7_1048_yilmaz.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 4, No. 2, 2011, 152-161 ISSN 1307-5543 – www.ejpam.com On a Semi Symmetric Metric Connection with a Special Condition on a Riemannian Manifold Hülya Bağdatlı Yılmaz1,∗, Füsun Özen Zengin2, and S. Aynur Uysal3 1 Department of Mathematics, Faculty of Sciences and Letters, Marmara University, Istanbul, Turkey 2 Department of Mathematics, Faculty of Sciences and Letters, Istanbul Technical University, Istan- bul, Turkey 3 Department of Mathematics, Faculty of Sciences and Letters, Dogus University, Istanbul, Turkey Abstract. 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. 2000 Mathematics Subject Classifications: 53B15, 53B20, 53C15 Key Words and Phrases: Semi symmetric metric connection, Generalized quasi -Einstein manifold, Mixed generalized quasi constant curvature manifold, Mixed generalized quasi-Einstein manifold 1. Introduction The 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 Ricci tensor Rk j is not identically zero and satisfies the condition Rk j = αgk j + βuku j + γvkv j where α,β ,γ are non-zero scalars and uk and vk are covariant vectors such that uk and vk are orthogonal to each other vector fields on M . The mixed generalized quasi Einstein manifold was 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), 152-161 153 mixed generalized quasi Einstein manifold if its Ricci tensor Rk j is non-zero and satisfies the condition Rk j = αgk j + βaka j + γbk b j + ϑ � ak b j + bka j � (1) where α,β ,γ,ϑ are non-zero scalars and ak and bk are covariant vectors such that ak and bk are orthogonal unit vector fields on M . Moreover, it is stated that a Riemannian manifold is of a mixed generalized quasi constant curvature if the curvature tensor of this manifold satisfies the condition Rik jm = p � gk j gim− gi j gkm � (2) + q � gimaka j − gkmaia j + gk jaiam− gi jakam � + s � gim bk b j − gkmbi b j + gk j bi bm− gi j bk bm � + t �¦ ak b j + bka j © gim− ¦ ai b j + bia j © gkm + � ai bm+ biam gk j − � ak bm+ bkam gi j � where p,q, r, s, t are non-zero scalars and ak and bk are covariant vectors such that ak and bk are 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 form T (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 connection with the torsion tensor of the form (4) is a semi symmetric metric connection. In [11], Yano proved that in order that a Riemannian manifold admits a semi symmetric metric connection whose curvature tensor vanishes, it is necessary and sufficient that the Riemannian manifold be conformally flat, for some properties of Riemannian manifolds with a semi symmetric metric connection, see also [4, 6, 8, 10] The components of semi symmetric metric connection are given by Γl ik = ¨ l ik « + δl i wk − gikw l (5) where wt and w l = wt g t l are covariant and contravariant components of a vector field, re- spectively and ∇kw j =∇kw j −wkw j +wgk j, w = wt w t (6) H. Yılmaz, F. Zengin, S. Uysal / Eur. J. Pure Appl. Math, 4 (2011), 152-161 154 By using (5), we obtain, Rik jm = Rik jm− gimπk j + gkmπi j − gk jπim+ gi jπkm (7) where Rik jm and Rik jm are the Riemannian curvature tensors of ∇ and ∇, respectively [11]. And π is a tensor field of type (0,2) defined by πk j =∇kw j −wkw j + 1 2 gk jw (8) Transvecting the equation (7) with g im, we get Rk j = Rk j − (n− 2)πk j −πgk j (9) where Rk j and Rk j are the Ricci tensors for the connections ∇ and ∇, respectively and π= πim g im. Multiplying (9) by gk j, we obtain R= R− 2(n− 1)π (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 Metric Connection De 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 torsion tensor 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 torsion T satisfies the following condition ∇ j T l ik = a j T l ik + b j b l gik +δ l j biak (11) where bl = bt g t l . The equation (4) can be written in the following form T l ik = δ l i wk − δ l kwi Contracting on l and i in the last equation, we get T l lk = (n− 1)wk (12) Thus, we can find ∇ j T l lk = (n− 1)∇ jwk (13) Moreover, by using (11), we obtain ∇ j T l lk = a j T l lk+ b j bk + b jak (14) H. Yılmaz, F. Zengin, S. Uysal / Eur. J. Pure Appl. Math, 4 (2011), 152-161 155 From (12)-(14), it is found that ∇ jwk = a jwk + 1 n− 1 b j bk + 1 n− 1 b 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 find πk j =∇kw j − 1 2 gk jw (17) Again, using (15) and (17) , we obtain πk j = akw j + 1 n− 1 bk b j + 1 n− 1 bka j − 1 2 gk jw (18) Then, if we substitute (18) in (7), we get Rik jm = Rik jm (19) +w � gim gk j − gkmgi j � − gim � akw j + 1 n− 1 bk b j + 1 n− 1 bka j � + gkm � aiw j + 1 n− 1 bi b j + 1 n− 1 bia j � − gk j � aiwm+ 1 n− 1 bi bm+ 1 n− 1 biam � + gi j � akwm+ 1 n− 1 bk bm+ 1 n− 1 bkam � From (19), we have the following theorem: Theorem 1. The curvature tensor of a Riemannian manifold admitting a semi symmetric metric connection 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 the associated 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 symmetric metric connection ∇ to be symmetric is that the curvature tensor R of (0,4) type with respect to the connection ∇ satisfies one of the following two conditions: i Rik jm = R jmik H. Yılmaz, F. Zengin, S. Uysal / Eur. J. Pure Appl. Math, 4 (2011), 152-161 156 ii Rik jm+ Rk jim+ R jikm = 0. From (19), we can write R jmik = R jmik (20) +w � g jk gim− gmk g ji � − g jk � amwi + 1 n− 1 bm bi + 1 n− 1 bmai � + gmk � a jwi + 1 n− 1 b j bi + 1 n− 1 b jai � − gmi � a jwk + 1 n− 1 b j bk + 1 n− 1 b jak � + g ji � amwk + 1 n− 1 bm bk + 1 n− 1 bmak � 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. In virtue of Theorem 2, the Ricci tensor of a Riemannian manifold with a semi symmetric metric connection is symmetric. Thus, due to Theorem 3, we get R jmik = Rik jm (21) In case the equation (21) is satisfied, we find 0=gim � a j � wk − 1 n− 1 bk � − ak � w j − 1 n− 1 b j �� (22) + gkm � ai � w j − 1 n− 1 b j � − a j � wi − 1 n− 1 bi �� + gk j � am � wi − 1 n− 1 bi � − ai � wm − 1 n− 1 bm �� + gi j � ak � wm − 1 n− 1 bm � − am � wk − 1 n− 1 bk �� Transvecting (22) with g im , we get (2− n) � ak � w j − 1 n− 1 b j � − a j � wk − 1 n− 1 bk �� = 0 (23) Since n> 2, we get ak � w j − 1 n− 1 b j � = a j � wk − 1 n− 1 bk � (24) Now, permutating the indices and adding the three equations side by side, we obtain Rik 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 − 1 n− 1 bk � − ak � w j − 1 n− 1 b j �� + gkm � ai � w j − 1 n− 1 b j � − a j � wi − 1 n− 1 bi �� + g jm � ak � wi − 1 n− 1 bi � − ai � wk − 1 n− 1 bk �� 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 Bianchi Identity. 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 semi symmetric 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 Riemannian manifold 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 + 1 n− 1 b j bk + 1 n− 1 b jak +wkw j − g jkw (26) Subtracting the corresponding equation found by interchanging k and j in (26) from (26), we get the equation (24). Thus, by using Theorem 2, Theorem 3 and Theorem 4, we have the following Theorem: Theorem 5. In a Riemannian manifold with a semi symmetric metric connection whose the torsion 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 of block symmetry, ii The curvature tensor with respect to the connection ∇ of this manifold satisfies the first Bianchi Identity, iii The Ricci tensor of this manifold is symmetric. 