| Title:
|
Algebraic aspects of web geometry (English) |
| Author:
|
Akivis, Maks A. |
| Author:
|
Goldberg, Vladislav V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
41 |
| Issue:
|
2 |
| Year:
|
2000 |
| Pages:
|
205-236 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Algebraic aspects of web geometry, namely its connections with the quasigroup and loop theory, the theory of local differential quasigroups and loops, and the theory of local algebras are discussed. (English) |
| Keyword:
|
quasigroup |
| Keyword:
|
loop |
| Keyword:
|
web |
| Keyword:
|
group |
| Keyword:
|
local quasigroup |
| Keyword:
|
local loop |
| Keyword:
|
Akivis algebra |
| Keyword:
|
$n$-quasigroup |
| MSC:
|
17D99 |
| MSC:
|
20N05 |
| MSC:
|
22A30 |
| MSC:
|
53A60 |
| idZBL:
|
Zbl 1042.53007 |
| idMR:
|
MR1780866 |
| . |
| Date available:
|
2009-01-08T19:00:28Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119158 |
| . |
| Reference:
|
[Ac 65] Aczel J.: Quasigroups, nets and nomograms.Adv. Math. 1 (1965), 3 383-450. Zbl 0135.03601, MR 0193174 |
| Reference:
|
[A 69a] Akivis M.A.: The canonical expansions of the equations of a local analytic quasigroup.Dokl. Akad. Nauk SSSR 188 (1969), no. 5, 967-970 (Russian); English transl.: Soviet Math. Dokl. 10 (1969), no. 5, 1200-1203. MR 0262413 |
| Reference:
|
[A 69b] Akivis M.A.: Three-webs of multidimensional surfaces.Trudy Geometr. Sem. 2 (1969), 7-31 (Russian). MR 0254760 |
| Reference:
|
[A 73a] Akivis M.A.: The local differentiable quasigroups and three-webs that are determined by a triple of hypersurfaces.Sibirsk. Mat. Zh. 14 (1973), no. 3, 467-474 (Russian); English transl.: Siberian Math. J. 14 (1973), no. 3, 319-324. MR 0324559 |
| Reference:
|
[A 73b] Akivis M.A.: Local differentiable quasigroups and three-webs of multidimensional surfaces.in ``Studies in the Theory of Quasigroups and Loops'', pp.3-12 (Russian), Izdat. ``Shtiintsa'', Kishinev, 1973. MR 0370391 |
| Reference:
|
[A 75] Akivis M.A.: Closed $G$-structures on a differentiable manifold.Problems in Geometry, Vol. 7, 69-79 (Russian), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1975. Zbl 0549.53032, MR 0478055 |
| Reference:
|
[A 76] Akivis M.A.: The local algebras of a multidimensional three-web.Sibirsk. Mat. Zh. 17 (1976), no. 1, 5-11 (Russian); English transl.: Siberian Math. J. 17 (1976), no. 1, 3-8. MR 0405261 |
| Reference:
|
[A 78] Akivis M.A.: Geodesic loops and local triple systems in a space with an affine connection.Sibirsk. Mat. Zh. 19 (1978), no. 2, 243-253 (Russian); English transl.: Siberian Math. J. 19 (1978), no. 2, 171-178. MR 0487840 |
| Reference:
|
[AG 74] Akivis M.A., Goldberg V.V.: The $4$-web and the local differentiable ternary quasigroup that are determined by a quadruple of surfaces of codimension two.Izv. Vyssh. Uchebn. Zaved. Mat. 1974, no. 5(144), 12-24 (Russian); English transl.: Soviet Math. (Iz. VUZ) 18 (1974), no. 5, 9-19. MR 0355847 |
| Reference:
|
[AG 00] Akivis M.A., Goldberg V.V.: Differential geometry of webs.Chapter 1 in ``Handbook of Differential Geometry'', Vol. 1, Elsevier, 2000, pp.1-152. Zbl 0968.53001, MR 1736852 |
| Reference:
|
[AS 71] Akivis M.A., Shelekhov A.M.: Local differentiable quasigroups and connections that are associated with a three-web of multidimensional surfaces.Sibirsk. Mat. Zh. 12 (1971), no. 6, 1181-1191 (Russian); English transl.: Siberian Math. J. 12 (1971), no. 6, 845-852. MR 0288681 |
