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The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research


Author: I. Kh. Sabitov
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 77 (2016), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2016, 149-175
MSC (2010): Primary 01A55, 01A60, 01A61, 01A74, 53-03
DOI: https://doi.org/10.1090/mosc/257
Published electronically: November 28, 2016
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Abstract: We describe the history of the development of geometric studies related to the work of the Moscow Mathematical Society from its early years to the present day. The main focus is on papers on ``geometry in the large''.


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  • 122. I. Kh. Sabitov, Local theory of the bendings of surfaces, Current problems in mathematics. Fundamental directions, Vol. 48 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 196–270 (Russian). MR 1039820
  • 123. 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. I. Ivanova-Karatopraklieva and I. Kh. Sabitov, Deformation of surfaces. I, Problems in geometry, Vol. 23 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1991, pp. 131–184, 187 (Russian). Translated in J. Math. Sci. 70 (1994), no. 2, 1685–1716. MR 1152588
  • 125. I. Kh. Sabitov, On relations between infinitesimal bendings of different orders, Ukrain. Geom. Sb. 35 (1992), 118–124, 165 (Russian, with Russian summary); English transl., J. Math. Sci. 72 (1994), no. 4, 3237–3241. MR 1267540, https://doi.org/10.1007/BF01249526
  • 126. È. G. Poznyak, An example of a closed surface with singular point, having a countable fundamental system of infinitesimal deformations, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 3(75), 363–367 (Russian). MR 0090835
  • 127. È. G. Poznyak, On nonrigidity of the second order, Uspekhi Mat. Nauk 16 (1961), no. 1, 157-161. (Russian)
  • 128. I. H. Sabitov, Infinitesimal bendings of troughs of revolution. I, Mat. Sb. (N.S.) 98(140) (1975), no. 1 (9), 113–129, 159 (Russian). MR 0405299
  • 129. 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. 5 (1986), 29–36, 99 (Russian). MR 872261
  • 130. 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 (Russian, with Russian summary); English transl., Sb. Math. 204 (2013), no. 9-10, 1516–1547. MR 3137162
  • 131. I. Kh. Sabitov, Second-order infinitesimal bendings of surfaces of revolution with flattenings at the poles, Mat. Sb. 205 (2014), no. 12, 111–140 (Russian, with Russian summary); English transl., Sb. Math. 205 (2014), no. 11-12, 1787–1814. MR 3309393
  • 132. 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, with English and Russian summaries). MR 1789365
  • 133. I. H. 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 (Russian). MR 0420522
  • 134. I. Ivanova-Karatopraklieva and I. Kh. Sabitov, Bending of surfaces. II, J. Math. Sci. 74 (1995), no. 3, 997–1043. Geometry, 1. MR 1330961, https://doi.org/10.1007/BF02362831
  • 135. I. Ivanova-Karatopraklieva, P. E. Markov, and I. Kh. Sabitov, Bending of surfaces. III, Fundam. Prikl. Mat. 12 (2006), no. 1, 3–56 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 149 (2008), no. 1, 861–895. MR 2249679, https://doi.org/10.1007/s10958-008-0033-0
  • 136. 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 (Russian). MR 0150702
  • 137. N. V. Efimov, Generation of singularites on surfaces of negative curvature, Mat. Sb. (N.S.) 64 (106) (1964), 286–320 (Russian). MR 0167938
  • 138. È. 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. Remembering Nikolaĭ Vladimirovich Efimov..., Moscow Centre for Contin. Math. Educ., Moscow, 2014. (Russian)
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  • 142. A. N. Tikhonov, A. A. Samarskiĭ, O. A. Oleĭnik et al., Èduard Gendrikhovich Poznyak (on the occasion of his seventieth birthday), Uspekhi Mat. Nauk 48 (1993), no. 4(292), 245–247 (Russian); English transl., Russian Math. Surveys 48 (1993), no. 4, 267–269. MR 1257896, https://doi.org/10.1070/RM1993v048n04ABEH001065