3. Conformally Flat Manifolds with Semi Symmetric Metric Connection Satisfying some Special Condition In this section, we shall investigate a Riemannian manifold M admitting a semi symmetric metric connection whose the torsion tensor satisfies a special condition in the case of confor- mally flat. Firstly, we consider the condition (11). Then, ∇ j T l ik = a j T l ik + b j b l gik +δ l j biak (27) H. Yılmaz, F. Zengin, S. Uysal / Eur. J. Pure Appl. Math, 4 (2011), 152-161 158 where ak and bk be orthogonal to each other. The conformal curvature tensor is given by Cik jm = Rik jm− 1 n− 2 � Rimgk j − Rkmgi j + Rk j gim− Ri j gkm � (28) + R (n− 1)(n− 2) � gim gk 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 Riemannian manifold be conformally flat. Suppose that this manifold is conformally flat. Hence, we can write Rik jm = 0 (29) Therefore, due to (7) and (29), we obtain Rik jm = gimπk j − gkmπi j + gk jπim− gi jπkm (30) Multiplying (29) by g im, we get the corresponding identity Rk j = 0 (31) Transvecting (19) with g im and using (31), we have Rk j = � (1− n)w + � amwm+ 1 n− 1 b+ 1 n− 1 bmam �� gk j (32) + (n− 2) � akw j + 1 n− 1 bk b j + 1 n− 1 bka j � where am = ai g im, b = bmbm 6= 0. Since ak and bk are the orthogonal vector fields, it can be written Rk j = � (1− n)w +φ + 1 n− 1 b � gk j + (n− 2) � akw j + 1 n− 1 bk b j + 1 n− 1 bka j � (33) where amwm = φ is a non-zero scalar function. Subtracting (33) from the corresponding equation found by interchanging k and j in (33), we get (24). Transvecting (24) with a j bk , we find bkwk = ab n− 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 get w j = θa j + 1 n− 1 b j (35) H. Yılmaz, F. Zengin, S. Uysal / Eur. J. Pure Appl. Math, 4 (2011), 152-161 159 where θ = φ a 6= 0. By using (35), we find that a j is not orthogonal to w j . Substituting (29) and (35) in (19), we obtain Rik jm = w � gkmgi j − gimgk j � (36) + θ � gimaka j − gkmaia j + gk jaiam− gi jakam � + 1 n− 1 � gim bk b j − gkmbi b j + gk j bi bm− gi j bk bm � + 1 n− 1 � gim � ak b j + bka j � − gkm � ai b j + bia j � +gk j � ai bm+ biam � − gi j � ak bm+ bkam � � If w = θφ + ab (n−1)2 6= 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 quasi constant curvature. Multiplying (36) by g im,we obtain Rk j = µgk j + (n− 2)θaka j + � n− 2 n− 1 � � bk b j + ak b j + bka j � (37) where µ = (1− n)w + θa+ b n− 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 obtain Rk j = � p(n− 1) + qa+ bs � gk j + q(n− 2)aka j (39) + s(n− 2)bk b j + t(n− 2) � ak b j + bka j � Transvecting (39) with gk j , we find R= (n− 1) � np+ 2qa+ 2sb � (40) Let us substitute (2) , (39) and (40) in (28).Then, if w = −p,θ = q and t = s = 1 n−1 , we get Cik jm = 0 We may now establish the following theorem: Theorem 7. In a Riemannian manifold with a semi symmetric metric connection whose the torsion tensor satisfies the condition (27), a necessary and sufficient condition that this manifold to 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 manifold is a mixed generalized quasi Einstein manifold. Thus, we can state the following theorem: REFERENCES 160 Theorem 8. A conformal flat Riemannian manifold with a semi symmetric metric connection whose 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 with constant sectional curvature, then this manifold is conformally flat. Thus, in virtue of Theorem 7, Theorem 8 and Theorem 9, we can establish the following theorems: Theorem 10. If the sectional curvature of a Riemannian manifold with a semi symmetric metric connection whose the torsion tensor satisfies the condition (27) is independent from the orienta- tion chosen, then i 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 metric connection whose the torsion tensor satisfies the condition (27) is independent from the orienta- tion chosen, then the condition (24) is satisfied. 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