| Reference:
|
[AS 92] Akivis M.A., Shelekhov A.M.: Geometry and Algebra of Multidimensional Three-Webs.translated from the Russian by V.V. Goldberg, Kluwer Academic Publishers, Dordrecht, 1992, xvii+358 pp. Zbl 0771.53001, MR 1196908 |
| Reference:
|
[Ba 90] Barlotti A.: Geometry of quasigroups.Chapter VIII in ``Quasigroups and Loops: Theory and Applications'', O. Chein, H.O. Pflugfelder, and J.D.H. Smith, Eds., Heldermann-Verlag, Berlin, 1990, pp.197-203. Zbl 0723.20041, MR 1125814 |
| Reference:
|
[BS 83] Barlotti A., Strambach K.: The geometry of binary systems.Adv. Math. 49 (1983), 1 1-105. Zbl 0518.20064, MR 0715128 |
| Reference:
|
[Be 67] Belousov V.D.: Foundations of the Theory of Quasigroups and Loops.Izdat. ``Nauka'', Moscow, 1967, 223 pp. (Russian). MR 0218483 |
| Reference:
|
[Be 71] Belousov V.D.: Algebraic Nets and Quasigroups.Izdat. ``Shtiintsa'', Kishinev, 1971, 165 pp. (Russian). MR 0340459 |
| Reference:
|
[Be 72] Belousov V.D.: $n$-ary Quasigroups.Izdat. ``Shtiintsa'', Kishinev, 1972, 227 pp. (Russian). MR 0354919 |
| Reference:
|
[BSa 66] Belousov V.D., Sandik M.D.: $n$-ary quasigroups and loops.Sibirsk. Mat. Zh. 7 (1966), no. 1, 31-54 (Russian); English transl.: Siberian Math. J. 7 (1966), no. 1, 24-42. MR 0204564 |
| Reference:
|
[Bl 28] Blaschke W.: Thomsens Sechseckgewebe. Zueinander diagonale Netze.Math. Z. 28 (1928), 150-157. MR 1544947 |
| Reference:
|
[Bl 47] Blaschke W.: Projective Geometrie.Wolfenbüttel, Hannover, 1947, 160 pp.; 2nd ed., Wolfenbüttel, Hannover, 1948, 160 pp.; 3rd ed., Birkhäuser, Basel-Stuttgart, 1954, 197 pp. |
| Reference:
|
[Bl 55] Blaschke W.: Einführung in die Geometrie der Waben.Birkhäuser-Verlag, Basel-Stuttgart, 1955, 108 pp.; Russian transl.: GITTL, Moskva, 1959, 144 pp. MR 0075630 |
| Reference:
|
[BB 38] Blaschke W., Bol G.: Geometrie der Gewebe.Springer-Verlag, Berlin, 1938, viii+339 pp. Zbl 0020.06701 |
| Reference:
|
[B 35] Bol G.: Über 3-Gewebe in vierdimensionalen Raum.Math. Ann. 110 (1935), 431-463. MR 1512949 |
| Reference:
|
[B 37] Bol G.: Gewebe und Gruppen.Math. Ann. 114 (1937), 414-431. Zbl 0016.22603, MR 1513147 |
| Reference:
|
[C 36] Chern S.S.: Eine Invariantentheorie der Dreigewebe aus $r$-dimensionalen Mannigfaltigkeiten in ${\bold R}_{2r}$.Abh. Math. Sem. Univ. Hamburg 11 (1936), 1-2 333-358. |
| Reference:
|
[F 78] Fedorova V.I.: A condition defining multidimensional Bol's three-webs.Sibirsk. Mat. Zh. 19 (1978), no. 4, 922-928 (Russian); English transl.: Siberian Math. J. 19 (1978), no. 4, 657-661. MR 0514146 |
| Reference:
|
[G 73] Goldberg V.V.: $(n+1)$-webs of multidimensional surfaces.Dokl. Akad. Nauk SSSR 210 (1973), no. 4, 756-759 (Russian); English transl.: Soviet Math. Dokl. 14 (1973), no. 3, 795-799. MR 0324567 |
| Reference:
|
[G 75a] Goldberg V.V.: An invariant characterization of certain closure conditions in ternary quasigroups.Sibirsk. Mat. Zh. 16 (1975), no. 1, 29-43 (Russian); English transl.: Siberian Math. J. 16 (1975), no. 1, 23-34. MR 0370392 |
| Reference:
|
[G 75b] Goldberg V.V.: Local ternary quasigroups that are connected with a four-web of multidimensional surfaces.Sibirsk. Mat. Zh. 16 (1975), no. 2, 247-263 (Russian); English transl.: Siberian Math. J. 16 (1975), no. 2, 190-202. MR 0372101 |
| Reference:
|
[G 75c] Goldberg V.V.: The $(n+1)$-webs defined by $n+ 1$ surfaces of codimension $n-1$.Probl. Geom., Vol. 7 (Russian), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1975, pp.173-195. Zbl 0547.53007, MR 0478042 |
