The objective of the present paper is to study the properties of special weakly Ricci symmetric and generalized Ricci recurrent trans Sasakian manifolds.
[1] Blair, D. E., (1976), Contact manifolds in Riemannian Geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin.
[2] Patterson, E. M., (1952), Some theorems on Ricci-recurrent spaces, J. London Math. Soc., 27, 287-295.
[3] Sitaramayya, M., (1973), Curvature tensors in K¨ahler manifolds, Trans. Amer. Math. Soc., 183, 341- 353.
[4] Janssens, D. and Vanhacke, L., (1981), Almost contact structures and curvature tensors, Kodai Math. J., 4, 1-27.
[5] Dragomir, S. and Ornea, L., (1998), Locally conformal K¨ahler geometry, Progress in Mathematics, 155, Birkhauser Boston, Inc., Boston, MA.
[6] Chen, B. Y. and Yano, K., (1972), Hypersurfaces of conformally flat spaces, Tensor N. S., 26, 318-322.
[7] Ojha, R. H., (1975), A note on the M-projective curvature tensor, Indian J. Pure Applied Math., 8, No. (1-2), 1531-1534.
[8] Ojha, R. H., (1973), On Sasakian manifold, Kyungpook Math. J., 13, 211-215.
[9] Pokhariyal, G. P. and Mishra, R. S., (1971), Curvature tensor and their relativistic significance II, Yokohama Mathematical Journal, 19, 97-103.
[10] Chaubey, S. K. and Ojha, R. H., (2010), On the m-projective curvature tensor of a Kenmotsu manifold, Differential Geometry-Dynamical Systems, 12, 52-60.
[11] Chaubey, S. K., (2011), Some properties of LP-Sasakian manifolds equipped with m−projective curvature tensor, Bull. of Math. Anal. and Appl., 3 (4), 50-58.
[12] Chaubey, S. K., Prakash, S. and Nivas, R., (2012), Some properties of m−projective curvature tensor in Kenmotsu manifolds, Bull. of Math. Anal. and Appl., Volume 4 (3), 48-56.
[13] Chaubey, S. K., (2012), On weakly m−projectively symmetric manifolds, Novi Sad J. Math., Vol. 42, No. 1, 67-79.
[14] Blair, D. E. and Oubina, J. A., 1990), Conformal and related changes of metric on the product of almost contact contact metric manifolds, Publications Matemaliques, 34, 199-207.
[15] Tanno, S., (1971), Curvature tensors and non-existence of Killing vectors, Tensor, N. S., 22, 387-394.
[16] Tanno, S., (1969), The automorphism groups of almost contact Riemannian manifolds, Tohoku Math. J., 21, 21-38.
[17] Gray, A. and Hervella, L. M. , (1980), The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl., 123 (4) (1980), 35-58.
[18] Kenmotsu, K., (1972), A class of almost contact Riemannian manifolds, Tohoku Math. J., 24, 93-103.
[19] Oubina, J. A., (1985), New classes of contact metric structures, Publ. Math. Debrecen, 32 (3-4), 187-193.
[20] China, D. and Gonzales, C., (1987), Curvature relations in trans-Sasakian manifolds, Proceedings of the XIIth Poruguese-Spanish Conference of Mathematics, Vol. II (Potuguese) (Braga, 1987), 564-571, Univ. Minho, Braga.
[21] De, U. C. and Sarkar, A., (2008), On three dimensional trans-Sasakian manifolds, Extracta Math., 23 (3), 265-277.
[22] Marrero, J. C., (1992), The local structure of trans-Sasakian manifolds, Ann. Mat. Pure Appl., 162 (4), 77-86.
[23] De, U. C. and Tripathi, M. M. , (2003), Ricci tensor in 3−dimensional trans-Sasakian manifolds, Kyungpook Math. J., 43, 1-9.
[24] Bagewadi, C. S. and Venkatesha, (2007), Some curvature tensors on trans-Sasakian manifolds, Turk. J. Math., 30, 1-11.
[25] Bagewadi, C. S. and Kumar, Girish , (2004), Note on trans-Sasakian manifolds, Tensor N. S., 65 (1), 80-88.
[26] Kim, J. S., Prasad, R. and Tripathi, M. M., (2002), On generalized Ricci-recurrent trans Sasakian manifolds, J. Korean Math. Soc., 39 (6), 953-961.
[27] De, U. C., Guha, N. and Kamilya, D., (1995), On generalized Ricci-recurrent manifolds, Tensor N. S., 56, 312-317.
