Erokhovets, Truncations of simple polytopes and applications, Trudy Mat. Inst. Steklova 289 (2015), 115-144; English transl., Proc. Steklov Inst. Math. 289 (2015), 104-133.
[174] s19 I. Kh. Sabitov, The volume of a polyhedron as a function of its metric, Fundam. Prikl. Mat. 2 (1996), no. 4, 1235-1246. (Russian) · Zbl 0904.52002
[175] s20 I. Kh. Sabitov, The generalized Heron-Tartaglia formula and some of its consequences, Mat. Sb. 189 (1998), no. 10, 105-134; English transl., Sb. Math. 189 (1998), nos. 9-10, 1533-1561. · Zbl 0941.52020
[176] s21 I. Kh. Sabitov, Solution of cyclic polygons, Math. Educ. 3rd Ser., no. 11, Moscow Centre Contin. Math. Educ., Moscow, 2010, 83-106. (Russian)
[177] G1 A. A. Gaifullin, Sabitov polynomials for volumes of polyhedra in four dimensions, Adv. in Math. 252 (2014), 586-611. · Zbl 1301.51027
[178] G2 A. A. Gaifullin, Generalization of Sabitov’s theorem to polyhedra of arbitrary dimensions, Discrete and Comput. Geometry, 52 (2014), no. 2, 195-220. · Zbl 1314.52008
[179] G3 A. A. Gaifullin, The analytic continuation of volume and the bellows conjecture in Lobachevski\uspaces, Mat. Sb. 206 (2015), no. 11, 61-112; English transl., Sb. Math. 206 (2015), nos. 11-12, 1564-1609. · Zbl 1370.52068
[180] G4 A. A. Gaifullin, Embedded flexible spherical cross-polytopes with nonconstant volumes, Trudy Mat. Inst. Steklova 288 (2015), 67-94; English transl., Proc. Steklov Inst. Math. 288 (2015), no. 1, 56-80. · Zbl 1322.52017
[181] s22 I. Kh. Sabitov, Algorithmic solution of the problem of the isometric realization of two-dimensional polyhedral metrics, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 2, 159-172; English transl., Izv. Math. 66 (2002), no. 2, 377-391. · Zbl 1076.51513
[182] s23 I. Kh. Sabitov, Algebraic methods for the solution of polyhedra, Uspekhi Mat. Nauk 66 (2011), no. 3, 3-66; English transl., Russian Math. Surveys 66 (2011), no. 3, 445-505. · Zbl 1230.52031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.