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Galaev S. V. Extended Structures on Codistributions of Contact Metric Manifolds. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2017, vol. 17, iss. 2, pp. 138-147. DOI: 10.18500/1816-9791-2017-17-2-138-147, EDN: ZEVWZH

# Extended Structures on Codistributions of Contact Metric Manifolds

In the paper, the notion of an AP-manifold is introduced. Such a manifold is an almost contact metric manifold that is locally equivalent to the direct product of a contact metric manifold and an Hermitian manifold. A normal AP-manifold with a closed fundamental form is a quasi-Sasakian manifold. A quasi-Sasakian AP-manifold is called in the paper a special quasi-Sasakian manifold (SQS-manifold). A SQS-manifold is locally equivalent to the product of a Sasakian manifold and a K¨ ahlerian manifold. As a subsidiary result, a proposition is proved stating that a contact metric space with a zero curvature distribution is a K–contact metric space. The codistribution D∗ of a contact metric structure (M, ~ξ, η, ϕ, g,D) is defined as the subbundle of the cotangent bundle T∗M, consisting of all 1-forms annihilating the structure vector ~ξ. On the codistributionD∗, the extended almost contact metric structure (D∗, ~u = ∂n, μ = η ◦ π∗, J,G, ˜D ). is defined. Structural equations are introduced. These equations were used to prove the statement that the extended almost contact metric structure defines a structure of an AP-manifold if and only if the Schouten tensor of the contact metric manifold M is equal to zero. Finally we prove the theorem stating that the extended almost contact metric structure is a SQS-structure if and only if the initial manifold is a Sasakian manifold with a zero curvature distribution.

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