In this study, we consider a manifold equipped with semi symmetric metric connection whose the torsion tensor satisfies a special condition. We investigate some properties of the Ricci tensor and the curvature tensor of this manifold . We obtain a necessary and sufficient condition for the mixed generalized quasi-constant curvature of this manifold. Finally, we prove that if the manifold mentioned above is conformally flat, then it is a mixed generalized quasi- Einstein manifold and we prove that if the sectional curvature of a Riemannian manifold with a semi symmetric metric connection whose the special torsion tensor is independent from orientation chosen, then this manifold is of a mixed generalized quasi constant curvature

Bağdatlı Yılmaz, Hülya

Uysal, Samiye Aynur

Özen Zengin, Füsun

European Journal of Pure and Applied Mathematics

Dogus University Institutional Repository

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICSVol. 4, No. 2, 2011, 152-161ISSN 1307-5543 – www.ejpam.comOn a Semi Symmetric Metric Connection with a SpecialCondition on a Riemannian ManifoldHülya Bag˘datlı Yılmaz1,∗, Füsun Özen Zengin2, and S. Aynur Uysal31 Department of Mathematics, Faculty of Sciences and Letters, Marmara University, Istanbul, Turkey2 Department of Mathematics, Faculty of Sciences and Letters, Istanbul Technical University, Istan-bul, Turkey3 Department of Mathematics, Faculty of Sciences and Letters, Dogus University, Istanbul, TurkeyAbstract. In this study, we consider a manifold equipped with semi symmetric metric connectionwhose the torsion tensor satisfies a special condition. We investigate some properties of the Ricci tensorand the curvature tensor of this manifold . We obtain a necessary and sufficient condition for the mixedgeneralized quasi-constant curvature of this manifold. Finally, we prove that if the manifold mentionedabove is conformally flat, then it is a mixed generalized quasi- Einstein manifold and we prove thatif the sectional curvature of a Riemannian manifold with a semi symmetric metric connection whosethe special torsion tensor is independent from orientation chosen, then this manifold is of a mixedgeneralized quasi constant curvature.2000 Mathematics Subject Classifications: 53B15, 53B20, 53C15Key Words and Phrases: Semi symmetric metric connection, Generalized quasi -Einstein manifold,Mixed generalized quasi constant curvature manifold, Mixed generalized quasi-Einstein manifold1. IntroductionThe notion of a generalized quasi- Einstein manifold was introduced by De and Ghosh [5].A non-flat Riemannian manifold M is called a generalized quasi Einstein manifold if its Riccitensor Rk j is not identically zero and satisfies the conditionRk j = αgk j + βuku j + γvkv jwhere α,β ,γ are non-zero scalars and uk and vk are covariant vectors such that uk and vk areorthogonal to each other vector fields on M . The mixed generalized quasi Einstein manifoldwas defined by Bhattacharyya and De [1]. A non-flat Riemannian manifold M is called a∗Corresponding author.Email addresses: hbagdatlimarmara.edu.tr (H. Yılmaz), fozenitu.edu.tr (F. Zengin),auysaldogus.edu.tr (S. Uysal)http://www.ejpam.com 152 c© 2011 EJPAM All rights reserved.H. Yılmaz, F. Zengin, S. Uysal / Eur. J. Pure Appl. Math, 4 (2011), 152-161 153mixed generalized quasi Einstein manifold if its Ricci tensor Rk j is non-zero and satisfies theconditionRk j = αgk j + βaka j + γbkb j + ϑakb j + bka