On Generalized *φ*-Recurrent Sasakian Manifolds

ABSTRACT

The object of the present paper is to introduce the notion of generalized*φ*-recurrent Sasakian manifold and study its various geometric properties with the existence of such notion. Among others we study generalized concircularly *φ*-recurrent Sasakian manifolds. The existence of generalized *φ*-recurrent Sasakian manifold is given by a proper example.

The object of the present paper is to introduce the notion of generalized

KEYWORDS

Locally*φ*-Symmetric Sasakian Manifold,
*φ*-Recurrent Sasakian Manifold,
Generalized *φ*-Recurrent Sasakian Manifold,
Scalar Curvature

Locally

Cite this paper

nullA. Shaikh and H. Ahmad, "On Generalized*φ*-Recurrent Sasakian Manifolds," *Applied Mathematics*, Vol. 2 No. 11, 2011, pp. 1317-1322. doi: 10.4236/am.2011.211184.

nullA. Shaikh and H. Ahmad, "On Generalized

References

[1] A. G. Walker, “On Ruses Spaces of Recurrent Curvature,” Proceedings London Mathematical Society, Vol. 52, No. 1, 1950, pp. 36-64. doi:10.1112/plms/s2-52.1.36

[2] Z. I. Szab?, “Structure Theorems on Riemannian Spaces Satisfying R(X, Y) R = 0,” I, The local version, Journal of Differential Geometry, Vol. 17, No. 4, 1982, pp. 531-582.

[3] M. C. Chaki, “On Pseudo Symmetric Manifolds,” Analele Stiintifice Ale Univeritatii, Alexandru Ioan Cuza, Din Iasi, Romania, Vol. 33, 1987, pp. 53-58.

[4] R. Deszcz, “On Pseudosymmetric Spaces,” Acta Mathematica Hungarica, Vol. 53, No. 3-4, 1992, pp. 185-190.

[5] L. Tamássy and T. Q. Binh, “On Weakly Symmetric and Weakly Projective Symmetric Rimannian Manifolds,” Colloquia Math.ematica Societatis, Vol. 50, 1989, pp. 663-670.

[6] A. Selberg, “Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces with Applications to Dirichlet Series,” Indian Mathematical Society, Vol. 20, 1956, pp. 47-87.

[7] T. Takahashi, “Sasakian φ-Symmetric Spaces,” Tohoku Mathematical Journal, Vol. 29, No. 1, 1977, pp. 91-113. doi:10.2748/tmj/1178240699

[8] U. C. De, A. A. Shaikh and S. Biswas, “On φ-Recurrent Sasakian Manifolds,” Novi Sad Journal of Mathematics, Vol. 33, 2003, pp. 13-48.

[9] A. Al-Aqeel, U. C. De and G. C. Ghosh, “On Lorentzian para-Sasakian Manifolds,” Kuwait Journal of Science & Engineering, Vol. 31, 2004, pp. 1-13.

[10] A. A. Shaikh and K. K. Baishya, “On φ-Symmetric LP-Sasakian Manifolds,” Yokohama Mathematic Journal, Vol. 52, 2005, pp. 97-112.

[11] A. A. Shaikh, T. Basu and K. K. Baishya, “On the Existence of Locally φ-Recurrent LP-Sasakian Manifolds,” Bulletin of the Allahabad Mathematical Society, Vol. 24, 2009, pp. 281-295.

[12] A. A. Shaikh, T. Basu and S. Eyasmin, “On Locally φ-Symmetric (LCS)n-Manifolds,” International Journal of Pure and Applied Mathematics, Vol. 48, No. 8, 2007, pp. 1161-1170.

[13] A. A. Shaikh, K. K. Baishya and S. Eyasmin, “On φ-Recurrent Generalized (k, μ)-Contact Metric Manifolds,” Lobachevski Journal of Mathematics, Vol. 27, 2007, pp. 3-13.

[14] R. S. D. Dubey, “Generalized Recurrent Spaces,” Indian Journal of Pure and Applied Mathematics, Vol. 10, 1979, pp. 1508-1513.

[15] U. C. De and N. Guha, “On Generalized Recurrent Manifolds,” Journal of National Academy Mathematics, Vol. 9, 1991, pp. 85-92.

[16] U. C. De, N. Guha and D. Kamilya, “On Generalized Ricci-Recurrent Manifolds,” Tensor New Series, Vol. 56, 1995, pp. 312-317.

[17] D. E. Blair, “Contact Manifolds in Riemannian Geometry,” Series Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1976.