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  • 147. È. G. Poznjak, Isometric imbedding of two-dimensional Riemannian metrics in Euclidean spaces, Uspehi Mat. Nauk 28 (1973), no. 4(172), 47–76 (Russian). MR 0394514
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  • 151. È. G. Poznyak and A. G. Popov, Geometry of the sine-Gordon equation, Problems in geometry, Vol. 23 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1991, pp. 99–130, 187 (Russian). Translated in J. Math. Sci. 70 (1994), no. 2, 1666–1684. MR 1152587
  • 152. N. N. Yanenko, Some questions of the theory of imbedding of Riemannian metrics in Euclidean spaces, Uspehi Matem. Nauk (N.S.) 8 (1953), no. 1(53), 21–100 (Russian). MR 0055758
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  • 154. I. Kh. Sabitov, Isometric immersions and embeddings of locally Euclidean metrics in 𝐑², Izv. Ross. Akad. Nauk Ser. Mat. 63 (1999), no. 6, 147–166 (Russian, with Russian summary); English transl., Izv. Math. 63 (1999), no. 6, 1203–1220. MR 1748564, https://doi.org/10.1070/im1999v063n06ABEH000270
  • 155. I. Kh. Sabitov, Isometric embedding of locally Euclidean metrics in 𝑅³, Sibirsk. Mat. Zh. 26 (1985), no. 3, 156–167, 226 (Russian). MR 792065
  • 156. S. N. Mikhalev and I. Kh. Sabitov, Isometric embeddings of locally Euclidean metrics in ℝ³ as conical surfaces, Math. Notes 95 (2014), no. 1-2, 212–223. Translation of Mat. Zametki 95 (2014), no. 2, 234–247. MR 3267209, https://doi.org/10.1134/S0001434614010234
  • 157. S. N. Mikhalev and I. Kh. Sabitov, Isometric embeddings in ℝ³ of an annulus with a locally Euclidean metric which are multivalued of cylindrical type, Mat. Zametki 98 (2015), no. 3, 378–385 (Russian, with Russian summary); English transl., Math. Notes 98 (2015), no. 3-4, 441–447. MR 3438494, https://doi.org/10.4213/mzm10815
  • 158. I. Kh. Sabitov, On developable ruled surfaces of low smoothness, Sibirsk. Mat. Zh. 50 (2009), no. 5, 1163–1175 (Russian, with Russian summary); English transl., Sib. Math. J. 50 (2009), no. 5, 919–928. MR 2603859, https://doi.org/10.1007/s11202-009-0102-8
  • 159. I. Kh. Sabitov, On the extrinsic curvature and the extrinsic structure of 𝐶¹-smooth normal developable surfaces, Mat. Zametki 87 (2010), no. 6, 900–906 (Russian, with Russian summary); English transl., Math. Notes 87 (2010), no. 5-6, 874–879. MR 2840384, https://doi.org/10.1134/S0001434610050275
  • 160. I. Kh. Sabitov, Isometric immersions and embeddings of locally Euclidean metrics, Reviews in Mathematics and Mathematical Physics, vol. 13, Cambridge Scientific Publishers, Cambridge, 2008. MR 2584444
  • 161. M. I. Shtogrin, Piecewise-smooth developable surfaces, Tr. Mat. Inst. Steklova 263 (2008), no. Geometriya, Topologiya i Matematicheskaya Fizika. I, 227–250 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 263 (2008), no. 1, 214–235. MR 2599382, https://doi.org/10.1134/S0081543808040160
  • 162. M. I. Shtogrin, Bending of a developable surface with preservation of its edge and generators, Tr. Mat. Inst. Steklova 266 (2009), no. Geometriya, Topologiya i Matematicheskaya Fizika. II, 263–271 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 266 (2009), no. 1, 251–259. MR 2603272, https://doi.org/10.1134/S0081543809030158
  • 163. M. I. Shtogrin, Isometric embeddings of the surfaces of Platonic solids, Uspekhi Mat. Nauk 62 (2007), no. 2(374), 183–184 (Russian); English transl., Russian Math. Surveys 62 (2007), no. 2, 395–397. MR 2352375, https://doi.org/10.1070/RM2007v062n02ABEH004406
  • 164. 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. M. I. Shtogrin, Bending of a piecewise developable surface, Tr. Mat. Inst. Steklova 275 (2011), no. Klassicheskaya i Sovremennaya Matematika v Pole Deyatel′nosti Borisa Nikolaevicha Delone, 144–166 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 275 (2011), no. 1, 133–154. MR 2962975, https://doi.org/10.1134/S0081543811080086
