International Journal of Analysis and Applications ISSN 2291-8639 Volume 7, Number 2 (2015), 153-161 http://www.etamaths.com ON A TYPE OF PROJECTIVE SEMI-SYMMETRIC CONNECTION S. K. PAL1,∗, M. K. PANDEY2 AND R. N. SINGH1 Abstract. In the present paper, we have studied some properties of curvature tensors of special projective semi-symmetric connection. We have shown that curvature tensor of such a connection satisfies Bianchi’s identities. 1. Introduction The idea of semi-symmetric connection was introduced by A. Friedmann and J. A. Schouten [2] in 1924. In 1932, H. A. Hayden [4] studied semi-symmetric metric- connection. It was K. Yano [10] who started systematic study of semi-symmetric metric connection and this was further studied by T. Imai [6], R. S. Mishra and S. N. Pandey [9], U. C. De and B. K. De [1] and several other mathematicians ([7], [11]). In 2001, P. Zhao and H. Song [12] studied a semi-symmetric connection which is projectively equivalent to Levi-Civita connection and such a connection is called as projective semi-symmetric connection. They found an invariant under the transformation of projective semi-symmetric connection and showed that this invariant could degenerate into the Weyl projective curvature tensor under certain conditions. After this various papers ([3], [5], [13]) on projective semi-symmetric metric connection have appeared. The organization of the paper is as follows. After introduction we give some preliminary results in section 2. In sections 3, we present a brief account of special projective semi-symmetric connection. Section 4 is devoted to the study of special projective semi symmetric connection with recurrent curvature tensor. 2. Preliminaries Let Mn be an n-dimensional (n > 2) Riemannian manifold equipped with a Riemannian metric g and ∇ be the Levi-Civita connection associated with metric g. A linear connection ∇̄ on Mn is called the semi symmetric metric connection [10 ], if the torsion tensor T̄ of the connection ∇̄, given by (2.1) T̄(X,Y ) = ∇̄XY −∇̄Y X − [X,Y ] satisfies the condition (2.2) T̄(X,Y ) = π(Y )X −π(X)Y 2010 Mathematics Subject Classification. 53C12. Key words and phrases. Projective semi-symmetric connection; curvature tensor. c©2015 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 153 154 PAL, PANDEY AND SINGH and (2.3) (∇̄Xg)(Y,Z) = 0, where π is a 1 - form on Mn associated with vector field ρ, i.e., (2.4) π(X) = g(X,ρ). If the geodesic with respect to ∇̄ are always consistent with those of ∇, then ∇̄ is called a connection projectively equivalent to ∇. If ∇̄ is projective equivalent connection to ∇ as well as the semi-symmetric, then ∇̄ is called projective semi- symmetric connection. We also call ∇̄ as projective semi- symmetric transforma- tion. In this paper, we study a type of projective semi-symmetric connection ∇̄ intro- duced by P. Zhao and H. Song [12]. The connection is given by (2.5) ∇̄XY = ∇XY + ψ(Y )X + ψ(X)Y + φ(Y )X −φ(X)Y, where 1-forms φ and ψ are given as (2.6) φ(X) = 1 2 π(X) and ψ(X) = n− 1 2(n + 1) π(X). It is easy to observe that torsion tensor of projective semi- symmetric transforma- tion is same as given by the equation (2.2) and also that (2.7) (∇̄Xg)(Y,Z) = 1 n + 1 [2π(X)g(Y,Z) −nπ(Y )g(Z,X) −nπ(Z)g(X,Y ), i.e., the connection ∇̄ is a non metric one. Let R̄ and R be the curvature tensor of the manifold relative to the projective semi-symmetric connection ∇̄ and Levi-Civita connection ∇ respectively. It is known that [12] (2.8) R̄(X,Y,Z) = R(X,Y,Z) + β(X,Y )Z + α(X,Z)Y −α(Y,Z)X, where β(X,Y ) and α(X,Y ) are given by the following relations (2.9) β(X,Y ) = Ψ′(X,Y ) − Ψ′(Y,X) + Φ′(Y,X) − Φ′(X,Y ), (2.10) α(X,Y ) = Ψ′(X,Y ) + Φ′(Y,X) −ψ(X)φ(Y ) −φ(X)ψ(Y ), (2.11) Ψ′(X,Y ) = (∇Xψ)(Y ) −ψ(X)ψ(Y ) and (2.12) Φ′(X,Y ) = (∇Xφ)(Y ) −φ(X)φ(Y ). Contracting X in the equation (2.8), we get a relation between Ricci tensors R̄ic(Y,Z) and Ric(Y,Z) of manifold with respect to connections ∇̄ and ∇ respec- tively (2.13) R̄ic(Y,Z) = Ric(Y,Z) + β(Y,Z) − (n− 1)α(Y,Z). If r̄ and r are scalar curvatures of manifold with respect to connection ∇̄ and ∇ respectively, then from the equation (2.13), we get (2.14) r̄ = r + b− (n− 1)a, ON A TYPE OF PROJECTIVE SEMI-SYMMETRIC CONNECTION 155 where b = n∑ i = 1 β(ei,ei) and a = n∑ i = 1 α(ei,ei). The Weyl-projective curvature tensor W , conharmonic curvature tensor P and concircular curvature tensor I are given by [9] (2.15) W(X,Y,Z) = R(X,Y,Z) + 1 n− 1 {Ric(X,Z)Y −Ric(Y,Z)X}, P(X,Y,Z) = R(X,Y,Z)− 1 n− 2 [Ric(Y,Z)X −Ric(X,Z)Y + g(Y,Z)QX −g(X,Z)QY ], (2.16) where (2.17) g(QX,Y ) = Ric(X,Y ) and (2.18) I(X,Y,Z) = R(X,Y,Z) − r n− 1 [g(Y,Z)X −g(X,Z)Y ]. 3. Special Projective Semi-Symmetric Connection In this section, we consider a projective semi-symmetric connection ∇̄ given by the equation (2.5) whose associated 1-form π is closed, i.e., (3.1) (∇̄Xπ)Y = (∇̄Y π)X. Such a connection ∇̄ is called special projective semi-symmetric connection [12]. It is easy to verify that both the 1-forms φ and ψ are closed as the 1-form π is closed and also that the tensors Φ′ and Ψ′ are symmetric. Consequently, we get (3.2) β(X,Y ) = 0 and (3.3) α(X,Y ) = α(Y,X). In view of the equations (3.1) and (3.2), the expressions (2.8), (2.13) and (2.14) reduces to (3.4) R̄(X,Y,Z) = R(X,Y,Z) + α(X,Z)Y −α(Y,Z)X, (3.5) R̄ic(Y,Z) = Ric(Y,Z) − (n− 1)α(Y,Z) and (3.6) r̄ = r − (n− 1)a. It is easy to observe that the Ricci tensor R̄ic(Y,Z) is symmetric. Now, we prove the following theorems: Theorem 3.1. Curvature tensor of special projective semi-symmetric connection satisfies Bianchi’s first identity. 156 PAL, PANDEY AND SINGH Proof : Writing two more equations by cyclic permutations of X, Y and Z from equation (3.4), we get R̄(Y,Z,X) = R(Y,Z,X) + α(Y,X)Z −α(Z,X)Y, and R̄(Z,X,Y ) = R(Z,X,Y ) + α(Z,Y )X −α(X,Y )Z. Adding these equations to the equation (3.4), we get result. Theorem 3.2. Curvature tensor of special projective semi-symmetric connection satisfies Bianchi’s second identity if α is parallel tensor with respect to Levi-Civita connection ∇. Proof : Suppose α is a parallel tensor with respect to Levi-Civita connection ∇, i.e., ∇α = 0. Now differentiating the equation (3.4) covariantly with respect to the connection ∇, we have (3.7) (∇XR̄)(Y,Z,U) = (∇XR)(Y,Z,U). Writing two more equations by cyclic permutations of X, Y and Z in above equa- tion, we get (3.8) (∇Y R̄)(Z,X,U) = (∇Y R)(Z,X,U), and (3.9) (∇ZR̄)(X,Y,U) = (∇ZR)(X,Y,U). Adding the equations (3.7), (3.8) and (3.9), we get (∇XR̄)(Y,Z,U) + (∇Y R̄)(Z,X,U) + (∇ZR̄)(X,Y,U) = 0. This shows that the curvature tensor of special projective semi-symmetric connec- tion satisfies Bianchi’s second identity. Theorem 3.3. The Weyl-projective curvature tensor of Riemannian manifold with respect to the special projective semi-symmetric connection ∇̄ satisfies W(X,Y,Z) + W(Y,Z,X) + W(Z,X,Y ) = 0. Proof : The Weyl-projective curvature tensor of Riemannian Manifold with respect to special projective semi-symmetric connection ∇̄ is given by (3.10) W(X,Y,Z) = R̄(X,Y,Z) − 1 n− 1 [R̄ic(Y,Z)X − R̄ic(X,Z)Y ]. Writing two more equations by cyclic permutations of X, Y and Z in above equation, we get (3.11) W(Y,Z,X) = R̄(Y,Z,X) − 1 n− 1 [R̄ic(Z,X)Y − R̄ic(Y,X)Z], (3.12) W(Z,X,Y ) = R̄(Z,X,Y ) − 1 n− 1 [R̄ic(X,Y )Z − R̄ic(Z,Y )X]. Adding the equations (3.10), (3.11) and (3.12), we get