On Generalized φ-Recurrent Sasakian Manifolds

Absos Ali Shaikh, Helaluddin Ahmad

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DOI: 10.4236/am.2011.211184                

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.

Keywords

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

Conflicts of Interest

The authors declare no conflicts of interest.

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