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

Sabitov, I.

doi:10.1090/mosc/257

The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research
HTML articles powered by AMS MathViewer

by I. Kh. Sabitov
Translated by: E. Khukhro

Trans. Moscow Math. Soc. 2016, 149-175

DOI: https://doi.org/10.1090/mosc/257

Published electronically: November 28, 2016

PDF | Request permission

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”.

References

A. N. Kolmogorov and S. P. Novikov (eds.), Issledovaniya po metricheskoĭ teorii poverkhnosteĭ, Matematika: Novoe v Zarubezhnoĭ Nauke [Mathematics: Recent Publications in Foreign Science], vol. 18, “Mir”, Moscow, 1980 (Russian). Papers translated from the English and from the French by I. H. Sabitov. MR 608632
A. M. Vasil′ev, N. V. Efimov, and P. K. Raševskiĭ, Research in differential geometry at Moscow University in the Soviet period, Vestnik Moskov. Univ. Ser. I Mat. Meh. 22 (1967), no. 5, 12–23 (Russian). MR 0222775
Mathematics in the USSR over 40 years (1917–1957), vol. 1: Survey talks, GIFML, Moscow, 1959. (Russian)
P. K. Rashevskiĭ, Tensor differential geometry, Mathematics in the USSR over 30 years, GITTL, Moscow–Leningrad, 1948, 283–918.
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)
Geometry and topology, Sovrem. Probl. Mat. Mekh. 3 (2009), no. 2. (Russian)
Soviet mathematics over 20 years, Uspekhi Mat. Nauk 1938 (1938), no. 4, 3–13. (Russian)
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)
Mathematics. Science in the USSR over fifteen years (1917–1932), GTTI, Moscow, 1932. (Russian)
D. F. Egorov, Successes in mathematics in the USSR, Science and technology in the USSR (1917–1927), Rabotnik Prosvesch. 1 (1928), 223–234.
L. A. Lyusternik, “Matematicheskiĭ Sbornik”, Uspekhi Mat. Nauk 1 (1946), no. 1, 242–247. (Russian)
A. F. Lapko and L. A. Ljusternik, From the history of Soviet mathematics. I, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 13–140 (Russian). MR 0218186
A. D. Aleksandrov, Geometry and topology in the Soviet Union. II, Uspehi Matem. Nauk (N.S.) 2 (1947), no. 5(21), 9–92 (Russian). MR 0027147
L. Euler, Opera postuma, Mathematica et Physica, vol. I, Academiae Scientiarum Petropolitanae, St. Petersburg, 1862, 494–495.
A.-M. Legendre, Eléments de géométrie, Paris, 1794.
A. Cauchy, Sur les polygones et les polyèdres: second mémoire, J. d’Ecole Polytechnique, IX (1813), cahier XVI, 87–98.
A. L-v, On N. I. Lobachevskiĭ’s theory of parallel lines, Mat. Sb. 3 (1868), no. 2, 78–120. (Russian)
J. Bertrand, Sur la somme des angles du triangle, Mat. Sb. 4 (1870), no. 4, 198–207. (Russian)
V. Ya. Bunyakovskiĭ, A note concerning the question on parallel lines, Mat. Sb. 6 (1872), no. 1, 77–82. (Russian)
L. K. Lakhtin, On life and scientific works of Nikolaĭ Ivanovich Lobachevskiĭ (on the occasion of the centenary of his birth), Mat. Sb. 17 (1894), no. 3, 474–493. (Russian)
L. K. Lakhtin, On one concrete interpretation of the Lobachevskiĭ planimetry, Mat. Sb. 17 (1895), no. 4, 767–790. (Russian)
F. Klein, Über die sogenannte Nicht-Euklidische Geometrie, Math. Ann. 4 (1871), 573–625.
B. Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen, Gött. Abd. 13 (1868).