[28] De, U. C. and Mallick, S., (2012), Space times admitting m−projective curvature tensor, Bulg. J. Phys., 39, 331-338.
[29] Taleshian, A. and Asghari, N., (2012), On the m−projective tensor of P-Sasakian manifolds, J. of Phy. Sci., 16, 107-116.
[30] Singh, H. and Khan, Q., (2001), On special weakly symmetric Riemannian manifolds, Publ. Math. Debrecen, 58 (3), 523-536.
[31] Khan, Q., (2004), On conharmonically and special weakly Ricci symmetric Sasakian manifolds, Novi Sad J. Math., 34 (1), 71-77.
[32] Deszcz, R.,Verstraelen L. and Yaprak, S., (1994), Warped products realizing a certain condition of pseudo-symmetric type imposed on the curvature tensor, Chin. J. Math., 22 (2), 139-157.
[33] Vranceanu, Gh., (1968), Lecons des Geometric Differential, 4, Ed. de I’Academie, Bucharest.
[34] Mocanu, A. L., (1987), Les varietes a courbure quasi-constant de type Vranceanu, Lucr. Conf. Nat. de Geom. Si Top, Tirgoviste.
[35] Chaubey, S. K. and Prasad, C. S., (2015), On Generalized φ−Recurrent Kenmotsu Manifolds, TWMS J. App. Eng. Math. 5 (1), 1-9.
[36] Chaubey, S. K., (2017), Existence of N(k)−quasi Einstein manifolds, Facta Universitatis (NIS), Se. Math. Inform, (Accepted).
[37] Prakash, A., (2010), m−projectively recurrent LP-Sasakian manifolds, Indian Journal of Mathematics, 52 (1), 47-56.
[39] Singh, A., Kumar, R., Prakash, A. and Khare, S. (2015), On a Pseudo projective φ−recurrent Sasakian manifolds, Journal of Mathematics and Computer Science, 14(4), 309-314.
[40] Narain, D., Prakash, A. and Prasad, B. (2009), A pseudo projective curvature tensor on a Lorentzian para-Sasakian manifolds, Analele Stiintifice Ale Universitatii Al.I.Cuza Din Iasi (S.N.) Mathematica, Tomul LV fase.2, 275-284.
Year 2019,
Volume: 9 Issue: 2, 305 - 314, 01.06.2019
[1] Blair, D. E., (1976), Contact manifolds in Riemannian Geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin.
[2] Patterson, E. M., (1952), Some theorems on Ricci-recurrent spaces, J. London Math. Soc., 27, 287-295.
[3] Sitaramayya, M., (1973), Curvature tensors in K¨ahler manifolds, Trans. Amer. Math. Soc., 183, 341- 353.
[4] Janssens, D. and Vanhacke, L., (1981), Almost contact structures and curvature tensors, Kodai Math. J., 4, 1-27.
[5] Dragomir, S. and Ornea, L., (1998), Locally conformal K¨ahler geometry, Progress in Mathematics, 155, Birkhauser Boston, Inc., Boston, MA.
[6] Chen, B. Y. and Yano, K., (1972), Hypersurfaces of conformally flat spaces, Tensor N. S., 26, 318-322.
[7] Ojha, R. H., (1975), A note on the M-projective curvature tensor, Indian J. Pure Applied Math., 8, No. (1-2), 1531-1534.
[8] Ojha, R. H., (1973), On Sasakian manifold, Kyungpook Math. J., 13, 211-215.
[9] Pokhariyal, G. P. and Mishra, R. S., (1971), Curvature tensor and their relativistic significance II, Yokohama Mathematical Journal, 19, 97-103.
[10] Chaubey, S. K. and Ojha, R. H., (2010), On the m-projective curvature tensor of a Kenmotsu manifold, Differential Geometry-Dynamical Systems, 12, 52-60.
[11] Chaubey, S. K., (2011), Some properties of LP-Sasakian manifolds equipped with m−projective curvature tensor, Bull. of Math. Anal. and Appl., 3 (4), 50-58.
[12] Chaubey, S. K., Prakash, S. and Nivas, R., (2012), Some properties of m−projective curvature tensor in Kenmotsu manifolds, Bull. of Math. Anal. and Appl., Volume 4 (3), 48-56.
[13] Chaubey, S. K., (2012), On weakly m−projectively symmetric manifolds, Novi Sad J. Math., Vol. 42, No. 1, 67-79.