| Reference:
|
[G 76] Goldberg V.V.: Reducible, group, and $(2n+2)$-hedral $(n+1)$-webs of multidimensional surfaces.Sibirsk. Mat. Zh. 17 (1976), no. 1, 44-57 (Russian); English transl.: Siberian Math. J. 17 (1976), no. 1, 34-44. MR 0417940 |
| Reference:
|
[G 77] Goldberg V.V.: On the theory of four-webs of multidimensional surfaces on a differentiable manifold $X_{2r}$.Izv. Vyssh. Uchebn. Zaved. Mat. 1977, no. 11(186), 15-22 (Russian); English transl.: Soviet Math. (Iz. VUZ) 21 (1977), no. 11, 97-100. |
| Reference:
|
[G 80] Goldberg V.V.: On the theory of four-webs of multidimensional surfaces on a differentiable manifold $X_{2r}$.Serdica 6 (1980), 2 105-119 (Russian). MR 0601349 |
| Reference:
|
[G 82a] Goldberg V.V.: Multidimensional four-webs on which the Desargues and triangle figures are closed.Geom. Dedicata 12 (1982), 3 267-285. Zbl 0488.53009, MR 0661533 |
| Reference:
|
[G 82b] Goldberg V.V.: Grassmann and algebraic four-webs in a projective space.Tensor (N.S.) 38 (1982), 179-197. Zbl 0513.53009, MR 0832646 |
| Reference:
|
[G 88] Goldberg V.V.: Theory of Multicodimensional $(n+1)$-Webs.Kluwer Academic Publishers, Dordrecht, 1988, xxii+466 pp. Zbl 0668.53001, MR 0998774 |
| Reference:
|
[G 90] Goldberg V.V.: Local differentiable quasigroups and webs.Chapter X in ``Quasigroups and Loops: Theory and Applications'' O. Chein, H.O. Pflugfelder, and J.D.H. Smith, Eds., Heldermann-Verlag, Berlin, 1990, pp.263-311. Zbl 0737.53015, MR 1125816 |
| Reference:
|
[Go 99] Goursat E.: Sur les équations du second ordre à $n$ variables, analogues à l'équation de Monge-Ampère.Bull. Soc. Math. France 27 (1899), 1-34. MR 1504329 |
| Reference:
|
[HS 86a] Hofmann K.H., Strambach K.: Lie's fundamental theorems for local analytical loops.Pacific J. Math. 123 (1986), 2 301-327. Zbl 0596.22002, MR 0840846 |
| Reference:
|
[HS 86b] Hofmann K.H., Strambach K.: The Akivis algebra of a homogeneous loop.Mathematika 33 (1986), 1 87-95. Zbl 0601.22002, MR 0859501 |
| Reference:
|
[HS 90] Hofmann K.H., Strambach K.: Topological and analytic loops.Chapter IX in ``Quasigroups and Loops: Theory and Applications'', O. Chein, H.O. Pflugfelder, and J.D.H. Smith, Eds., Heldermann-Verlag, Berlin, 1990, pp.205-262. Zbl 0747.22004, MR 1125815 |
| Reference:
|
[Kn 32] Kneser H.: Gewebe und Gruppen.Abh. Math. Sem. Univ. Hamburg 9 (1932), 147-151. Zbl 0006.03302 |
| Reference:
|
[Ku 70] Kuz'min E.N.: La relation entre les algebras de Malcev et les boucles de Moufang analytiques.C.R. Acad. Sci. Paris Sér. A 271 (1970), 23 1152-1155. MR 0269766 |
| Reference:
|
[Ku 71] Kuz'min E.N.: On relation between Mal'tsev algebras and analytic Moufang loops.Algebra i Logika 10 (1971), 1 3-22 (Russian) English transl.: Algebra and Logic 10 (1971), 1 1-14. MR 0297916 |
| Reference:
|
[Lie 93] Lie S.: Vorlesungen über kontinuierliche Gruppen mit geometrischen und anderen Anwendungen.Teubner, Leipzig, 1893, xii+810 pp.; reprint Chelsea Publishing Co., Bronx, N.Y., 1971, xii+810 pp. MR 0392458 |
| Reference:
|
[Ma 55] Mal'cev A.I.: Analytical loops.Mat. Sb. 36 (1955), 569-576 (Russian). MR 0069190 |
| Reference:
|
[Mi 96] Mikheev P.O.: On a problem of Chern-Akivis-Shelekhov on hexagonal three-webs.Aequationes Math. 51 (1996), 1-2 1-11. Zbl 0845.53011, MR 1372779 |
| Reference:
|