A. P. Norden (ed.), On foundations of geometry. A collection of classical papers on Lobachevskiĭ geometry and the development of its ideas, GITTL, Moscow, 1956. (Russian)
S. A. Bogomolov, Introduction to Riemann’s non-Euclidean geometry, Leningrad–Moscow: ONTI GTTI, 1934. (Russian)
K. M. Peterson, On relations and affinities between curved surfaces, Mat. Sb. 1 (1866), no. 1, 391–438. (Russian)
O. Bonnet, Mémoire sur la théorie des surfaces applicables sur une surface donnée, J. d’Ecole Polytechnique. Première Partie XXIV (1865), no. 41, 209–230; Deuxième Partie XXV (1867), no. 42, 1–151
Alexander 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., vol. 51, Math. Soc. Japan, Tokyo, 2008, pp. 1–53. MR 2509789, DOI 10.2969/aspm/05110001
I. Kh. Sabitov, Isometric surfaces with common mean curvature and the problem of Bonnet pairs, Mat. Sb. 203 (2012), no. 1, 115–158 (Russian, with Russian summary); English transl., Sb. Math. 203 (2012), no. 1-2, 111–152. MR 2933095, DOI 10.1070/SM2012v203n01ABEH004216
K. M. Peterson, On the bending of surfaces of the 2nd order, Mat. Sb. 10 (1883), no. 4, 476–523. (Russian)
B. K. Mlodzeevskiĭ, Karl Mikhaĭlovich Peterson and his geometric works, Mat. Sb. 24 (1903), no. 1, 1–21. (Russian)
Élie Cartan, Sur les couples de surfaces applicables avec conservation des courbures principales, Bull. Sci. Math. (2) 66 (1942), 55–72, 74–85 (French). MR 9876
K. Peterson, Über Kurven und Flächen, Moskau und Leipzig, 1868.
S. D. Rossinskiĭ, Obituary: Karl Mihaĭlovič Peterson (1828–1881), Uspehi Matem. Nauk (N.S.) 4 (1949), no. 5(33), 3–13 (1 plate) (Russian). MR 0034346
D. F. Egorov, Towards a general theory of the correspondence of surfaces, Mat. Sb. 18 (1896), no. 1, 86–107. (Russian)
B. K. Mlodzeevskiĭ, On surfaces related to Peterson surfaces, Mat. Sb. 21 (1900), no. 3, 450–460. (Russian)
B. K. Mlodzeevskiĭ, On bending of Peterson surfaces, Mat. Sb. 24 (1900), no. 3, 417–473. (Russian)
B. K. Mlodzeevskiĭ, On one transformation of infinitesimal bendings, Mat. Sb. 28 (1911), no. 1, 205–214. (Russian)
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)
S. S. Byushgens, On bending of surfaces over a principal base, Mat. Sb. 28 (1912), no. 4, 507–528. (Russian)
S. P. Finikov, On bending of surfaces of the 2nd order over a principal base, Mat. Sb. 28 (1912), no. 4, 529–543. (Russian)
S. S. Byushgens, On cyclic congruences and Bianchi surfaces, Mat. Sb. 30 (1916), no. 2, 296–313. (Russian)
S. P. Finikov, General problem of bending over a principal base, Moscow, 1917. (Russian)
S. S. Byushgens, Bending over a principal base, Moscow, 1918. (Russian)
D. Th. Egoroff, Sur les surfaces, engendrées par la distrubution des lignes d’une famille donnée, Mat. Sb. 31 (1923), no. 3, 153–184.
S. Finikoff, Sur les surfaces de M. Bianchi, Mat. Sb. 32 (1924), no. 1, 249–254.
S. P. Finikov, On one case of special bending of a congruence, Mat. Sb. 32 (1924), no. 1, 241–248. (Russian)
S. Bucheguennce, Sur certaines familles invariables de courbes, Mat. Sb. 32 (1925), no. 2, 348–352.
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)
S. Bucheguennce, Sur une class des hypersurfaces, Mat. Sb. 32 (1925), no. 4, 625–631.
S. Bucheguennce, Sur les surfaces ayant une famille des parallèles planes ou sphériques, Mat. Sb. 32 (1925), no. 4, 632–645.
S. Finikoff, Sur la déformation des surfaces à réseaux cinématiquemant conjugés persistant, Mat. Sb. 33 (1926), no. 3, 129–160.
