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The scientific heritage of I. P. Egorov (July 25, 1915 – October 2, 1990). (English) Zbl 0862.01020
The article under review describes the scientific heritage of I. P. Egorov (July 25, 1915 – October 2, 1990). The principal object of Egorov’s research was the theory of lacunas in motion group orders and the geometries of their corresponding lacunary spaces. Starting with the Fubini theorem stating that there are no Riemannian spaces of dimension \(n\) with a complete isometry group \(G_r\) of order \(r=n(n+1)-1\) Egorov obtained his famous result that there exists no affine connection space admitting complete isometry group \(G_r\) of orders \(n^2< r<n^2+n\), finding an interval of “forbidden” isometry group orders named lacuna. Later Egorov determined the exact boundaries of some lacunas and characterized the spaces of the corresponding lacunarities. The article under review gives a complete account of these researches and corresponding methods based on Egorov’s study of isometry equation integrability conditions as well as some other Egorov’s results.

01A70 Biographies, obituaries, personalia, bibliographies
01A60 History of mathematics in the 20th century
Biographic References:
Egorov, I. P.
Full Text: DOI
[1] I. P. EGOROV ”Isometry groups of affine connection space,” Cand. Thesis, Kazan, (1945).
[2] – ”On the order of isometry groups of affine connection spaces,”Dokl. Akad. Nauk SSSR,57, No. 9, 867–870 (1947).
[3] – ”On collineations of projective connection spaces,”Dokl. Akad. Nauk SSSR,61, No. 4 605–608 (1948). · Zbl 0032.35401
[4] – ”On isometry groups of nonsymmetrical affine connection spaces,”Dokl. Akad. Nauk SSSR,64, No. 5, 621–624 (1949).
[5] – ”On strengthening the Fubini theorem about the order of the Riemannian space isometry group,”Dokl. Akak. Nauk SSSR,66, No. 5, 793–796 (1949). · Zbl 0038.34601
[6] – ”Isometry groups of affine connection spaces,”Izv. Kazan Fiz.-Mat. Ob-va, (Kazan’),14, No. 3, 53–72 (1949).
[7] – ”On isometry groups of spaces of general nonsymmetrical affine connection spaces,”Dokl. Akad. Nauk SSSR,72, No. 2, 265–267 (1950).
[8] I. P. EPGOROV ”On collineations of projective connection spaces” [in Russian], In:Trudy Seminara po Vectornomu i Tenzornomu Analizu, Moscow-Leningrad,8, No. 6 (1950), pp. 9–10.
[9] – ”Collineations of projective connection spaces,”Dokl. Akad. Nauk SSSR,80, No. 5, 709–712 (1951).
[10] – ”Tensor characteristic of maximally mobileA n of nonzero curvature,”Dokl. Akad. Nauk SSSR,84, 2, 209–212 (1952).
[11] – ”Maximally mobile semisymmetrical connectionL n ,”Dokl. Akad. Nauk SSSR,84, No. 3, 433–435 (1952).
[12] – ”Isometries in affine connection spaces”,Dokl. Akad. Nauk SSSR,87, No. 5, 693–696 (1952). · Zbl 0049.11904
[13] – ”On isometries in affine connection spaces,”Dokl. Akad. Nauk SSSR,89, No. 5, 781–784 (1953). · Zbl 0051.13202
[14] – ”Isometries in affine connection spaces,”Uspekhi Mat. Nauk,4 (56), 182–183 (1953).
[15] I. P. EGOROV, ”Isometries in affine connection spaces,”Doctoral Thesis, Moscow, (1955). · Zbl 0064.33602
[16] I. P. EGOROV, ”Equiaffine third-lacunarity spaces,” In:Tr. III Vsesoyuz. Mat. Sjezda, Moscow, Akad. Nauk SSSR Press (1956) pp. 151–152. · Zbl 0073.38903
[17] – ”Maximally mobile nonconstant curvature Riemannian spacesV n ,”Dokl. Akad. Nauk SSSR,103, No. 1, 9–12 (1955).
[18] – ”Equiaffine third-lacunarity spaces,”Dokl. Akad. Nauk SSSR,108, No. 6, 1007–1010 (1956). · Zbl 0073.38903
[19] – ”Second lacunarity Riemannian spaces,”Dokl. Akad. Nauk SSSR,111, No. 2, 276–279 (1955). · Zbl 0073.17102
[20] I. P. EGOROV Review of Yano, ”Lie theory of derivation and its applications,” In:Novyye Knigi za Rubezhom,8, 11–15 (1958).
[21] I. P. EGOROV ”Isometries and homotheties in the Riemannian spaces,” In:Tr. 2 Konferentsii Mat. Kafedr Pedinstitutov Povolzhya, Kuibyshev (1962), pp. 138–145.
[22] – ”Maximally mobile Einsteinian spaces of nonconstant curvature,”Dokl. Akad. Nauk SSSR,145, No. 5, 975–978 (1962).
[23] – ”On the Riemannian spaces of three first lacunarities in the homothetic sense,”Dokl. Akad. Nauk SSSR,145, No. 4, 730–732 (1963).