[18] U. C. De and A. A. Shaikh, “Complex Manifolds and Contact Manifolds,” Narosa Publishing House Pvt. Ltd., New Delhi, 2009.

[19] K. Yano and M. Kon, “Structures on Manifolds,” World Scienti?c Publishing, Singapore, 1984.

[20] K. Yano, “Concircular Geometry I,” Proceedings of the Imperial Academy, Vol. 16, No. 6, 1940, pp. 195-200. doi:10.3792/pia/1195579139

[1] A. G. Walker, “On Ruses Spaces of Recurrent Curvature,” Proceedings London Mathematical Society, Vol. 52, No. 1, 1950, pp. 36-64. doi:10.1112/plms/s2-52.1.36

[2] Z. I. Szab?, “Structure Theorems on Riemannian Spaces Satisfying R(X, Y) R = 0,” I, The local version, Journal of Differential Geometry, Vol. 17, No. 4, 1982, pp. 531-582.

[3] M. C. Chaki, “On Pseudo Symmetric Manifolds,” Analele Stiintifice Ale Univeritatii, Alexandru Ioan Cuza, Din Iasi, Romania, Vol. 33, 1987, pp. 53-58.

[4] R. Deszcz, “On Pseudosymmetric Spaces,” Acta Mathematica Hungarica, Vol. 53, No. 3-4, 1992, pp. 185-190.

[5] L. Tamássy and T. Q. Binh, “On Weakly Symmetric and Weakly Projective Symmetric Rimannian Manifolds,” Colloquia Math.ematica Societatis, Vol. 50, 1989, pp. 663-670.

[6] A. Selberg, “Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces with Applications to Dirichlet Series,” Indian Mathematical Society, Vol. 20, 1956, pp. 47-87.

[7] T. Takahashi, “Sasakian φ-Symmetric Spaces,” Tohoku Mathematical Journal, Vol. 29, No. 1, 1977, pp. 91-113. doi:10.2748/tmj/1178240699

[8] U. C. De, A. A. Shaikh and S. Biswas, “On φ-Recurrent Sasakian Manifolds,” Novi Sad Journal of Mathematics, Vol. 33, 2003, pp. 13-48.

[9] A. Al-Aqeel, U. C. De and G. C. Ghosh, “On Lorentzian para-Sasakian Manifolds,” Kuwait Journal of Science & Engineering, Vol. 31, 2004, pp. 1-13.

[10] A. A. Shaikh and K. K. Baishya, “On φ-Symmetric LP-Sasakian Manifolds,” Yokohama Mathematic Journal, Vol. 52, 2005, pp. 97-112.

[11] A. A. Shaikh, T. Basu and K. K. Baishya, “On the Existence of Locally φ-Recurrent LP-Sasakian Manifolds,” Bulletin of the Allahabad Mathematical Society, Vol. 24, 2009, pp. 281-295.

[12] A. A. Shaikh, T. Basu and S. Eyasmin, “On Locally φ-Symmetric (LCS)n-Manifolds,” International Journal of Pure and Applied Mathematics, Vol. 48, No. 8, 2007, pp. 1161-1170.

[13] A. A. Shaikh, K. K. Baishya and S. Eyasmin, “On φ-Recurrent Generalized (k, μ)-Contact Metric Manifolds,” Lobachevski Journal of Mathematics, Vol. 27, 2007, pp. 3-13.

[14] R. S. D. Dubey, “Generalized Recurrent Spaces,” Indian Journal of Pure and Applied Mathematics, Vol. 10, 1979, pp. 1508-1513.

[15] U. C. De and N. Guha, “On Generalized Recurrent Manifolds,” Journal of National Academy Mathematics, Vol. 9, 1991, pp. 85-92.

[16] U. C. De, N. Guha and D. Kamilya, “On Generalized Ricci-Recurrent Manifolds,” Tensor New Series, Vol. 56, 1995, pp. 312-317.

[17] D. E. Blair, “Contact Manifolds in Riemannian Geometry,” Series Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1976.

[18] U. C. De and A. A. Shaikh, “Complex Manifolds and Contact Manifolds,” Narosa Publishing House Pvt. Ltd., New Delhi, 2009.

[19] K. Yano and M. Kon, “Structures on Manifolds,” World Scienti?c Publishing, Singapore, 1984.

[20] K. Yano, “Concircular Geometry I,” Proceedings of the Imperial Academy, Vol. 16, No. 6, 1940, pp. 195-200. doi:10.3792/pia/1195579139