A. Th. Masloff, Sur la déformation des surfaces avec conservation d’un système conjugué, Mat. Sb. 33 (1926), no. 1, 43–48.
A. Th. Masloff, Sur la déformation continue d’une classe des surfaces, Mat. Sb. 33 (1926), no. 4, 367–370.
S. Finikoff, Sur la congruence rectiligne de roulement d’une infinité de manières, Mat. Sb. 34 (1927), no. 1, 49–54.
L. N. Sretenskiĭ, On the bending of surfaces, Mat. Sb. 36 (1929), no. 2, 19–111. (Russian)
S. Finikoff, Sur les quadriques de Lie et les congruences de M. Demoulin, Mat. Sb. 38 (1931), nos. 1–2, 48–97.
S. P. Finikov, Teoriya kongurènciĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). MR 0040753
S. P. Finikov, Theory of pairs of congruences, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956 (Russian). MR 0089450
S. P. Finikov, Bending over a principal base and related geometric problems, ONTI NKTP SSSR, Moscow–Leningrad, 1937. (Russian)
N. Delaunay, Sur les surfaces n’ayant qu’un côté et sur les points singuliers des courbes planes, Bull. Soc. Math. France 26 (1898), 43–52 (French). MR 1504303
L. K. Lakhtin, A note on one-sided surfaces, Mat. Sb. 24 (1904), no. 2, 178–193. (Russian)
B. K. Mlodzeevskiĭ, Investigation into the bending of surfaces, Uchen. Zapiski Moskov. Univ. Otdel Fiz.-Mat. Nauk 1886 (1886), no. 7, 1–141. (Russian)
P. K. Rashevskiĭ, Course of differential geometry, GITTL, Moscow, 1956. (Russian)
D. F. Egorov, Boleslav Kornelievich Mlodzeevskiĭ (Obituary), Mat. Sb. 32 (1925), no. 3, 449–452. (Russian)
S. D. Rossinskiĭ, Boleslav Kornelievič Mlodzeevskiĭ, 1858–1923, Izdat. Moskov. Univ., Moscow, 1950 (Russian). MR 0041781
V. I. Bogachev, On the history of the discovery of the Egorov and Luzin theorems, Istor.-Mat. Issled. (2) 13(48) (2009), 54–67, 377 (Russian, with English summary). MR 2814590
D. F. Egorov, On one class of orthogonal systems, Uchen. Zapiski Moskov. Univ. 1901 (1901), no. 18, 1–239. (Russian)
D. F. Egorov, Raboty po differentsial′noĭ geometrii, Izdat. “Nauka”, Moscow, 1970 (Russian). Edited by S. P. Finikov. MR 0275296
G. Darboux, Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, Paris, 1910.
P. I. Kuznecov, Dmitriĭ Fedorovič Egorov (on the centenary of his birthday), Uspehi Mat. Nauk 26 (1971), no. 5(161), 169–206. (1 plate) (Russian). MR 0384434
I. M. Nikonov, A. T. Fomenko, and A. I. Shafarevich, On the works of D. F. Egorov in differential geometry, Istor.-Mat. Issled. (2) 13(48) (2009), 49–53, 377 (Russian, with English summary). MR 2814589
Yu. M. Kolyagin and O. A. Savvina, Dmitriĭ Fëdorovich Egorov. The path of a scientist and a Christian, PSTGU, Moscow, 2010. (Russian)
D. Th. Egorov, Sur les systèmes orthogonaux admettant un groupe continu de transformations de Combescure, C. R. Acad. Sci. Paris 131 (1900), 668–671; ibid. 132 (1901), 74–77.