  • 166. N. P. Dolbilin, M. A. Shtan′ko, and M. I. Shtogrin, Nonbendability of the covering of a sphere by squares, Dokl. Akad. Nauk 354 (1997), no. 4, 443–445 (Russian). MR 1472082
  • 167. 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(328), 167–168 (Russian); English transl., Russian Math. Surveys 54 (1999), no. 4, 839–840. MR 1741291, https://doi.org/10.1070/rm1999v054n04ABEH000187
  • 168. M. I. Shtogrin, Rigidity of the covering of a pretzel-shaped surface by squares, Uspekhi Mat. Nauk 54 (1999), no. 5(329), 183–184 (Russian); English transl., Russian Math. Surveys 54 (1999), no. 5, 1044–1045. MR 1741682, https://doi.org/10.1070/rm1999v054n05ABEH000218
  • 169. I. Kh. Sabitov, On a class of inflexible polyhedra, Sibirsk. Mat. Zh. 55 (2014), no. 5, 1175–1183 (Russian, with Russian summary); English transl., Sib. Math. J. 55 (2014), no. 5, 961–967. MR 3289119
  • 170. I. Kh. Sabitov, Volumes of polyhedra, 2nd ed., Moscow Centre for Contin. Math. Educ., Moscow, 2009. (Russian)
  • 171. Victor Alexandrov, An example of a flexible polyhedron with nonconstant volume in the spherical space, Beiträge Algebra Geom. 38 (1997), no. 1, 11–18. MR 1447982
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  • 173. V. M. Buchstaber and N. Yu. Erokhovets, Truncations of simple polytopes and applications, Proc. Steklov Inst. Math. 289 (2015), no. 1, 104–133. MR 3486778, https://doi.org/10.1134/S0081543815040070
  • 174. I. Kh. Sabitov, The volume of a polyhedron as a function of its metric, Fundam. Prikl. Mat. 2 (1996), no. 4, 1235–1246 (Russian, with English and Russian summaries). Research papers dedicated to the memory of B. V. Gnedenko (Russian). MR 1785783
  • 175. I. Kh. Sabitov, The generalized Heron-Tartaglia formula and some of its consequences, Mat. Sb. 189 (1998), no. 10, 105–134 (Russian, with Russian summary); English transl., Sb. Math. 189 (1998), no. 9-10, 1533–1561. MR 1691297, https://doi.org/10.1070/SM1998v189n10ABEH000354
  • 176. I. Kh. Sabitov, Solution of cyclic polygons, Math. Educ. 3rd Ser., no. 11, Moscow Centre Contin. Math. Educ., Moscow, 2010, 83-106. (Russian)
  • 177. Alexander A. Gaifullin, Sabitov polynomials for volumes of polyhedra in four dimensions, Adv. Math. 252 (2014), 586–611. MR 3144242, https://doi.org/10.1016/j.aim.2013.11.005
  • 178. Alexander A. Gaifullin, Generalization of Sabitov’s theorem to polyhedra of arbitrary dimensions, Discrete Comput. Geom. 52 (2014), no. 2, 195–220. MR 3249379, https://doi.org/10.1007/s00454-014-9609-2
  • 179. A. A. Gaĭfullin, The analytic continuation of volume and the bellows conjecture in Lobachevskiĭ spaces, Mat. Sb. 206 (2015), no. 11, 61–112 (Russian, with Russian summary); English transl., Sb. Math. 206 (2015), no. 11-12, 1564–1609. MR 3438569, https://doi.org/10.4213/sm8522
  • 180. Alexander A. Gaifullin, Embedded flexible spherical cross-polytopes with nonconstant volumes, Proc. Steklov Inst. Math. 288 (2015), no. 1, 56–80. MR 3485701, https://doi.org/10.1134/S0081543815010058
  • 181. 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 (Russian, with Russian summary); English transl., Izv. Math. 66 (2002), no. 2, 377–391. MR 1918847, https://doi.org/10.1070/IM2002v066n02ABEH000382
  • 182. I. Kh. Sabitov, Algebraic methods for the solution of polyhedra, Uspekhi Mat. Nauk 66 (2011), no. 3(399), 3–66 (Russian, with Russian summary); English transl., Russian Math. Surveys 66 (2011), no. 3, 445–505. MR 2859189, https://doi.org/10.1070/RM2011v066n03ABEH004748

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Additional Information

I. Kh. Sabitov
Affiliation: Moscow State University
Email: isabitov@mail.ru

DOI: https://doi.org/10.1090/mosc/257
Keywords: Moscow Mathematical Society, journal ``Matematicheski\u{\i} Sbornik'', Presidents-geometers, characteristic results and surveys on geometric studies, ``geometry on the whole''.
Published electronically: November 28, 2016
Article copyright: © Copyright 2016 American Mathematical Society