[MS 90] Mikheev P.O., Sabinin L.V.: Quasigroups and differential geometry.Chapter XII in ``Quasigroups and Loops: Theory and Applications'', O. Chein, H.O. Pflugfelder, and J.D.H. Smith, Eds., Heldermann-Verlag, Berlin, 1990, pp.357-430. Zbl 0721.53018, MR 1125818 |
| Reference:
|
[Mo 35] Moufang R.: Zur Struktur von Alternativ Körpern.Math. Ann. 110 (1935), 416-430. MR 1512948 |
| Reference:
|
[NS 94] Nagy P., Strambach K.: Loops as invariant sections in groups, and their geometry.Canad. J. Math. 46 (1994), 5 1027-1056. Zbl 0814.20055, MR 1295130 |
| Reference:
|
[P 75] Pickert G.: Projective Ebenen.2nd edition, Springer-Verlag, Berlin, 1975, ix+371 pp. MR 0370350 |
| Reference:
|
[Ra 60] Rado F.: Generalisation of space webs for certain algebraic structures.Studia Univ. Babeş-Bolyai Ser. Math. Phys. 1960 1, pp.41-55 (Roumanian). MR 0182913 |
| Reference:
|
[Re 28] Reidemeister K.: Gewebe und Gruppen.Math. Z. 29 (1928), 427-435. |
| Reference:
|
[Re 68] Reidemeister K.: Vorlesungen Über Grundlagen der Geometrie.Springer-Verlag, Berlin-New York, 1968, x+148 pp. Zbl 0167.18701, MR 0223957 |
| Reference:
|
[Sab 98] Sabinin L.V.: Quasigroups, geometry and nonlinear geometric algebra.Acta Appl. Math. 50 (1998), 1-2 45-66. Zbl 0903.20038, MR 1608635 |
| Reference:
|
[SM 82] Sabinin L.V., Mikheev P.O.: Analytical Bol loops.Webs and Quasigroups (Russian), Kalinin. Gos. Univ., Kalinin, 1982, pp.102-109. Zbl 0499.20044, MR 0674312 |
| Reference:
|
[SM 85] Sabinin L.V., Mikheev P.O.: The Theory of Smooth Bol Loops.Univ. Druzhby Narodov, Moscow, 1985, 81 pp. MR 0831661 |
| Reference:
|
[Sag 61] Sagle A.A.: Mal'tsev algebras.Trans. Amer. Math. Soc. 101 (1961), 3 426-458. MR 0143791 |
| Reference:
|
[San 65] Sandik M.D.: Completely reducible $n$-quasigroups.Izv. Akad. Nauk Moldav. SSR Ser. Fiz.-Tekhn. Mat. Nauk 7 (1965), 55-67 (Russian). MR 0194542 |
| Reference:
|
[S 89a] Shelekhov A.M.: Classification of multidimensional three-webs according to closure conditions.Probl. Geom. 21, 109-154, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989 (Russian); English transl.: Soviet Math. 55 (1991), no. 6, 2140-2168. Zbl 0705.53014, MR 1027856 |
| Reference:
|
[S 89b] Shelekhov A.M.: The $G$-structures associated to a hexagonal $3$-web is closed.J. Geom. 35 (1989), 169-176. MR 1002653 |
| Reference:
|
[S 89c] Shelekhov A.M.: On the closed $G$-structure associated to a hexagonal $3$-web.in ``Differential Geometry and Its Applications'', Dubrovnik, 1988, Univ. Novi Sad, Novi Sad, 1989, pp.323-326. Zbl 0699.53024, MR 1040080 |
| Reference:
|
[Sh 99a] Shestakov I.P.: Any Akivis algebra is linear.Geom. Dedicata 77 (1999), 2 215-223. MR 1713296 |
| Reference:
|
[Sh 99b] Shestakov I.P.: Linear representability of Akivis algebras.Dokl. Akad. Nauk 368 (1999), 1 21-23. Zbl 1034.17500, MR 1726261 |
| Reference:
|
[Sh ta] Shestakov I.P.: Speciality and deformation of algebras.Proceedings of the Intern. Algebraic Conf. dedicated to the memory of A.G. Kurosh, Moscow, 1998, to appear. MR 1754680 |
| Reference:
|
[Su 37] Sushkevich A.K.: Theory of Generalized Groups.GNTI, Kharkiv-Kiev, 1937, 176 pp. |
| Reference:
|
[Sm 88] Smith J.D.H.: Multilinear algebras and Lie's theorem for formal $n$-loops.Arch. Math. (Basel) 51 (1988), 169-177. Zbl 0627.22003, MR 0959394 |
| Reference:
|
[T 27] Thomsen G.: Un teoreme topologico sulle schiere di curve e una caratterizzazione geometrica delle superficie isotermo-asintotiche.Boll. Un. Mat. Ital. Bologna 6 (1927), 80-85. |
| . |