j(1)where α,β ,γ,ϑ are non-zero scalars and ak and bk are covariant vectors such that ak and bkare orthogonal unit vector fields on M . Moreover, it is stated that a Riemannian manifold is ofa mixed generalized quasi constant curvature if the curvature tensor of this manifold satisfiesthe conditionRik jm = pgk j gim− gi j gkm(2)+ qgimaka j − gkmaia j + gk jaiam− gi jakam+ sgimbkb j − gkmbi b j + gk jbi bm− gi j bkbm+ t¦akb j + bka j©gim−¦ai b j + bia j©gkm+ai bm+ biam	gk j −akbm+ bkam	gi jwhere p,q, r, s, t are non-zero scalars and ak and bk are covariant vectors such that ak and bkare orthonormal unit vector fields on M [1].Let ∇ be a linear connection on M . The torsion tensor is given by,T (X ,Y ) =∇X Y −∇Y X − [X ,Y ]The connection ∇ is symmetric if its torsion tensor T vanishes, otherwise it is non-symmetric.If there is a Riemannian metric g in M such that∇g = 0 (3)then the connection ∇ is a metric connection, otherwise it is non-metric [12]. A linear con-nection is said to be a semi symmetric connection if its torsion tensor T is of the formT (X ,Y ) = w(Y )X −w(X )Y (4)where w(X ) = g(X ,U) and U is a vector field. In [9], Pak showed that a Hayden connectionwith the torsion tensor of the form (4) is a semi symmetric metric connection. In [11], Yanoproved that in order that a Riemannian manifold admits a semi symmetric metric connectionwhose curvature tensor vanishes, it is necessary and sufficient that the Riemannian manifoldbe conformally flat, for some properties of Riemannian manifolds with a semi symmetricmetric connection, see also [4, 6, 8, 10]The components of semi symmetric metric connection are given byΓlik =¨lik«+ δliwk − gikwl (5)where wt and wl = wt gt l are covariant and contravariant components of a vector field, re-spectively and∇kw j =∇kw j −wkw j +wgk j,w = wtwt (6)H. Yılmaz, F. Zengin, S. Uysal / Eur. J. Pure Appl. Math, 4 (2011), 152-161 154By using (5), we obtain,Rik jm = Rik jm− gimpik j + gkmpii j − gk jpiim+ gi jpikm (7)where Rik jm and Rik jm are the Riemannian curvature tensors of ∇ and ∇, respectively [11].And pi is a tensor field of type (0,2) defined bypik j =∇kw j −wkw j +12gk jw (8)Transvecting the equation (7) with g im, we getRk j = Rk j − (n− 2)pik j −pigk j (9)where Rk j and Rk j are the Ricci tensors for the connections ∇ and ∇, respectively andpi= piimgim.Multiplying (9) by gk j, we obtainR= R− 2(n− 1)pi (10)where R and R are the scalar curvatures of semi symmetric metric connection and the Levi-Civita connection, respectively.2. A Riemannian Manifold Admitting a Special Semi Symmetric MetricConnectionDe and Sengupta considered a semi symmetric metric connection and studied some prop-erties of an almost contact manifold of a semi symmetric metric connection whose the torsiontensor satisfies a special condition different from the following condition [2]. In this section,we consider a manifold equipped with a semi symmetric metric connection whose the torsionT satisfies the following condition∇ jTlik = a jTlik + b j bl gik +δlj biak (11)where bl = bt gt l . The equation (4) can be written in the following formT lik = δliwk − δlkwiContracting on l and i in the last equation, we getT llk = (n− 1)wk (12)Thus, we can find∇ jTllk = (n− 1)∇ jwk (13)Moreover, by using (11), we obtain∇ jTllk = a jTllk+ b j bk + b jak (14)H. Yılmaz, F. Zengin, S. Uysal / Eur. J. Pure Appl. Math, 4 (2011), 152-161 155From (12)-(14), it is found that∇ jwk = a jwk +1n− 