×

About \(A\)-properties of some elastic three-webs. (English. Russian original) Zbl 1386.53014

Russ. Math. 60, No. 7, 19-28 (2016); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2016, No. 7, 23-33 (2016).
Summary: We consider two families of multidimensional three-webs and prove that these webs are webs \(E\), i.e., in their coordinate loops the elasticity identity holds true. We also show that these webs have \(A\)-properties.

MSC:

53A60 Differential geometry of webs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Moufang, R. “Zur Struktur von Alternativkörpern”, Math. Ann. 110, 416-430 (1935). · JFM 60.0093.02 · doi:10.1007/BF01448037
[2] Maltsev, A. I. “Analytic Loops”, Mat. Sb. 36, No. 3, 569-575 (1955) [in Russian].
[3] Sabinin, L. V. Analytic Quasigroups and Geometry (Peoples Friendship Univ., Moscow, 1991) [in Russian]. · Zbl 0832.22003
[4] Mikheev, P. O., Sabinin, L. V. “Smooth Quasigroups and Geometry”, J. Sov. Math. 51, No. 6, 2642-2666 (1990). · Zbl 1267.17040 · doi:10.1007/BF01095429
[5] Akivis, M. A., Shelekhov, A. M. Multidimensional Three-Webs and Their Applications (Tver Gos. Univ., 2010) [in Russian]. · Zbl 0792.53009
[6] Belousov, V. D. Fundamentals of the Theory of Quasigroups and Loops (Nauka, Moscow, 1967) [in Russian].
[7] Shelekhov, A. M. “On Analytic Solutions of the Functional Equation <Emphasis Type=”Italic“>x(<Emphasis Type=”Italic“>yx) = (<Emphasis Type=”Italic“>xy)<Emphasis Type=”Italic“>x”, Mat. Zametki 50, No. 4, 132-140 (1991) [in Russian]. · Zbl 0739.39012
[8] Balandina, G. A.; Shelekhov, A. M., On General Theory of ElasticWebs, 622-74 (1995) · Zbl 0844.53009
[9] Dzhukashev, K. R., About Three-Webs with Flexible Coordinate Loops, 52-80 (2013)
[10] Khasina, V. I. “Multidimensional Three-Webs with ElasticW-Algebras”, Sib.Mat. Zh. 17, 945-949 (1976).
[11] Fedorova, V. I. “Three-Webs with Partially Skew-Symmetric Curvature Tensions”, Soviet Mathematics (Iz. VUZ) 20, No. 11, 114-117 (1976).
[12] Fedorova, V. I. “On a Class W6 of Three-Webs with a Partially Skew-Symmetric Curvature Tensor”, Ukr. Geom. Sb. 20, 115-124 (1977) [in Russian].
[13] Fedorova, V. I. “A Condition DefiningMultidimensional Bol Three-Webs”, Sib.Mat.Zh. 19, 922-928 (1978) [in Russian].
[14] Fedorova, V. I., Six-Dimensional Bol Three-Webs with Symmetric Tensor aij, 1102-123 (1981)
[15] Antipova, M. V. “About Three-Webs of Bol with Almost Null Curvature Tensor”, Izv. Penz. Gos. Pedagog. Univ., Fiz.-Math. i Tekhn. Nauki, No. 26, 28-34 (2011) [in Russian].
[16] Antipova, M. V., Shelekhov, A. M. “Eight-Dimensional Bol Webs with Almost Zero Curvature Tensor”, Russian Mathematics (Iz. VUZ) 57, 2, 1-12 (2013). · Zbl 1266.53015
[17] Murdoch, D. G. “Quasigroups Which Satisfy Certain Generalized Associative Laws”, Amer. J.Math. 61, No. 2, 509-522 (1939). · Zbl 0020.34702 · doi:10.2307/2371517
[18] Osborn, M. J. “A Theorem on <Emphasis Type=”Italic“>A-Loops”, Proc. Amer.Math. Soc. 9, No. 3, 347-349 (1958). · Zbl 0097.25302
[19] Phillips, J. D. “On Moufang <Emphasis Type=”Italic“>A-Loops”, Comment.Math. Univ. Carolinae 41, No. 2, 371-349 (2000). · Zbl 1038.20050
[20] Drápal, A. “<Emphasis Type=”Italic“>A-Loops Close to Code Loops are Groups”, Comment. Math. Univ. Carolinae 41, No. 2, 245-249 (2000). · Zbl 1038.20046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.