B. A. Dubrovin and S. P. Novikov, Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov-Whitham averaging method, Dokl. Akad. Nauk SSSR 270 (1983), no. 4, 781–785 (Russian). MR 715332
S. P. Tsarëv, Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type, Dokl. Akad. Nauk SSSR 282 (1985), no. 3, 534–537 (Russian). MR 796577
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(270), 29–98, 203 (Russian); English transl., Russian Math. Surveys 44 (1989), no. 6, 35–124. MR 1037010, DOI 10.1070/RM1989v044n06ABEH002300
S. P. Tsarëv, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 5, 1048–1068 (Russian); English transl., Math. USSR-Izv. 37 (1991), no. 2, 397–419. MR 1086085, DOI 10.1070/IM1991v037n02ABEH002069
O. I. Mokhov and E. V. Ferapontov, Nonlocal Hamiltonian operators of hydrodynamic type that are connected with metrics of constant curvature, Uspekhi Mat. Nauk 45 (1990), no. 3(273), 191–192 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 3, 218–219. MR 1071942, DOI 10.1070/RM1990v045n03ABEH002351
E. V. Ferapontov, Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type, Funktsional. Anal. i Prilozhen. 25 (1991), no. 3, 37–49, 95 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 3, 195–204 (1992). MR 1139873, DOI 10.1007/BF01085489
E. V. Ferapontov, Hamiltonian systems of hydrodynamic type and their realizations on hypersurfaces of a pseudo-Euclidean space, Problems in geometry, Vol. 22 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990, pp. 59–96, 219 (Russian). Translated in J. Soviet Math. 55 (1991), no. 5, 1970–1995. MR 1099220
I. M. Krichever, Algebraic-geometric $n$-orthogonal curvilinear coordinate systems and the solution of associativity equations, Funktsional. Anal. i Prilozhen. 31 (1997), no. 1, 32–50, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 31 (1997), no. 1, 25–39. MR 1459831, DOI 10.1007/BF02466001
O. I. Mokhov, Compatible and almost compatible pseudo-Riemannian metrics, Funktsional. Anal. i Prilozhen. 35 (2001), no. 2, 24–36, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 35 (2001), no. 2, 100–110. MR 1850431, DOI 10.1023/A:1017575131419
M. V. Pavlov and S. P. Tsarev, Tri-Hamiltonian structures of Egorov systems of hydrodynamic type, Funktsional. Anal. i Prilozhen. 37 (2003), no. 1, 38–54, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 37 (2003), no. 1, 32–45. MR 1988008, DOI 10.1023/A:1022971910438
V. M. Bukhshtaber, D. V. Leĭkin, and M. V. Pavlov, Egorov hydrodynamic chains, the Chazy equation, and the group $\textrm {SL} (2,\Bbb C)$, Funktsional. Anal. i Prilozhen. 37 (2003), no. 4, 13–26, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 37 (2003), no. 4, 251–262. MR 2083228, DOI 10.1023/B:FAIA.0000015576.05085.bc
Vladimir E. Zakharov, Description of the $n$-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type. I. Integration of the Lamé equations, Duke Math. J. 94 (1998), no. 1, 103–139. MR 1635908, DOI 10.1215/S0012-7094-98-09406-6
S. P. Finikov, Sergeĭ Sergeevič (on his seventieth birthday), Uspehi Matem. Nauk (N.S.) 8 (no year given), no. 4(56), 185–192 (1 plate) (Russian). MR 0056518
A. M. Vasil′ev and G. F. Laptev, Sergeĭ Pavlovič Finikov, Uspehi Mat. Nauk 19 (1964), no. 4 (118), 155–162. (1 plate) (Russian). MR 0176904
V. T. Bazylev, On the ninetieth anniversary of the birth of S. P. Finikov, Trudy Geometr. Sem. 6 (1974), 17–24 (Russian, with English summary). MR 0432360
C. Burstin, Ein Beitrag zum Problem der Einbettung der Riemannschen Räume in euklidischen Räumen, Mat. Sb. 38 (1931), no. 3–4, 74–85.