[14] Blair, D. E. and Oubina, J. A., 1990), Conformal and related changes of metric on the product of almost contact contact metric manifolds, Publications Matemaliques, 34, 199-207.
[15] Tanno, S., (1971), Curvature tensors and non-existence of Killing vectors, Tensor, N. S., 22, 387-394.
[16] Tanno, S., (1969), The automorphism groups of almost contact Riemannian manifolds, Tohoku Math. J., 21, 21-38.
[17] Gray, A. and Hervella, L. M. , (1980), The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl., 123 (4) (1980), 35-58.
[18] Kenmotsu, K., (1972), A class of almost contact Riemannian manifolds, Tohoku Math. J., 24, 93-103.
[19] Oubina, J. A., (1985), New classes of contact metric structures, Publ. Math. Debrecen, 32 (3-4), 187-193.
[20] China, D. and Gonzales, C., (1987), Curvature relations in trans-Sasakian manifolds, Proceedings of the XIIth Poruguese-Spanish Conference of Mathematics, Vol. II (Potuguese) (Braga, 1987), 564-571, Univ. Minho, Braga.
[21] De, U. C. and Sarkar, A., (2008), On three dimensional trans-Sasakian manifolds, Extracta Math., 23 (3), 265-277.
[22] Marrero, J. C., (1992), The local structure of trans-Sasakian manifolds, Ann. Mat. Pure Appl., 162 (4), 77-86.
[23] De, U. C. and Tripathi, M. M. , (2003), Ricci tensor in 3−dimensional trans-Sasakian manifolds, Kyungpook Math. J., 43, 1-9.
[24] Bagewadi, C. S. and Venkatesha, (2007), Some curvature tensors on trans-Sasakian manifolds, Turk. J. Math., 30, 1-11.
[25] Bagewadi, C. S. and Kumar, Girish , (2004), Note on trans-Sasakian manifolds, Tensor N. S., 65 (1), 80-88.
[26] Kim, J. S., Prasad, R. and Tripathi, M. M., (2002), On generalized Ricci-recurrent trans Sasakian manifolds, J. Korean Math. Soc., 39 (6), 953-961.
[27] De, U. C., Guha, N. and Kamilya, D., (1995), On generalized Ricci-recurrent manifolds, Tensor N. S., 56, 312-317.
[28] De, U. C. and Mallick, S., (2012), Space times admitting m−projective curvature tensor, Bulg. J. Phys., 39, 331-338.
[29] Taleshian, A. and Asghari, N., (2012), On the m−projective tensor of P-Sasakian manifolds, J. of Phy. Sci., 16, 107-116.
[30] Singh, H. and Khan, Q., (2001), On special weakly symmetric Riemannian manifolds, Publ. Math. Debrecen, 58 (3), 523-536.
[31] Khan, Q., (2004), On conharmonically and special weakly Ricci symmetric Sasakian manifolds, Novi Sad J. Math., 34 (1), 71-77.
[32] Deszcz, R.,Verstraelen L. and Yaprak, S., (1994), Warped products realizing a certain condition of pseudo-symmetric type imposed on the curvature tensor, Chin. J. Math., 22 (2), 139-157.
[33] Vranceanu, Gh., (1968), Lecons des Geometric Differential, 4, Ed. de I’Academie, Bucharest.
[34] Mocanu, A. L., (1987), Les varietes a courbure quasi-constant de type Vranceanu, Lucr. Conf. Nat. de Geom. Si Top, Tirgoviste.
[35] Chaubey, S. K. and Prasad, C. S., (2015), On Generalized φ−Recurrent Kenmotsu Manifolds, TWMS J. App. Eng. Math. 5 (1), 1-9.
[36] Chaubey, S. K., (2017), Existence of N(k)−quasi Einstein manifolds, Facta Universitatis (NIS), Se. Math. Inform, (Accepted).
[37] Prakash, A., (2010), m−projectively recurrent LP-Sasakian manifolds, Indian Journal of Mathematics, 52 (1), 47-56.
[39] Singh, A., Kumar, R., Prakash, A. and Khare, S. (2015), On a Pseudo projective φ−recurrent Sasakian manifolds, Journal of Mathematics and Computer Science, 14(4), 309-314.
[40] Narain, D., Prakash, A. and Prasad, B. (2009), A pseudo projective curvature tensor on a Lorentzian para-Sasakian manifolds, Analele Stiintifice Ale Universitatii Al.I.Cuza Din Iasi (S.N.) Mathematica, Tomul LV fase.2, 275-284.