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On an algebraical computation of the tensor and the curvature for 3-web. (English) Zbl 1250.53011

Summary: Algebraic methods are used in web geometry, in particular in 3-web geometry. Along this line, we suggest a new, alternative algebraic method for computation of the quantities \(\overset{1}{\nabla}_l a^i_{jk}, \overset{2}{\nabla}_l a^i_{jk}\), and \(d^i_{jklm}\) by means of the embedding of local loops into Lie groups.

MSC:

53A60 Differential geometry of webs
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[1] G. Darboux, “Sur le contact des courbes et des surfaces,” Bulletin de la Société Mathématique de France, vol. 4, pp. 348-384, 1880. · JFM 12.0590.01
[2] G. Thomsen, “Un teoreme topologico sulle schiere di curve e una caratteriz-zazione geometrica sulle superficie isotermoasintotiche,” Bollettino della Unione Matematica Italiana, vol. 6, pp. 80-85, 1927. · JFM 53.0653.02
[3] J. V. Pereira and L. Pirio, “The classification of exceptional CDQL webs on compact complex surfaces,” International Mathematics Research Notices, no. 12, pp. 2169-2282, 2010. · Zbl 1208.53011 · doi:10.1093/imrn/rnp208
[4] J. V. Pereira and C. Perrone, “Germs of integrable forms and varieties of minimal degree,” Bulletin des Sciences Mathématiques, vol. 134, no. 1, pp. 1-11, 2010. · Zbl 1197.32013 · doi:10.1016/j.bulsci.2009.09.005
[5] J. V. Pereira and P. Sad, “Rigidity of fibrations,” in Equations Differentielles et Singularites, Asterisque, Number 323, pp. 291-299, Amer Mathematical Society, 2009. · Zbl 1203.37082
[6] F. Cukierman, J. V. Pereira, and I. Vainsencher, “Stability of foliations induced by rational maps,” Annales de la Faculté des Sciences de Toulouse. Mathématiques, vol. 18, no. 4, pp. 685-715, 2009. · Zbl 1208.32029 · doi:10.5802/afst.1221
[7] J. V. Pereira and L. Pirio, An Invitation to Web Geometry. From Abel’s Addition Theorem to the Algebraization of Codimension One Webs, IMPA, Rio de Janeiro, Brazil, 2009. · Zbl 1184.53002
[8] J. V. Pereira, “Algebraization of Codimension One Webs,” in Sminaire Bourbaki, vol. 2006/2007 of Astrisque No. 317, IMPA, 2008.
[9] J. V. Pereira, “Luc Classification des tissus exceptionnels quasilinaires compltement dcomposables,” Comptes Rendus. Mathématique. Académie des Sciences, Paris, vol. 346, no. 19-20, pp. 1093-1098, 2008. · Zbl 1153.53014 · doi:10.1016/j.crma.2008.09.013
[10] J. V. Pereira and C. Perrone, “Germs of integrable forms and varieties of minimal degree,” Bulletin des Sciences Mathématiques, vol. 134, no. 1, pp. 1-11, 2010. · Zbl 1197.32013 · doi:10.1016/j.bulsci.2009.09.005
[11] V. V. Goldberg and V. V. Lychagin, “Samuelson webs,” Doklady Akademii Nauk, vol. 430, no. 3, pp. 305-309, 2010. · Zbl 1202.53020 · doi:10.1134/S1064562410010138
[12] V. V. Goldberg and V. V. Lychagin, “Geodesic webs on a two-dimensional manifold and Euler equations,” Acta Applicandae Mathematicae, vol. 109, no. 1, pp. 5-17, 2010. · Zbl 1207.53014 · doi:10.1007/s10440-009-9437-1
[13] V. V. Goldberg and V. V. Lychagin, “Geodesic webs of hypersurfaces,” Doklady Akademii Nauk, vol. 425, no. 6, pp. 737-740, 2009. · Zbl 1183.53013 · doi:10.1134/S1064562409020355
[14] V. V. Goldberg, “On the existence of paratactic three-webs,” Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika, no. 4, pp. 22-27, 2008. · Zbl 1168.53009 · doi:10.3103/S1066369X08040038
[15] M. A. Akivis and V. V. Goldberg, “Differential geometry of Lagrange-type webs,” Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika, no. 12, pp. 19-32, 2007. · Zbl 1298.53015 · doi:10.3103/S1066369X0712002X
[16] T. B. Bouetou and J. P. Dufour, “Veronese curves and webs: interpolation,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 93142, 11 pages, 2006. · Zbl 1128.53013 · doi:10.1155/IJMMS/2006/93142
[17] M. A. Akivis and V. V. Goldberg, “Differential geometry of Veronese-type webs,” Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika, no. 10, pp. 3-28, 2007 (Russian). · Zbl 1298.53014 · doi:10.3103/S1066369X07100015
[18] M. A. Akivis, “Differential geometry of webs,” in Problems in Geometry, vol. 15 of Itogi Nauki i Tekhniki, pp. 187-213, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, Russia, 1983. · Zbl 0567.53014 · doi:10.1007/BF02105053