C. Burstin, Beiträge der Verbiegung von Hyperflächen in euklidischen Räumen, Mat. Sb. 38 (1931), no. 3–4, 86–93.
I. Kh. Sabitov, On the history of an interpretation of bendings on a main base, Istor.-Mat. Issled. (2) 5(40) (2000), 164–166, 382 (Russian, with English summary). MR 1930587
V. F. Kagan, Foundations of the theory of surfaces. Part 2, OGIZ, GITTL, Moscow–Leningrad, 1948. (Russian)
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)
A. D. Aleksandrov, On infinitesimal bendings of surfaces, Mat. Sb. 1 (1936), no. 3, 307–322. (Russian)
S. Cohn-Vossen, Bendability of surfaces in the large, Uspekhi Mat. Nauk 1936 (1936), no. 1, 33–76. (Russian)
A. G. Dorfman, Solution of the bending equation for certain classes of surfaces, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 2(74), 147–150 (Russian). MR 0090077
N. V. Efimov, Bending of a neighbourhood of a parabolic point on a surface, Mat. Sb. 6 (1939), no. 3, 427–474. (Russian)
N. V. Efimov, Study of bending of a surface with a point of flattening, Mat. Sb. 19 (1946), no. 3, 461–488. (Russian)
N. V. Efimov, Research on the deformation of surfaces containing points with zero Gaussian curvature, Mat. Sbornik N.S. 23(65) (1948), 89–125 (Russian). MR 0027165
N. V. Efimov, Qualitative problems of the theory of deformations of surfaces in the small, Trudy Mat. Inst. Steklov. 30 (1949), 128 (Russian). MR 0039327
N. V. Efimov, Qualitative problems of the theory of deformation of surfaces, Uspehi Matem. Nauk (N.S.) 3 (1948), no. 2(24), 47–158 (Russian). MR 0027567
N. Efimov, On rigidity in the small, Doklady Akad. Nauk SSSR (N.S.) 60 (1948), 761–764 (Russian). MR 0027568
Encyclopedia of elementary mathematics. Book 5: Geometry, Nauka, Moscow, 1966; German transl., VEB Deutscher Verlag der Wissenschaften, Berlin, 1971.
D. V. Alekseevskij, A. M. Vinogradov, and V. V. Lychagin, Basic ideas and concepts of differential geometry, Geometry, I, Encyclopaedia Math. Sci., vol. 28, Springer, Berlin, 1991, pp. 1–264. MR 1300019
D. V. Alekseevskij, È. B. Vinberg, and A. S. Solodovnikov, Geometry of spaces of constant curvature, Geometry, II, Encyclopaedia Math. Sci., vol. 29, Springer, Berlin, 1993, pp. 1–138. MR 1254932, DOI 10.1007/978-3-662-02901-5_{1}
È. B. Vinberg and O. V. Shvartsman, Discrete groups of motions of spaces of constant curvature, Geometry, II, Encyclopaedia Math. Sci., vol. 29, Springer, Berlin, 1993, pp. 139–248. MR 1254933, DOI 10.1007/978-3-662-02901-5_{2}
Yu. D. Burago, Geometry of surfaces in Euclidean spaces, 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. 5–97 (Russian). MR 1039818
Yu. G. Reshetnyak, Two-dimensional manifolds of bounded curvature, Geometry, IV, Encyclopaedia Math. Sci., vol. 70, Springer, Berlin, 1993, pp. 3–163, 245–250. MR 1263964, DOI 10.1007/978-3-662-02897-1_{1}
V. N. Berestovskiĭ and I. G. Nikolaev, Multidimensional generalized Riemannian spaces, Geometry, 4 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 190–277, 279 (Russian). MR 1099203
I. K. Babenko, Closed geodesics, asymptotic volume and the characteristics of growth of groups, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 4, 675–711, 895 (Russian); English transl., Math. USSR-Izv. 33 (1989), no. 1, 1–37. MR 966980, DOI 10.1070/IM1989v033n01ABEH000811
I. K. Babenko, Asymptotic invariants of smooth manifolds, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 707–751 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 1–38. MR 1208148, DOI 10.1070/IM1993v041n01ABEH002181