1b j bk +1n− 1b jak (15)After that, from the covariant derivative of wk with respect to ∇, we get the following∇ jwk =∇ jwk +wkw j − g jkw (16)Substituting (16) in (8), we findpik j =∇kw j −12gk jw (17)Again, using (15) and (17) , we obtainpik j = akw j +1n− 1bkb j +1n− 1bka j −12gk jw (18)Then, if we substitute (18) in (7), we getRik jm = Rik jm (19)+wgimgk j − gkmgi j− gimakw j +1n− 1bkb j +1n− 1bka j+ gkmaiw j +1n− 1bi b j +1n− 1bia j− gk jaiwm+1n− 1bi bm+1n− 1biam+ gi jakwm+1n− 1bkbm+1n− 1bkamFrom (19), we have the following theorem:Theorem 1. The curvature tensor of a Riemannian manifold admitting a semi symmetric metricconnection whose the torsion tensor satisfies the condition (11) is of the form (19).Now, we recall some theorems which will be used in this section:Theorem 2. [3] The Ricci tensor S(X ,Y ) of a semi symmetric metric connection ∇ with theassociated 1-form w will be symmetric if and only if w is closed.Theorem 3. [3] A necessary and sufficient condition that the Ricci tensor of the semi symmetricmetric connection ∇ to be symmetric is that the curvature tensor R of (0,4) type with respect tothe connection ∇ satisfies one of the following two conditions:i Rik jm = R jmikH. Yılmaz, F. Zengin, S. Uysal / Eur. J. Pure Appl. Math, 4 (2011), 152-161 156ii Rik jm+ Rk jim+ R jikm = 0.From (19), we can writeR jmik = R jmik (20)+wg jkgim− gmkg ji− g jkamwi +1n− 1bmbi +1n− 1bmai+ gmka jwi +1n− 1b j bi +1n− 1b jai− gmia jwk +1n− 1b j bk +1n− 1b jak+ g jiamwk +1n− 1bmbk +1n− 1bmakwe assume that the associated 1-form w of a Riemannian manifold admitting a semi sym-metric metric connection whose the torsion tensor satisfies the condition (11) is closed. Invirtue of Theorem 2, the Ricci tensor of a Riemannian manifold with a semi symmetric metricconnection is symmetric. Thus, due to Theorem 3, we getR jmik = Rik jm (21)In case the equation (21) is satisfied, we find0=gima jwk −1n− 1bk− akw j −1n− 1b j(22)+ gkmaiw j −1n− 1b j− a jwi −1n− 1bi+ gk jamwi −1n− 1bi− aiwm −1n− 1bm+ gi jakwm −1n− 1bm− amwk −1n− 1bkTransvecting (22) with g im , we get(2− n)akw j −1n− 1b j− a jwk −1n− 1bk= 0 (23)Since n> 2, we getakw j −1n− 1b j= a jwk −1n− 1bk(24)Now, permutating the indices and adding the three equations side by side, we obtainRik jm+Rk jim+ R jikm (25)H. Yılmaz, F. Zengin, S. Uysal / Eur. J. Pure Appl. Math, 4 (2011), 152-161 157= gima jwk −1n− 1bk− akw j −1n− 1b j+ gkmaiw j −1n− 1b j− a jwi −1n− 1bi+ g jmakwi −1n− 1bi− aiwk −1n− 1bkConversely, let us assume that (24) is satisfied. Then, the expression on the right side of (25)vanishes. It means that the curvature tensor of the connection ∇ satisfies the first BianchiIdentity. Due to Theorem 3, the Ricci tensor with respect to the connection ∇ is symmet-ric. Because of Theorem 2, the associated 1-form w of a Riemannian manifold with a semisymmetric metric connection is closed. Hence, we can establish the following theorem:Theorem 4. A necessary and sufficient condition that the associated 1-form w of a Riemannianmanifold with a semi symmetric metric connection whose the torsion tensor satisfies the condition(11) to be closed is that the condition (24) is satisfied.Suppose that w is closed. Substituting (15) in (16), we get∇ jwk = a jwk +1n− 1b j bk +1n− 1b jak +wkw j − g jkw (26)Subtracting the corresponding equation found by interchanging k and j in (26) from (26), weget the equation (24). Thus, by using Theorem 2, Theorem 3 and