Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Galaev S. V. Admissible Hypercomplex Structures on Distributions of Sasakian Manifolds. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 3, pp. 263-272. DOI: 10.18500/1816-9791-2016-16-3-263-272, EDN: WMIIFT

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
14.09.2016
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Russian
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WMIIFT

Admissible Hypercomplex Structures on Distributions of Sasakian Manifolds

Autors: 
Galaev Sergei Vasil'evich, Saratov State University
Abstract: 

The notions of admissible (almost) hypercomplex structure and almost contact hyper-Kahlerian structure are introduced. On a ¨ manifold M with an almost contact metric structure (M, ~ξ, η, ϕ, D) an interior symmetric connection ∇ is defined. In the case of a contact manifold of dimension bigger than or equal to five, it is proved that the curvature tensor of the connection ∇ is zero if and only if there exist adapted coordinate charts with respect to that the coefficients of the interior connection are zero. On the distribution D of an almost contact structure as on the total space of the vector bundle (D, π, M), an admissible almost hypercomplex structure (D, J, J ˜ 1, J2,~u, λ = η ◦ π∗, D) is defined. Under the condition that the admissible almost complex structure ϕ is integrable, it is proved that the constructed almost hypercomplex structure is integrable if and only if the distribution D is a distribution of zero curvature. In the case of a Sasakian structure (M, ~ξ, η, ϕ, g, D), the conditions that imply that the admissible hypercomplex structure (D, J, J ˜ 1, J2,~u, λ = η ◦ π∗, g, D˜ ) is an almost contact hyper-Kahlerian structure. 

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Received: 
20.04.2016
Accepted: 
27.08.2016
Published: 
30.09.2016