A. T. Fomenko, The multidimensional Plateau problem in Riemannian manifolds, Mat. Sb. (N.S.) 89(131) (1972), 475–519, 534 (Russian). MR 0348599, DOI 10.1017/s0025557200178672
A. O. Ivanov and A. A. Tuzhilin, The one-dimensional Gromov minimal filling problem, Mat. Sb. 203 (2012), no. 5, 65–118 (Russian, with Russian summary); English transl., Sb. Math. 203 (2012), no. 5-6, 677–726. MR 2977099, DOI 10.1070/SM2012v203n05ABEH004239
O. I. Mokhov, Realization of Frobenius manifolds as submanifolds in pseudo-Euclidean spaces, Tr. Mat. Inst. Steklova 267 (2009), no. Osobennosti i Prilozheniya, 226–244 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 267 (2009), no. 1, 217–234. MR 2723953, DOI 10.1134/S008154380904018X
I. H. Sabitov and S. Z. Šefel, Connections between the order of smoothness of a surface and that of its metric, Sibirsk. Mat. Ž. 17 (1976), no. 4, 916–925 (Russian). MR 0425855
M. L. Gromov and V. A. Rohlin, Imbeddings and immersions in Riemannian geometry, Uspehi Mat. Nauk 25 (1970), no. 5 (155), 3–62 (Russian). MR 0290390
I. H. Sabitov, Regularity of convex domains with a metric that is regular on Hölder classes, Sibirsk. Mat. Ž. 17 (1976), no. 4, 907–915 (Russian). MR 0425854
È. 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. Obshch. 53 (1990), 171–191, 261 (Russian); English transl., Trans. Moscow Math. Soc. (1991), 177–197. MR 1097996
I. H. Sabitov, The rigidity of “corrugated” surfaces of revolution, Mat. Zametki 14 (1973), 517–522 (Russian). MR 355911
N. V. Efimov and Z. D. Usmanov, Infinitesimal bendings of a surface with a point of flattening, Dokl. Akad. Nauk SSSR 208 (1973), 28–31 (Russian). MR 0315603
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
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)
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
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, DOI 10.1007/BF01249526
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
È. G. Poznyak, On nonrigidity of the second order, Uspekhi Mat. Nauk 16 (1961), no. 1, 157–161. (Russian)
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
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
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, DOI 10.1070/sm2013v204n10abeh004347
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, DOI 10.1070/sm2014v205n12abeh004440
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
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 420522
I. Ivanova-Karatopraklieva and I. Kh. Sabitov, Bending of surfaces. II, J. Math. Sci. 74 (1995), no. 3, 997–1043. Geometry, 1. MR 1330961, DOI 10.1007/BF02362831
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, DOI 10.1007/s10958-008-0033-0
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
N. V. Efimov, Generation of singularites on surfaces of negative curvature, Mat. Sb. (N.S.) 64 (106) (1964), 286–320 (Russian). MR 0167938
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)
Remembering Nikolaĭ Vladimirovich Efimov..., Moscow Centre for Contin. Math. Educ., Moscow, 2014. (Russian)
È. R. Rozendorn, The construction of a bounded, complete surface of nonpositive curvature, Uspehi Mat. Nauk 16 (1961), no. 2 (98), 149–156 (Russian). MR 0131847
È. R. Rozendorn, Surfaces of negative curvature, 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. 98–195 (Russian). MR 1039819
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, DOI 10.1070/RM1993v048n04ABEH001065
È. 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.
