TRANSVERSAL HYPERSURFACES OF ALMOST HYPERBOLIC CONTACT MANIFOLDS WITH A QUARTER SYMMETRIC NON METRIC CONNECTION

Yıl 2013, Cilt: 3 Sayı: 1, 108 - 116, 01.06.2013

Shamsur Rahman

Öz

Transversal hypersurfaces of trans hyperbolic contact manifolds endowed with a quarter symmetric non metric connection are studied. It is proved that transversal hypersurfaces of almost hyperbolic contact manifold with a quarter symmetric non metric connection admits an almost product structure and each transversal hypersurfaces of almost hyperbolic contact metric manifold with a quarter symmetric non metric connection admits an almost product semi-Riemannian structure. The fundamental 2- form on the transversal hypersurfaces of cosymplectic hyperbolic manifold and α, 0 trans hyberbolic Sasakian manifold with hyperbolic f, g, u, v, α -structure are closed. It is also proved that transversal hypersurfaces of trans hyperbolic contact manifold with a quarter symmetric non metric connection admits a product structure. Some properties of transversal hypersurfaces with a quarter symmetric non metric connection are proved.

Anahtar Kelimeler

Hypersurfaces, Quarter Symmetric semi-metric connection, Quasi-sasakianmanifold, Gass and Weingarten equations

Kaynakça

• [1] Gray, A. and Hervella, L. M., (1980), The sixteen classes of almost Hermitian manifolds and their liner invariants, Ann, Mat. Pura Appl., 123 (4), 35-58.
• [2] Chen, B. Y., (1981), Geometry of submanifolds and its applications, Sci. Univ. Tokyo, Tokyo.
• [3] Blair, D. E., (1976), Contact manifolds in Riemannian geometry, Lecture Notes in Math., 509, Springer Verlag.
• [4] Blair, D. E. and Oubina, J. A., (1990), Conformal and related changes of metric on the product of two almost contact metric manifolds, Publications Matematiques 34, 199-207.
• [5] Chinea, D. and Perestelo, P. S., (1991), Invariant submnifolds of a trans-Contact manifold, Publ. Math. Debrecen, 38, 103-109, MR 92 g: 53025.
• [6] Janssens, D. and Vanhecke, L., (1981), Almost contact strucures and curvature tensors, Kodai Math. J., 4, 1-27.
• 7] Oubina, J. A., (1985), New classes of almost contact metic structures, Publ. Math. Debrecen, 32, 187-193.
• [8] Marrero, J. C., (1992), The local structure of trans-Contact manifolds, Ann. Mat. Pura Appl., 162 (4), 77-86, MR 93j:53044.
• [9] Sengupta, J. and Biswas, B., (2003), Quarter-symmetric non-metric connection on a Sasakian manifold, Bull. Cal. Math. Soc., 95 (2), 169-176.
• [10] Kenmotsu, K., (1972), A class of almost contact Riemannian manifolds, Tohoku Math. J., 24, 93-103.
• [11] Yano, K. and Imai, T., (1982), Quarter-symmetric metric connections and their curvature tensors, Tensor, (N. S.), 38, 13-18.
• [12] Bhatt, L. and Dubey, K. K., (2003), On CR-submanifold of trans hyperbolic Contact manifold, Acta Cieme. Indica, 29 (1), 91-96.
• [13] Upadhyay, M. D. and Dubey, K. K., (1976), Almost Contact hyperbolic (f,g,η,ξ)- structure, Acta Math. Acad. Scient Hung., 28, 13-15.
• [14] Singh, R. N. and Pandey, M. K., (2007), On a type of quarter-symmetric non-metric connection in a Kenmotsu manifold, Bull. Cal. Math. Soc., 99 (4), 433-444.
• [15] Mishra, R. S. and Pandey, S. N., (1980), On quarter symmetric metric F-connections, Tensor, (N. S.), 34 (1), 1-7.
• [16] Mishra, R. S., (1991), Almost contact metric manifolds, Monograph 1, Tensor society of India, Lucknow.
• [17] Prasad, R., Tripathi, M. M. and Kim, J.-S., (2002), On generalized recci-recurrent trans contact manifolds, J. Korean Math. Soc., 39 (6), 953-961.
• [18] Prasad, R., Tripathi, M. M. and Kim, J.-S., (2002), J-H Cho, On automorphism groups of an 2 framed manifold, Commun. Korean Math. Soc., 17 (4), 635-645.
• [19] Prasad, R. and Tripathi, M. M., (2003), ξ-horizontal hyper surfaces of Kemmotsu manifolds, Bull. Cal. Math. Soc., 95 (2), 121-126.
• [20] Prasad, R. and Tripathi, M. M., (2002), On non-invariant hyper surface of Trans-Contact manifolds, J. Int. Acad. Physical. Sci., 6 (1), 33-40.
• [21] Prasad, R. and Tripathi, M. M., (2003), Transversal hyper surface of Kenmotsu manifold, Indian J. Pure Appl. Math., 34 (3), 443-452.
• [22] Biswas, S. C. and De, U. C., (1997), Quarter-symmetric metric connection in an SP-Sasakian manifold, Common. Fac. Sci. Univ. Ank. Al., 46, 49.
• [23] Golab, S., (1975), On semi-symmetric and quarter symmetric linear connections, Tensor, (N.S.), 29 (3), 249-254.
• [24] Rastogi, S. C., (1978), On quarter symmetric non metric connection, C. R. Acad. Bulgare Sci., 31 (7), 811-814.
• [25] Rastogi, S. C., (1987), On quarter-symmetric metric connection, Tensor, (N. S.), 44, 133-141.
• [26] Kanemaki, S., (1984), On quasi-Contact manifolds, Diff. Geometry Banach Center Publications, 12, 95-125.
• [27] Mukhopadhyay, S., Ray, A. K. and Barua, B., (1991), Some properties of a quarter-symmetric metric connection on a Riemannian manifold, Soochow J. of Math., 17 (2), 205.
• [28] Tanno, S., (1969), The automorphism groups of almost contact Riemannian manifolds, Tohoku Math. J., 21, 21-38.
• [29] Ki, U. H., Pak, J. S. and Suh, H. B., (1975), On (f, g, u(k), α(k))-structures, Kodai Math. Sem. Rep, 26, 160-175.