#### For citation:

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.

- Salimov A. A., Agca F. On para-Nordenian structures. Ann. Polon. Math., 2010, vol. 99, no. 2, pp. 193–200. DOI: https://doi.org/10.4064/ap99-2-6.
- Salimov A. A, Agca F. Some properties of Sasakian metrics in cotangent bundles. Mediterr. J. Math., 2011, vol. 8, iss. 2, pp. 243–255. DOI: https://doi.org/10.1007/s00009-010-0080-x.
- Yano K., Ishihara S. Tangent and cotangent bundles. New York, Marcel Dekker, 1973. 434 p.
- Aso K. Notes on some properties of the sectional curvature of the tangent bundle. Yokohama Math. J., 1981, vol. 5, pp. 1–5.
- Gudmundsson S., Kappos E. On the geometry of the tangent bundles. Expo. Math., 2002, vol. 20, iss. 1, pp. 1–41.
- Kowalski O. Curvature of the induced Riemannian metric on the tangent bundle of Riemannian manifold. J. Reine Angew. Math., 1971, vol. 250, pp. 124–129.
- Musso E., Tricerri F. Riemannian metric on tangent bundles. Ann. Math. Pura. Appl., 1988, vol. 150, iss. 1, pp. 1–19. DOI: https://doi.org/10.1007/BF01761461.
- Salimov A. A. Tensor operators and their applications. New York, Nova Science Publ., 2013. 692 p.
- Sasaki S. On the Differential geometry of tangent bundles of Riemannian manifols. Tohoku Math. J., 1958, vol. 10, no. 3, pp. 338–358.
- Bukusheva A. V., Galaev S. V. Almost contact metric structures defined by connection over distribution with admissible Finslerian metric. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2012, vol. 12, iss. 3, pp. 17–22 (in Russian).
- Bukusheva A. V. Foliation on distribution with Finslerian metric. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 3, pp. 247—251 (in Russian).
- Bukusheva A. V., Galaev S. V. Connections on distributions and geodesic sprays. Russian Math., 2013, vol. 57, iss. 4, pp. 7–13. DOI: https://doi.org/10.3103/S1066369X13040026.
- Galaev S. V. Geometric interpretation of the Wagner curvature tensor in the case of amanifold with contact metric structure. Sib. Math. J., 2016, vol. 57, no. 3, pp. 498–504. DOI: https://doi.org/10.1134/S0037446616030101.
- Galaev S. V. Admissible hypercomplex structures on distributions of Sasakian manifolds. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2016, vol. 16, iss. 3, pp. 263–272 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2016-16-3-263-272.
- Bukusheva A. V. The geometry of the contact metric spaces ϕ-connection. Belgorod State University Scientific Bulletin. Mathematics & Physics, 2015, no. 17 (214), iss. 40, pp. 20–24 (in Russian).
- Galaev S. V. N-extended symplectic connections in almost contact metric spaces. Russian Math., 2017, iss. 3, pp. 15–23.
- Vagner V. V. The geometry of an (n − 1)-dimensional nonholonomic manifold in an ndimensional space. Trudy Sem. Vektor. Tenzor. Analiza [Proc. of the Seminar on Vector and Tensor Analysis]. Moscow, Moscow Univ. Press, 1941, iss. 5, pp. 173–255 (in Russian).
- Vagner V. V. Geometric interpretation of the motion of nonholonomic dynamical systems. Trudy Sem. Vektor. Tenzor. Analiza [Proc. of the Seminar on Vector and Tensor Analysis]. Moscow, Moscow Univ. Press, 1941, iss. 5, pp. 301–327 (in Russian).

- 1121 reads