E. V. Šikin, Isometric imbeddings in $E^{3}$ of noncompact domains of nonpositive curvature, Problems in geometry, Vol. 7 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1975, pp. 249–266, 302. (errata insert) (Russian, with English summary). MR 0500744
E. V. Shikin, On the equations of isometric immersions of two-dimensional manifolds of negative curvature in a three-dimensional Euclidean space, Mat. Zametki 31 (1982), no. 4, 601–612, 655 (Russian). MR 657721
D. V. Tunitskiĭ, Regular isometric immersion in $E^3$ of unbounded domains of negative curvature, Mat. Sb. (N.S.) 134(176) (1987), no. 1, 119–134, 143 (Russian); English transl., Math. USSR-Sb. 62 (1989), no. 1, 121–138. MR 912415, DOI 10.1070/SM1989v062n01ABEH003230
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
È. G. Poznjak and E. V. Šikin, Surfaces of negative curvature, Algebra. Topology. Geometry, Vol. 12 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1974, pp. 171–207 (Russian). MR 0407777
È. G. Poznjak and D. D. Sokolov, Isometric immersions of Riemannian spaces in Euclidean spaces, Algebra. Topology. Geometry, Vol. 15 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1977, pp. 173–210 (errata and summary inserts) (Russian). MR 0482596
È. G. Poznyak and E. V. Shikin, Small parameter in the theory of isometric imbeddings for two-dimensional Riemannian manifolds into Euclidean spaces, J. Math. Sci. 74 (1995), no. 3, 1078–1116. Geometry, 1. MR 1330963, DOI 10.1007/BF02362833
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
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
N. N. Yanenko, On the theory of the imbedding of surfaces in a multi-dimensional Euclidean space, Trudy Moskov. Mat. Obšč. 3 (1954), 89–180 (Russian). MR 0063096
I. Kh. Sabitov, Isometric immersions and embeddings of locally Euclidean metrics in $\mathbf R^2$, 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, DOI 10.1070/im1999v063n06ABEH000270
I. Kh. Sabitov, Isometric embedding of locally Euclidean metrics in $\textbf {R}^3$, Sibirsk. Mat. Zh. 26 (1985), no. 3, 156–167, 226 (Russian). MR 792065
S. N. Mikhalev and I. Kh. Sabitov, Isometric embeddings of locally Euclidean metrics in $\Bbb R^3$ as conical surfaces, Math. Notes 95 (2014), no. 1-2, 212–223. Translation of Mat. Zametki 95 (2014), no. 2, 234–247. MR 3267209, DOI 10.1134/S0001434614010234
S. N. Mikhalev and I. Kh. Sabitov, Isometric embeddings in $\Bbb R^3$ 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, DOI 10.4213/mzm10815
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, DOI 10.1007/s11202-009-0102-8
I. Kh. Sabitov, On the extrinsic curvature and the extrinsic structure of $C^1$-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, DOI 10.1134/S0001434610050275
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
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, DOI 10.1134/S0081543808040160
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, DOI 10.1134/S0081543809030158
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, DOI 10.1070/RM2007v062n02ABEH004406
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)
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, DOI 10.1134/S0081543811080086
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
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, DOI 10.1070/rm1999v054n04ABEH000187
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, DOI 10.1070/rm1999v054n05ABEH000218
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, DOI 10.1134/s0037446614050140
I. Kh. Sabitov, Volumes of polyhedra, 2nd ed., Moscow Centre for Contin. Math. Educ., Moscow, 2009. (Russian)
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
M. I. Shtogrin, On flexible polyhedral surfaces, Proc. Steklov Inst. Math. 288 (2015), no. 1, 153–164. Translation of Tr. Mat. Inst. Steklova 288 (2015), 171–183. MR 3485708, DOI 10.1134/S0081543815010125
V. M. Buchstaber and N. Yu. Erokhovets, Truncations of simple polytopes and applications, Proc. Steklov Inst. Math. 289 (2015), no. 1, 104–133. Translation of Tr. Mat. Inst. Steklova 289 (2015), 115–144. MR 3486778, DOI 10.1134/S0081543815040070
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
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, DOI 10.1070/SM1998v189n10ABEH000354
I. Kh. Sabitov, Solution of cyclic polygons, Math. Educ. 3rd Ser., no. 11, Moscow Centre Contin. Math. Educ., Moscow, 2010, 83–106. (Russian)
Alexander A. Gaifullin, Sabitov polynomials for volumes of polyhedra in four dimensions, Adv. Math. 252 (2014), 586–611. MR 3144242, DOI 10.1016/j.aim.2013.11.005
Alexander A. Gaifullin, Generalization of Sabitov’s theorem to polyhedra of arbitrary dimensions, Discrete Comput. Geom. 52 (2014), no. 2, 195–220. MR 3249379, DOI 10.1007/s00454-014-9609-2
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, DOI 10.4213/sm8522
Alexander A. Gaifullin, Embedded flexible spherical cross-polytopes with nonconstant volumes, Proc. Steklov Inst. Math. 288 (2015), no. 1, 56–80. Published in Russian in Tr. Mat. Inst. Steklova 288 (2015), 67–94. MR 3485701, DOI 10.1134/S0081543815010058
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, DOI 10.1070/IM2002v066n02ABEH000382
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, DOI 10.1070/RM2011v066n03ABEH004748