Applied Mathematics, 2011, 2, 13171322 doi:10.4236/am.2011.211184 Published Online November 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM On Generalized φRecurrent Sasakian Manifolds Absos Ali Shaikh, Helaluddin Ahmad Department of Mathematics, The University of Burdwan, Bur dwan , India Email: aask2003@yahoo.co.in Received July 22, 2011; revised September 13, 2011; accepted September 20, 2011 Abstract The object of the present paper is to introduce the notion of generalized φrecurrent Sasakian manifold and study its various geometric properties with the existence of such notion. Among others we study generalized concircularly φrecurrent Sasakian manifolds. The existence of generalized φrecurrent Sasakian manifold is given by a proper example. Keywords: Locally φSymmetric Sasakian Manifold, φRecurrent Sasakian Manifold, Generalized φRecurrent Sasakian Manifold, Scalar Curvature 1. Introduction Let M be an ndimensional connected Riemannian mani fold with Riemannian metric g and LeviCivita connec tion . M is called locally symmetric if its curvature tensor is parallel with respect to . During the last ﬁve decades, the notion of locally symmetric manifold has been weakend many authors in different directions such as recurrent manifold by Walker [1], semisymmetric manifold by Szabό [2], pseudosymmetric manifold by Chaki [3], pseudosymmetric manifold by Deszcz [4], weakly symmetric manifold by Tamássy and Binh [5], weakly symmetric manifold by Selberg [6]. However, the notion of pseudosymmetry by Chaki and Deszcz are different and that of weak symmetry by Selberg and Ta mássy and Binh are also different. As a weaker version of local symmetry, in 1977 Takahashi [7] introduced the notion of local φsymmetry on a Sasakian manifold. By extending the notion of local φsymmetry of Takahashi [7], De et al. [8] introduced and studied the notion of φrecurrent Sasakian manifold. It may be mentioned that locally φsymmetric and φrecurrent LPSasakian, (LCS)n and (k, µ)contact metric manifolds are respectively stu died in [913]. Again, in 1979 Dubey [14] introduced the notion of generalized recurrent manifold and then such a manifold is studied by De and Guha [15]. The manifold M, n > 2, is called generalized recurrent [14] if its curvature tensor R of type (1,3) satisﬁes the condition RARBG , (1) where A and B are nowhere vanishing unique 1forms defined by A(·) = g(·, ρ1), B(·)=g(·, ρ2) and G is a tensor of type (1,3) given by ,, ,GXYZgYZX gXZY (2) for all vector fields ,, YZ M ; being the Lie algebra of all smooth vector fields on M and is the LeviCivita connection. Again M, n > 2, is called generalized Riccirecurrent manifold [16] if its Ricci tensor S of type (0, 2) satisﬁes the condition SASBg (3) where A and B are nowhere vanishing unique 1forms. The object of the present paper is to introduce a type of nonﬂat Sasakian manifolds called generalized φre current Sasakian manifold, which includes both the no tion of local φsymmetry of Takahashi [7] and also φ recurrence of De et al. [8] as particular cases. The paper is organized as follows. Section 2 deals with some pre liminaries of Sasakian manifolds. Section 3 is devoted to the study of generalized φrecurrent Sasakian manifolds and it is shown that such a manifold is generalized Riccirecurrent [16]. In Section 4, we study generalized concircularly φrecurrent Sasakian manifolds and it is shown that in a generalized concircularly φrecurrent Sasakian manifold the vector field ρ2 associted with the 1form B and the characterstic vector field ξ are codi rectional. We also introduce the notion of super genera lized Riccirecurrent manifolds and proved that a gener alized concircularly φrecurrent Sasakian manifold is such one. Also the existence of generalized φrecurrent Sasa kian manifold is ensured by a proper example in the last Section.