[19] M. A. Akivis and V. V. Goldberg, “On four-dimensional three-webs with integrable transversal distributions,” Rendiconti del Seminario Matematico di Messina. Serie II, vol. 5(20), pp. 33-52, 1998. · Zbl 0971.53009
[20] M. A. Akivis and V. V. Goldberg, “Algebraic aspects of web geometry,” Commentationes Mathematicae Universitatis Carolinae, vol. 41, no. 2, pp. 205-236, 2000. · Zbl 1042.53007
[21] M. A. Akivis and V. V. Goldberg, “Differential geometry of webs,” in Handbook of Differential Geometry, vol. 1, pp. 1-152, North-Holland, Amsterdam, The Netherlands, 2000. · Zbl 0968.53001
[22] M. A. Akivis and V. V. Goldberg, Projective Differential Geometry of Submanifolds, North-Holland Mathematical Library 49, North-Holland, Amsterdam, The Netherlands, 1998. · Zbl 0971.53012
[23] M. A. Akivis and A. M. Shelekhov, Geometry and Algebra of Multidimensional Three-Webs, vol. 82 of Mathematics and Its Applications (Soviet Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. · Zbl 0771.53001
[24] M. A. Akivis and A. M. Shelekhov, “On the alternator of fourth order of local analytic loop and three-webs of multidimensional surfaces,” Izvestiya VUZ Matematika, no. 4, pp. 12-16, 1989. · Zbl 0701.53027
[25] R. Baer, “Nets and groups. II,” Transactions of the American Mathematical Society, vol. 47, pp. 435-439, 1940. · Zbl 0023.21502 · doi:10.2307/1989962
[26] V. D. Belousov, Foundations of Theory of Quasigroups and Loops, Nauka, Moscow, Russia, 1969. · Zbl 0211.33101
[27] V. V. Goldberg, “Local differentiable quasigroups and webs,” in Quasigroups and Loops: Theory and Applications, vol. 8 of Sigma Ser. Pure Math., pp. 263-311, Heldermann, Berlin, Germany, 1990. · Zbl 0737.53015
[28] P. O. Mikheev, “Fourth-order alternators of a three-web associated with an analytic loop,” Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika, no. 6, pp. 36-38, 1991. · Zbl 0736.53010
[29] P. O. Mikheev and L. V. Sabinin, “Smooth quasigroups and geometry,” in Problems in Geometry, vol. 20 of Itogi Nauki i Tekhniki, pp. 75-110, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, Russia, 1988. · Zbl 1267.17040
[30] P. O. Mikheev and L. V. Sabinin, The Theory of Smooth Bol Loops. Lecture notes, Friendship of Nations University Press, Moscow, Russia, 1985. · Zbl 0584.53001
[31] P. O. Miheev and L. V. Sabinin, “Quasigroups and differential geometry,” in Quasigroups and Loops: Theory and Applications, vol. 8 of Sigma Ser. Pure Math., pp. 357-430, Heldermann, Berlin, Germany, 1990. · Zbl 0721.53018
[32] L. V. Sabinin, “The geometry of loops,” Matematicheskie Zametki, vol. 12, pp. 605-616, 1972 (Russian).
[33] L. V. Sabinin, “Differential geometry and quasigroups,” Trudy Institute Matematiki, vol. 14, pp. 208-221, 1989. · Zbl 0714.53016
[34] L. V. Sabinin, Smooth Quasigroups and Loops Forty-Five Years of Incredible Growth, Springer, Prague, Czech Republic, 1999. · Zbl 1038.20051
[35] L. V. Sabinin, Smooth Quasigroups and Loops, vol. 492 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. · Zbl 0939.20064
[36] R. D. Schafer, An Introduction to Nonassociative Algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York, NY, USA, 1966. · Zbl 0145.25601
[37] A. M. Shelekhov, “Calculation of the covariant derivatives of the curvature tensor of a multidimensional three-web,” in Webs and Quasigroups, pp. 96-103, Kalinin. Gos. Univ., Kalinin, Russia, 1986. · Zbl 0617.53028
[38] A. M. Shelekhov, “Higher-order differential geometric objects of a multidimensional three-web,” in Problems of Geometry, vol. 19 of Itogi Nauki i Tekhniki, pp. 101-154, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, Russia, 1987. · Zbl 0711.53013 · doi:10.1007/BF01098653
[39] A. M. Shelekhov, “Classification of multidimensional three-webs according to closure conditions,” in Problems in Geometry, vol. 21 of Itogi Nauki i Tekhniki, pp. 109-154, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, Russia, 1989. · Zbl 0705.53014
[40] A. M. Shelekhov and L. M. Pidzhakova, “On three-webs with covariantly constant torsion and curvature tensors,” in Webs & Quasigroups, 1998-1999, pp. 92-103, Tver. Gos. Univ., Tver, Russia, 1999. · Zbl 0928.53010
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