On local differentiable quasigroups and connections related to a three-tissue of multidimensional surfaces

Akivis, Maks A.; Shelekhov, A. M.

doi:10.1007/bf00966528

Cited by 5 publications

(5 citation statements)

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“…In view of (27) we see that either λ 2 + µ 2 = 1, or X = 0. We find all purely imaginary antiquaternions J forming almost Hermitian structure with a metric of such kind.…”

Section: Isoclinic Distributionsmentioning

confidence: 93%

“…In the present paper the problems mentioned are investigated and as well as the role of geometry of generalized almost quaternionic structures as generalization of three-web geometry also widely discussed at present (see, for example, [24,25]). Namely, the papers on three-web theory give many examples of threewebs having a number of remarkable properties and showing close relations of the theory with other areas of mathematics, such as, algebraic geometry [26], quasi-group and loop theory [27], etc. This gives hope that the theory of almost quaternionic structures of hyperbolic type, or almost antiquaternionic structures, generalizing directly the three-web theory and being part of the theory of generalized almost quaternionic structures studied here will become the object of further serious investigation.…”

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confidence: 99%

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Generalized Gray-Hervella classes and holomorphically projective transformations of generalized almost-Hermitian structures

Kirichenko¹

2005

Izv. Math.

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The fibre bundles adjoint to generalized almost quaternionic structures are studied. The most important classes of generalized almost quaternionic manifolds are considered. nionic affine space. A great contribution to the study of πAQ-structures was made by M.Obata [10] who investigated the structure of affine connections preserving πAQ-structure in parallel translations, as well as properties of transformations preserving πAQ-structure and by S.Ishihara who studied special types of πAQ-manifolds transformations [11].Soon the reseachers realized that the class of πAQ-structures is too narrow for constructing a self-contained quaternionic geometry: even a quaternionic projective space HP n does not possess the structure of such type. So, there was proposed (and it is generally accepted at present) a broader understanding of almost quaternionic structure as a subbundle of tensor bundle of type (1,1) on manifold whose type fibre is a quaternion algebra (or, equivalently, as GL(n, H·Sp(1)-structures on manifold [12]). Some authors called the structures almost quaternionic [13]. But the study of such structures by traditional methods was difficult as the structures are not strictly and globally defined by a given system of tensor fields like, for example, Riemannian, or almost Hermitian, or almost contact structures. Thus, the question of integrability of almost quaternionic structures is not answered at present (unlike πAQ-structures). Moreover, it is not clear yet in what terminology the integrability criterion could be formulated. Nevertheless, several interesting results in this direction were received, for example, Kulkarni theorem asserting that a compact simply-connected integrable quaternionic manifold is isometric to the quaternionic projective space [14].The most important results in the theory of almost quaternionic manifolds were received for quaternionic, quaternionic-Hermitian, quaternionic-Kaehler and hyper-Kaehler manifolds. Quaternionic manifolds were first considered by S.Salamon [15]. They are quaternionic counterpart of complex manifolds. Nevertheless, their geometry differs considerably from complex geometry, i.e. unlike Kaehler structures, a quaternionic-Kaehler structure is not always integrable. The basic property of quaternionic manifolds is integrability of a canonical almost complex structure on the space of their twistor bundle for manifolds of dimension greater than 4 [7] (in case of 4-dimension its integrability is known to be equivalent to manifold self-duality [16]). M.Berger proved [17] that quaternionic-Kaehler manifold of dimension greater then 4 is an Einsteinian manifold. It is Ricci-flat if and only if it is locally hyper-Kaehler. Otherwise, it is not even locally reducible. The author also showed [18] that compact oriented quaternionic-Kaehler manifold with positive sectional curvature is isometric to the canonical quaternionic projective space. Hyper-Kaehler structures were thoroughly investigated by A.Beaquville [19] who stated their close connection with complexsymplectic stru...

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“…In view of (27) we see that either λ 2 + µ 2 = 1, or X = 0. We find all purely imaginary antiquaternions J forming almost Hermitian structure with a metric of such kind.…”

Section: Isoclinic Distributionsmentioning

confidence: 93%

mentioning

confidence: 99%

Generalized Gray-Hervella classes and holomorphically projective transformations of generalized almost-Hermitian structures

Kirichenko¹

2005

Izv. Math.

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“…where here, and in the sequel, σ denotes the sum over cyclic permutation of x, y, z. [2]; see also [4] for a survey of the subject).…”

Section: Preliminariesmentioning

confidence: 99%

“…Akivis algebras were introduced by M. A. Akivis ([1], [2], [3]) as a tool in the study of some aspects of web geometry and its connection with loop theory. These algebras were originally called "W -algebras" [3].…”

Section: Introductionmentioning

confidence: 99%

Hom-Akivis algebras

Issa¹

2010

Preprint

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Hom-Akivis algebras are introduced. The commutator-Hom-associator algebra of a non-Hom-associative algebra (i.e. a Hom-nonassociative algebra) is a Hom-Akivis algebra. It is shown that non-Hom-associative algebras can be obtained from nonassociative algebras by twisting along algebra automorphisms while Hom-Akivis algebras can be obtained from Akivis algebras by twisting along algebra endomorphisms. It is pointed out that a Hom-Akivis algebra associated to a Homalternative algebra is a Hom-Malcev algebra.

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“…The tangent algebras of the local analytic Moufang (Bol) loops turn out to be the Mal'tsev (Bol) algebras [8] ( [12,13,14]). These cases are quite remarkable for the following reason (generalized converse third Lie theorem): every real finite-dimensional Mal'tsev (Bol) algebra is the tangent algebra of some analytic Moufang (local Bol) loop [15,16,17] ( [12,13,14]). The converse third Lie theorem for the general Akivis algebras has been discussed in [10,18].…”

Section: Akivis Algebrasmentioning

confidence: 99%

Geodesic multiplication as a tool for classical and quantum gravity

Kuusk,

Paal

2008

Preprint

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On local differentiable quasigroups and connections related to a three-tissue of multidimensional surfaces

Cited by 5 publications

References 1 publication

Generalized Gray-Hervella classes and holomorphically projective transformations of generalized almost-Hermitian structures

Generalized Gray-Hervella classes and holomorphically projective transformations of generalized almost-Hermitian structures

Hom-Akivis algebras

Geodesic multiplication as a tool for classical and quantum gravity

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