A. A. SHAIKH ET AL. 1318 0 g 2. Sasakian Manifolds An n= (2m + 1)dimensional C∞ manifold M is said to be a contact manifold if it carries a global 1form η such that everywhere on the manifold. Given a contact form η, it is wellknown that there exists a unique vector field ξ, called the characteristic vector field of η, satisfying and for any vector field X on M. A Riemannian metric g is said to be an associated metric if there exists a tensor field φ of type (1,1) such that 0 m d 1 ,dX 2, ,, d,,Ig (4) 0,0, ,,gg (5) ,,gg . (6) Then the structure ,,, on M is called a con tact metric stucture and the manifold M equipped with such a stucture is called a contact metric manifold [17]. Given a contact metric manifold M we deﬁne a (1,1) tensor field h by 1£ 2 h , where denotes the ope £ rator of Lie differentiation. Then h is symmetric. The vector field ξ is a Killing vector field with respect to g if and only if . A contact metric manifold M for which ξ is a Killing vector is said to be a Kcontact ma nifold. A contact structure on M gives rise to an almost complex structure J on the product 0h defined by dd ,,( dd JXfX fX tt ), where f is a real valued function, is integrable, then the structure is said to be normal and the manifold M is a Sasakian manifold. Equivalently, a contact metric mani fold is Sasakian if and only if ,RXYYX XY (7) holds for all X, Y where R denotes the curvature tensor of the manifold. In an ndimensional Sasakian manifold M the follow ing relations hold [1719]: ,, ,, X RXYYgXY YX RX Y (8) ,, XX YgXY , (9) ,, ,RXYZgYZXgXZY 1, , (10) ,1,, SXnX Sn (11) ,,1SXY SXYnXY , (12) ,, 1, WSYSYWn gYW ,, ,, WRXY gWYX , WXYRXYW (14) ,, ,,, WRXZgXZ W ZWXRXWZ (15) for all vector fields ,,, YZWM . Deﬁnition 1. [7] A Sasakian manifold is said to be lo cally φsymmetric if 2, WRXYZ 0 (16) for all vector fields X, Y, Z, W orthogonal to ξ. Deﬁnition 2. [8] A Sasakian manifold is said to be φrecurrent if there exists a nowhere vanishing unique 1form A such that 2,, WRXYZ AWRXYZ (17) for all vector fields ,,, YZWM . Especially, if the 1form A vanishes and the vector fields are horizontal, then the manifold turns to be a lo cally φsymmetric Sasakian manifold [7]. 3. Generalized φRecurrent Sasakian Manifolds Deﬁnition 3. An ndimensional, n ≥ 3, Sasakian mani fold M is said to be a generalized φrecurrent if its cur vature tensor satisfies the relation 22 2 ,, , WR XYZAWRXYZ BWG XY Z (18) for all ,,, YZWM , where A and B are nowhere vanishing unique 1forms such that 1 ,AX gX , 2 ,gXBX and G(X, Y )Z is defined in (2). We consider a Sasakian manifold M, n ≥ 3, which is generalized φrecurrent. Then by virtue of (4), (18) yields ,, ,, ,,, WW R XYZR XYZ AWR XYZRXY Z BW GXYZGXYZ (19) from which it follows that ,, , ,, , ,, , . WW RXYZURXYZ U AWg RXYZUR XYZU BWgG XY ZUG XY ZU (20) Let :1,2,, i ei n be an orthonormal basis of the tangent space at any point of the manifold. Then putting , (13) Copyright © 2011 SciRes. AM
A. A. SHAIKH ET AL.1319 X = U = ei in (20) and taking summation over i, 1≤ i ≤ n, and using (15), (10), (8) and (2), we get ,, 2, . WSYZ AWSYZ nBWAWgYZ AW BWYZ (21) Setting Z = ξ in (19) and using (7), (2), (14) and (10) we obtain , WRXYAW BWYXXY . (22) From (14) and (22), we obtain ,,, . WYXgW XYRXYW AWBWYXX Y (23) Taking inner product of (23) with Z and then taking contraction over X and Z, we get ,(1),SY WnAWBWYgWY . (24) Putting Y in (24) we get 0AW BW for all W. (25) This leads to the following. Theorem 1. In a generalized φrecurrent Sasakian manifold M, n ≥ 3, the associated 1forms A and B are related by the relation . 