Theorem 4, we have thefollowing Theorem:Theorem 5. In a Riemannian manifold with a semi symmetric metric connection whose thetorsion tensor satisfies the condition (11), a necessary and sufficient condition that the condition(24) to be satisfied is that it is satisfied any one of the following properties:i The curvature tensor with respect to the connection ∇ of this manifold has the properity ofblock symmetry,ii The curvature tensor with respect to the connection ∇ of this manifold satisfies the firstBianchi Identity,iii The Ricci tensor of this manifold is symmetric.3. Conformally Flat Manifolds with Semi Symmetric Metric ConnectionSatisfying some Special ConditionIn this section, we shall investigate a Riemannian manifold M admitting a semi symmetricmetric connection whose the torsion tensor satisfies a special condition in the case of confor-mally flat. Firstly, we consider the condition (11). Then,∇ jTlik = a jTlik + b j bl gik +δlj biak (27)H. Yılmaz, F. Zengin, S. Uysal / Eur. J. Pure Appl. Math, 4 (2011), 152-161 158where ak and bk be orthogonal to each other. The conformal curvature tensor is given byCik jm = Rik jm−1n− 2Rimgk j − Rkmgi j + Rk j gim− Ri j gkm(28)+R(n− 1)(n− 2)gimgk j − gkmgi jNow, we remember that it is well known the following Theorem:Theorem 6. [11] In order that a Riemannian manifold admits a semi symmetric metric con-nection curvature tensor vanishes, it is necessary and sufficient condition that the Riemannianmanifold be conformally flat.Suppose that this manifold is conformally flat. Hence, we can writeRik jm = 0 (29)Therefore, due to (7) and (29), we obtainRik jm = gimpik j − gkmpii j + gk jpiim− gi jpikm (30)Multiplying (29) by g im, we get the corresponding identityRk j = 0 (31)Transvecting (19) with g im and using (31), we haveRk j =(1− n)w +amwm+1n− 1b+1n− 1bmamgk j (32)+ (n− 2)akw j +1n− 1bkb j +1n− 1bka jwhere am = ai gim, b = bmbm 6= 0. Since ak and bk are the orthogonal vector fields, it can bewrittenRk j =(1− n)w +φ +1n− 1bgk j + (n− 2)akw j +1n− 1bkb j +1n− 1bka j(33)where amwm = φ is a non-zero scalar function. Subtracting (33) from the correspondingequation found by interchanging k and j in (33), we get (24). Transvecting (24) with a j bk ,we findbkwk =abn− 1(34)where amam = a 6= 0. From (34), it is seen that bk can not be orthogonal to wk . Again,multiplying (24) by ak, we getw j = θa j +1n− 1b j (35)H. Yılmaz, F. Zengin, S. Uysal / Eur. J. Pure Appl. Math, 4 (2011), 152-161 159where θ = φa6= 0. By using (35), we find that a j is not orthogonal to w j . Substituting (29)and (35) in (19), we obtainRik jm = wgkmgi j − gimgk j(36)+ θgimaka j − gkmaia j + gk jaiam− gi jakam+1n− 1gimbkb j − gkmbi b j + gk jbi bm− gi j bkbm+1n− 1gimakb j + bka j− gkmai b j + bia j+gk j ai bm+ biam− gi j akbm+ bkamIf w = θφ + ab(n−1)26= 0, and since ak and bk are the orthogonal vector fields, the equation(36) is equivalent to (2). This implies that such a manifold is of a mixed generalized quasiconstant curvature.Multiplying (36) by g im,we obtainRk j = µgk j + (n− 2)θaka j +n− 2n− 1bkb j + akb j + bka j(37)whereµ = (1− n)w + θa+bn− 1(38)Suppose that µ 6= 0.Conversely, suppose that this manifold is of a mixed generalized quasi constant curvature.Multiplying (2) by g im , we obtainRk j =p(n− 1) + qa+ bsgk j + q(n− 2)aka j (39)+ s(n− 2)bkb j + t(n− 2)akb j + bka jTransvecting (39) with gk j , we findR= (n− 1)np+ 2qa+ 2sb(40)Let us substitute (2) , (39) and (40) in (28).Then, if w = −p,θ = q and t = s = 1n−1, we getCik jm = 0We may now establish the following theorem:Theorem 7. In a Riemannian manifold with a semi symmetric metric connection whose thetorsion tensor satisfies the condition (27), a necessary and sufficient condition that this manifoldto be of a mixed generalized quasi constant curvature is that it is conformally flat.When we compare (37) with (1), if p(n− 1) + qa+ bs 6= 0, we can say that this manifoldis a mixed generalized quasi Einstein manifold. Thus, we can state the following theorem:REFERENCES 160Theorem 8. A conformal flat Riemannian manifold with a semi symmetric metric connectionwhose the torsion tensor satisfies the condition (27) is a mixed generalized quasi Einstein mani-fold.Theorem 9. [13] If a Riemannian manifold admits a semi symmetric metric connection withconstant sectional curvature, then this manifold is conformally flat.Thus, in virtue of Theorem 7, Theorem 8 and Theorem 9, we can establish the followingtheorems:Theorem 10. If the sectional curvature of a Riemannian manifold with a semi symmetric metricconnection whose the torsion tensor satisfies the condition (27) is independent from the orienta-tion chosen, theni It is of a mixed generalized quasi constant curvature,ii It is a mixed generalized quasi Einstein manifold.Theorem 11. If the sectional curvature of a Riemannian manifold with a semi symmetric metricconnection whose the torsion tensor satisfies the condition (27) is independent from the orienta-tion chosen, then the condition (24) is satisfied.References[1] A Bhattacharyya and T De. On mixed generalized quasi-Einstein manifolds. Diff. Geo.-Dym Systm. A., 40-46, 9, 2007.[2] U C De and J Sengupta. On a type of semi symmetric metric connection on an almostcontact metric manifold, Facta Universitatis (NIŠ) , Ser. Math. Inform., 87-96, 16, 2001.[3] U C De and B K De. Some properties of a semi symmetric metric connection on a Rie-mannian manifold. Istanbul Univ. Fen Fak. Mat. Derg., pp. 111-117, 54, 1995.[4] U C De and S C Biswas. On a type of semi symmetric metric connection on a Riemannianmanifold. Publ. Inst. Math. (Beograd) (N. S.), 90-96, 61, 75, 1997.[5] U C De and G C Ghosh. On generalized quasi-Einstein manifolds. Kyungpook Math. J.,607-615, 44, 4, 2004.[6] T Imai. Notes on semi symmetric metric connections. Tensor (N.S.), 293-296, 24, 1972.[7] S Kobayashi and K Nomizu. Foundations of Differential Geometry. W. Interscience Pub-lishers, New York, 1963.[8] C Murathan and C Özgür. Riemannian manifolds with semi-symmmetric metric connec-tion satisfying some semisymmetry conditions. Proceedings of the Estonian Academy ofSciences, 210-216, 57, 4, 2008.REFERENCES 161[9] E Pak. On the pseudo-Riemannian spaces. J. Korean Math. Soc., 23-31, 6,1969.[10] L Tamássy and T Q Binh. On the non-existence of certain connections with torsion andof constant curvature. Publ. Math. Debrecen, 283-288, 36, 1989.[11] K Yano. On semi symmetric metric connection. Rev. Roumaine, Math. Pures Appl., 1579-1586,15, 1970.[12] K Yano and M Kon. Structures on manifolds. Series in Pure Math., World Scientific, 1984.[13] F Ö Zengin S A Uysal and S A Demirbag˘. On sectional curvature of a Riemannian mani-fold with semi-symmetric connection. Annales Polonici Mathematici, (in print).

On a semi symmetric metric connection with a special condition on a Riemannian manifold

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