[24] I. P. EGOROV ”On homothetic Riemannian spaces,” In:Trudy Pervoi Nauchnoi Sessii po Koordinatsii Nauchno-Iseledovatel’shikh Rabot”, Kazan’ (1963), pp. 78–79.
[25] – ”Romothetic Riemannian spaces,”Lit. Mat. Sb.,2, 223–224 (1963).
[26] – ”On homothetic isometries in nonreducible first-class symmetrical Riemannian spaces,”Volzhskii Mat. Sb., No. 1, 61–65 (1963).
[27] I. P. EGOROV ”Isometries and homotheties in Riemannian spaces,” In:Trudy Vtoroi Vsesoyuzn. Geometricheskoi Konferentsii, Khar’kov (1964). · Zbl 0139.39001
[28] – ”On a class of kernel functions,”Volzhskii Mat. Sb., No. 3–6, 145–148 (1965). · Zbl 0131.12903
[29] – ”On a class of kernel functions invariant under conformal metric mappings,”Volzhkii Mat. Sb., No. 4, 53–58 (1965).
[30] I. P. EGOROV ”On homothetic Keller-Shirokov metrics,” In:Materialy Vtoroi Pribaltiiskoi Geometricheskoi Konferentsii po Voprosam Differentsial’noi Geometrii, Tartu (1965), pp. 67–68.
[31] I. P. EGOROV ”Isometries in affine connection spaces,” In:Uchen. Zap. Penzen. Ped. In-ta, Kazan’, Izd. Kazanskogo Univ. (1965), pp. 3–179.
[32] I. P. EGOROV ”On a class of generalized differential geometric spaces of two first lacunarities,” In:Materialy 3 Pribaltiiskoi Geometricheskoi Konferentsii, Palanga (1968), pp. 61–63.
[33] – ”On some problems in the theory of Riemannian space isometrices,”Uchen. Zap. Penzen. Inst.,67, 187–191 (1967).
[34] I. P. EGOROV ”On homothetic isometries in Riemannian space dimensionless surface,” In:3 Mezhvuzovskaya Nauchnaya Konferentsiya po Problemam Geometrii, Kazan’ (1967), pp. 52–53.
[35] I. P. EGOROV ”On lacunas and lacunary spaces in isometry theory,” In:IV Vsesyuzn. Mezhvuz. Konferentsiya po Geometrii, Tbilisi (1969), pp. 69–72.
[36] – ”On isometries in Riemannian and affine connection spaces,”Uchen. Zap. Penzen. Ped. Inst.,124, 3–9 (1971).
[37] I. P. EGOROV ”On isometries in generalized affine connection spaces” (in collaboration with A. I. Egorov), In:Materialy 2oi Uyb. Nauchnoi Konferentsii Kirgiz. Univ., Frunze (1971), pp. 104–107.
[38] – ”On generalized affine connection spaces,” (in collaboration with A. I. Egorov),Uchen. Zap. Penzen. Ped. Inst.,124, 10–12 (1971).
[39] I. P. EGOROV ”On isometries and the problem of lacunarity in differential geometry spaces,” In:V Vsesouns. Konf. po Sovremennym Problemam Geometrii, Samarkand (1972), pp. 64.
[40] – ”On lacunary differential geometry spaces”,Rev. Roum. Math. Pures Appl.,15, No. 9 1365–1373 (1970). · Zbl 0215.51304
[41] I. P. EGOROV ”Isometries on spaces of linear and hypersurface elements of generalized affine connection” (in collaboration with A. I. Egorov), In:Materialy IV Pribaltiiskoi Geometricheskoi Konferentsii, Tartu (1973), pp. 35–36.
[42] I. P. EGOROV ”On global structure of maximally mobile nonplanar affine connection spaces,” In:IV Vsesoyuznaiya Geometricheskaya Konf. po Sovr. Probl. Geom., Vilnis (1975), pp. 89–91.
[43] I. P. EGOROVIsometries in Generalized Spaces [in Russian], Ryazan’ (1977).
[44] I. P. EGOROV ”Authomorphisms and homothetic transformations inC-spaces” (in collaboration with L. I. Egorova), In:Vsesoyuz. Nauchnaya Konferentsiya po Neevklidovoi Geometrii, Kazan’ (1976), p. 74.
[45] I. P. EGOROV ”Automorphisms of generalized spaces,” In:150 Let Geometrii Lobachevskogo, Kazan’ (1976), pp. 57–73.
[46] I. P. EGOROV ”On automorphisms in spaces of vector and covector densities,” In: VIIVsesoyuz. Konferentsiya po Sovremennym Problemam Geometrii, Minsk (1979), p. 64.
[47] –Geometry [in Russian], Prosveshchenie, Moscow (1979).
[48] I. P. EGOROV ”Some problems of automorphisms in generalized spaces” (in collaboration with A. I. Egorov), In:Dvizheniya v Obobshchennykh Prostranstvakh: Sb. Nauch. Trudov (1982), pp. 41–52.
[49] I. P. EGOROV ”On homothetic mobile Finsler spaces,” In: VIIIVsesoyuz. Nauch. Konferentsiya po Sovremennym Problemam Differentsial’noi Geometrii, Odessa (1984), p. 48.
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