0AB In view of (25), (21) turns into ,, WSYZAWSYZ bWgYZ , , (26) where . This leads to the follow ing. (3)bWn AW Theorem 2. A generalized φrecurrent Sasakian mani fold M, n ≥ 3, is generalized Riccirecurrent. 4. Generalized Concircularly φReccurent Sasakian Manifolds The concircular transformation on a Riemannian mani fold is a transformation under which geodesic circles re mains invariant [20]. The concircular curvature tensor of type (1,3) is given by [20] C (,) (,)(,) (1) r CXYZ RXYZGXYZ nn . (27) If the concircular curvature tensor C satisﬁes the re lation (18), then the manifold is said to be generalized concircularly φrecurrent Sasakian manifold. We also note that since conformal and projective curvature tensors are trace free, there do not exist any generalized conformally and projectively φreccurent Sasakian manifolds. Let us consider a generalized concircularly φrecurrent Sasakian manifold M, n ≥ 3. Hence the deﬁning condi tion of a generalized concircularly φrecurrent Sasakian manifold, yields by virtue of (27) that 22 2 ,, , (1) ,,, WRXYZAWRXYZ BWG XYZ rA WdrW nn gYZXXgYZgXZYYgXZ ,. (28) This leads to the following. Theorem 3. A generalized concircularly φrecurrent Sasakian manifold M, n ≥ 3, is generalized φrecurrent if and only if ,, (1) ,,0. rA WdrW YZ XXgYZ nn gXZYYgXZ (29) Now taking inner product of (29) with U we have ,, , (1) ,, , rA WdrW 0. YZgXUX gYZU nn gXZgYUYgXZ U Taking contraction over X and U we get (2) ,rA WdrWngYZYZ 0 . Again taking contraction over Y and Z we get (2)10rA WdrWn n . which implies that 1 WdrW r for all W and 0r i.e., 1 1 radr r , where 1 ,AW gW . This leads to the following. Theorem 4. If a generalized concircularly φrecurrent Sasakian manifold M, n ≥ 3, is a generalized φrecurrent Sasakian manifold, then the associated vector field cor responding to the 1form A is given by 1 1 radr r , r being the nonzero and nonconstant scalar curvature of the manifold. Now by virtue of (4), it follows from (28) that ,, ,, ,, ,, (1) ,,. WW RXYZ RXYZ AW RXYZRXYZ BW GXYZGXYZ rA WdrW YZ XX gYZ nn gXZY YgXZ (30) Copyright © 2011 SciRes. AM
A. A. SHAIKH ET AL. 1320 Taking inner product of (30) with U and then contra cting over X and U, and then using (2), (15), (10) and (8) we get ,, (2) , ()(2) , (1) (2) 1, (1) (1) . WSYZ AWSYZ nBWAWgYZ dr WngYZ YZ nn rnr WYZg nn nn BW YZ YZ (31) Again taking contraction over Y and Z in (31), we get 2 d(1)(1)rWrnnAWnnBW . (32) From (32), we can state the following. Theorem 5. In a generalized concircularly φrecurrent Sasakian manifold M, n ≥ 3, the associated 1forms A and B are related by the relation (32). Corollary 1. In a generalized concircularly φrecurrent Sasakian manifold M, n ≥ 3, with constant scalar curva ture, the associated 1forms A and B are related by 2 (1)(1) 0rnnAnn B . Now using (32) in (31) we get ,, (2)(1), . WSYZ AWSYZ nnBWnAWg Y Z nB WYZ (33) From (33), it follows that the Ricci tensor S satisﬁes the condition πSSg , (34) where , ,WAW (2)Wnn BW (1)nAW WnBW and π . By extending the notion of generalized Riccirecurrent manifold [16], we introduce the notion of super general ized Riccirecurrent manifold defined as follows. Deﬁnition 4. An ndimensional Riemannian manifold M, n > 2, is called a super generalized Riccirecurrent if its Ricci tensor S of type (0,2) satisﬁes the relation πSSg , where α, β, γ are nowhere vanishing unique 1forms and π . From (34), we can state the following: Theorem 6. A generalized concircularly φrecurrent Sasakian manifold M, n ≥ 3, is super generalized Ricci recurrent manifold. Now taking contraction of (33) over W and Z, we get 1 1d,(2) 2 (1) rY SYnnBY nAYnYB . By virtue of (32), the above relation takes the form 1 2 (1)(2) ,2 45 . 2 rn n SY AY nn n BYnY B (35) From (35), we can state the following. Theorem 7. In a generalized concircularly φrecurrent Sasakian manifold M, n ≥ 3, the Ricci tensor in the di rection of ρ1 is given by (35). Now setting Z in (33) and then using (13) and (11) we get ,( 1),( 1)SY WngY WnnBWY . (36) Replacing Y by φY in (36) and using (12) and (6) we have ,(1),SYWn gYW . (37) Replacing W by φW in (36) and then using (4) we get ,(1),(1)SYWngYWnnB WY . (38) From (37) and (38) we have 0BW , which implies that BWWB . This leads to the following. Theorem 8. In a generalized concircularly φrecurrent Sasakian manifold M, n ≥ 3, the vector field ρ2 associated with the 1form B and the characterstic vector field ξ are codirectional. 5. Example of Generalized φRecurrent Sasakian Manifold Example 1. We consider a 3dimensional manifold 3 ,,:0Mxyzy 3 , where (x, y, z) are the stan dard coordinates in . Let be a linearly independent global frame on M given by 123 ,,EEE 2 12 3 2,, .EExyE zy z Let g be the Riemannian metric defined by 1323 12 ,,,gEE gEE gEE0 , 112233 ,,,gEEgEEgE E1 . Copyright © 2011 SciRes. AM
A. A. SHAIKH ET AL.1321 Let η be the 1form defined by for any . Let φ be the (1,1) tensor field defined by 1221 3 ,UgUE UM ,EE EE and 30E . Then using the linearity of φ and g we have 1 3 E , and 3 UE 2UU ,, UW gUW ()( )UW for any . Thus for 3 ME,UW , (,,, ) deﬁnes an almost contact metric structure on M. Let be the Riemannian connection of g. Then we have 123 1323 ,2, ,0, ,0EEEEEEE . Using Koszul formula for the Riemannian metric g, we can easily calculate 11 1 1233 0,, , EE E EEEE 2 E , 0. 222 1323 1 ,0, EEE EEEE E 33 3 122 13 ,, EE E EEE EE From the above it can be easily seen that ,,, 3,,, is a Sasakian structure on M. Consequently g is a Sasakian manifold. Using the above relations, we can easily calculate the components of the curvature tensor as follows: 12 121221 12 3 ,3,, 3 ,0, RE EEERE EEE RE EE , , 13 1313213 31 ,3,, 0,,REE EEREE EREEEE 23 123232332 , 0,,,,RE EERE EEE RE EEE and the components which can be obtained from these by the symmetry properties. Since forms a basis of the Sasakian manifold, any vector field 123 ,,EE E ,, YZ M can be wri tten as 11 1213 , aE bEcE 212 223 ,YaEbE cE 31 3233 , aE bEcE where (the set of all positive real num bers), i = 1, 2, 3. Then ,, iii abc 312 21312211 31221312 212 31221312 213 ,3 3 RXYZc acacb ababE aab abcbc bc E aac acbbc bc E (39) and 232323 111213 1 31313212223 , . GX YZaabbccaEbEcE aabbcca EbEc E (40) By virtue of (39) we have the following: 1312 21312 211 312 212312 213 ,=4 44, ERXYZcab abbacacE aac acEaab abE (41) 231221 1 31221312212 31221 3 ,4 4 4, ERXYZbbc bcE cab ababcbcE bab abE (42) 3,= ERXYZ0. (43) From (39) and (40), we get 2 112 2 2 1122 , ,, RXYZpE pE GXYZqEqE (44) where 13122131221 2312213122 1213131 2323 221313 12323 3 3 . pbababcacac paababcbcbc qabbcc abbcc qbaacc baacc 1 Also from (41)(43), we obtain 2 1 , i Ei RXYZuE vE 2i for i = 1, 2, 3, (45) where 1312213122 131221 231221 23122131221 33 4 4 4 4 0, 0. ucababbacac vaacac ubbcbc vabcbccabab uv 1 Let us now consider the components of the 1forms as 21 12 21 () i i qu qv AE pq pq i for i = 1, 2 0 for i = 3 (46) and 1i2 i 12 21 p () i pv u BE pq pq for i = 1, 2 0 for i = 3 (47) where 1221 0pqp q 12 0 ii pu , and 21 0 ii qu qv pv for i = 1, 2. From (18), we have 22 2 ,, , i Ei i R XYZAERXYZ BE GXYZ (48) Copyright © 2011 SciRes. AM
A. A. SHAIKH ET AL. Copyright © 2011 SciRes. AM 1322 [9] A. AlAqeel, U. C. De and G. C. Ghosh, “On Lorentzian paraSasakian Manifolds,” Kuwait Journal of Science & Engineering, Vol. 31, 2004, pp. 113. for i = 1, 2, 3. By virtue of (44)(47), it can be easily shown that the manifold satisfies the relation (48). Hence the manifold under consideration is a generalized φre current Sasakian manifold, which is not φrecurrent. This leads to the following. [10] A. A. Shaikh and K. K. Baishya, “On φSymmetric LP Sasakian Manifolds,” Yokohama Mathematic Journal, Vol. 52, 2005, pp. 97112. Theorem 9. There exists a 3dimensional generalized φrecurrent Sasakian manifold, which is neither φsym metric nor φrecurrent. [11] A. A. Shaikh, T. Basu and K. K. Baishya, “On the Exis tence of Locally φRecurrent LPSasakian Manifolds,” Bulletin of the Allahabad Mathematical Society, Vol. 24, 2009, pp. 281295. 6. References [12] A. A. Shaikh, T. Basu and S. Eyasmin, “On Locally φ Symmetric (LCS)nManifolds,” International Journal of Pure and Applied Mathematics, Vol. 48, No. 8, 2007, pp. 11611170. [1] A. G. Walker, “On Ruses Spaces of Recurrent Curva ture,” Proceedings London Mathematical Society, Vol. 52, No. 1, 1950, pp. 3664. doi:10.1112/plms/s252.1.36 [13] A. A. Shaikh, K. K. Baishya and S. Eyasmin, “On φRe current Generalized (k, µ)Contact Metric Manifolds,” Lobachevski Journal of Mathematics, Vol. 27, 2007, pp. 313. [2] Z. I. Szabό, “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. [14] R. S. D. Dubey, “Generalized Recurrent Spaces,” Indian Journal of Pure and Applied Mathematics, Vol. 10, 1979, pp. 15081513. [3] M. C. Chaki, “On Pseudo Symmetric Manifolds,” Analele Stiintifice Ale Univeritatii, Alexandru Ioan Cuza, Din Iasi, Romania, Vol. 33, 1987, pp. 5358. [15] U. C. De and N. Guha, “On Generalized Recurrent Mani folds,” Journal of National Academy Mathematics, Vol. 9, 1991, pp. 8592. [4] R. Deszcz, “On Pseudosymmetric Spaces,” Acta Mathe matica Hungarica, Vol. 53, No. 34, 1992, pp. 185190. [16] U. C. De, N. Guha and D. Kamilya, “On Generalized RicciRecurrent Manifolds,” Tensor New Series, Vol. 56, 1995, pp. 312317. [5] L. Tamássy and T. Q. Binh, “On Weakly Symmetric and Weakly Projective Symmetric Rimannian Manifolds,” Colloquia Math.ematica Societatis, Vol. 50, 1989, pp. 663670. [17] D. E. Blair, “Contact Manifolds in Riemannian Geome try,” Series Lecture Notes in Mathematics, Springer Verlag, Berlin, 1976. [6] A. Selberg, “Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces with Applica tions to Dirichlet Series,” Indian Mathematical Society, Vol. 20, 1956, pp. 4787. [18] U. C. De and A. A. Shaikh, “Complex Manifolds and Contact Manifolds,” Narosa Publishing House Pvt. Ltd., New Delhi, 2009. [7] T. Takahashi, “Sasakian φSymmetric Spaces,” Tohoku Mathematical Journal, Vol. 29, No. 1, 1977, pp. 91113. doi:10.2748/tmj/1178240699 [19] K. Yano and M. Kon, “Structures on Manifolds,” World Scientiﬁc Publishing, Singapore, 1984. [20] K. Yano, “Concircular Geometry I,” Proceedings of the Imperial Academy, Vol. 16, No. 6, 1940, pp. 195200. doi:10.3792/pia/1195579139 [8] U. C. De, A. A. Shaikh and S. Biswas, “On φRecurrent Sasakian Manifolds,” Novi Sad Journal of Mathematics, Vol. 33, 2003, pp. 1348.