W(X,Y,Z) + W(Y,Z,X) + W(Z,X,Y ) = 0. ON A TYPE OF PROJECTIVE SEMI-SYMMETRIC CONNECTION 157 4. Special Projective Semi-Symmetric Connection with Recurrent Curvature Tensor In this section, we consider a special projective semi-symmetric connection ∇̄ whose curvature tensor R̄ is recurrent with respect to the Levi-Civita connection ∇, i.e., (4.1) (∇UR̄)(X,Y,Z) = B(U)R̄(X,Y,Z), where B is a non-zero 1-form. Differentiating the equation (3.4) covariantly with respect to the Levi-Civita con- nection ∇, we get (4.2) (∇UR̄)(X,Y,Z) = (∇UR)(X,Y,Z) + (∇Uα)(X,Z)Y − (∇Uα)(Y,Z)X. Contracting X in the above equation, we have (4.3) (∇UR̄ic)(Y,Z) = (∇URic)(Y,Z) − (n− 1)(∇Uα)(Y,Z). Putting Y = Z = ei in the above equation and taking summation over i, 1 ≤ i ≤ n, we get (4.4) (∇U r̄) = (∇Ur) − (n− 1)(∇Ua). Now the equations (3.4) and (4.2) together give (∇UR̄)(X,Y,Z) −B(U)R̄(X,Y,Z) =(∇UR)(X,Y,Z) −B(U)R(X,Y,Z) +[(∇Uα)(X,Z) −B(U)α(X,Z)]Y −[(∇Uα)(Y,Z) −B(U)α(Y,Z)]X, (4.5) which, in view of the equation (4.1), reduces to (∇UR)(X,Y,Z) −B(U)R(X,Y,Z) =[(∇Uα)(Y,Z) −B(U)α(Y,Z)]X −[(∇Uα)(X,Z) −B(U)α(X,Z)]Y. (4.6) Contracting X in above, we get (4.7) (∇URic)(Y,Z) −B(U)Ric(Y,Z) = (n− 1){(∇Uα)(Y,Z) −B(U)α(Y,Z)}. Further, we obtain (4.8) (∇Ur) −B(U)r = (n− 1){(∇Ua) −B(U)a}. Also, from the equation (2.17), we have (4.9) g((∇UQ)X,Y ) = (∇URic)(X,Y ), which can be written as (4.10) g((∇UQ)X −B(U)QX,Y ) = (∇URic)(X,Y ) −B(U)Ric(X,Y ). Now we prove following theorems: Theorem 4.1. If the curvature tensor of special projective semi-symmetric con- nection on a Riemannian manifold Mn is recurrent with respect to the Levi-Civita connection then manifold Mn is projectively recurrent. Proof : Differentiating the projective curvature tensor W given by (2.15) covari- antly with respect to Levi-Civita connection ∇, we have (4.11) (∇UW)(X,Y,Z) = (∇UR)(X,Y,Z)+ 1 n− 1 {(∇URic)(X,Z)Y −(∇URic)(Y,Z)X}. 158 PAL, PANDEY AND SINGH The above equation gives (∇UW)(X,Y,Z) −B(U)W(X,Y,Z) =(∇UR)(X,Y,Z) −B(U)R(X,Y,Z) + 1 n− 1 [{(∇URic)(X,Z) −B(U)Ric(X,Z)}Y −{(∇URic)(Y,Z) −B(U)Ric(Y,Z)}X]. (4.12) Using equation (4.6) and (4.7) in above, we get (∇UW)(X,Y,Z) = B(U)W(X,Y,Z), which proves the statement. Theorem 4.2. : A Riemannian manifold Mn admitting a special projective semi- symmetric connection whose curvature tensor and tensor α are recurrent with re- spect to the Levi-Civita connection, is conharmonically recurrent. Proof: Differentiating covariantly the equation (2.16) with respect to the Levi- Civita connection, we get (∇UP)(X,Y,Z) =(∇UR)(X,Y,Z) − 1 n− 2 [(∇URic)(Y,Z)X − (∇URic)(X,Z)Y. +g(Y,Z)(∇UQ)X −g(X,Z)(∇UQ)Y ], (4.13) From above, we have (∇UP)(X,Y,Z) −B(U)P(X,Y,Z) =(∇UR)(X,Y,Z) −B(U)R(X,Y,Z) − 1 n− 2 [{(∇URic)(Y,Z) −B(U)Ric(Y,Z)}X −{(∇URic)(X,Z) −B(U)Ric(X,Z)}Y +g(Y,Z){(∇UQ)X −B(U)QX} −g(X,Z){(∇UQ)Y −B(U)QY}]. (4.14) If the tensor α and the curvature tensor of the special projective semi-symmetric connection ∇̄ are recurrent with respect to the Levi-Civita connection ∇, then from the equations (4.6), (4.7) and (4.10), we get (∇UP)(X,Y,Z) = B(U)P(X,Y,Z), which shows that manifold is conharmonically recurrent. Theorem 4.3. A Riemannian manifold Mn admitting a special projective semi- symmetric connection whose curvature tensor and tensor α are recurrent with re- spect to Levi-Civita connection, is concircular recurrent. Proof: Differentiating the concircular curvature tensor I of Mn given by the equa- tion (2.18) covariantly with respect to the Levi- Civita connection ∇, we have (4.15) (∇UI)(X,Y,Z) = (∇UR)(X,Y,Z) − ∇Ur (n− 1) {g(Y,Z)X −g(X,Z)Y}. ON A TYPE OF PROJECTIVE SEMI-SYMMETRIC CONNECTION 159 From this, we have (∇UI)(X,Y,Z) −B(U)I(X,Y,Z) =(∇UR)(X,Y,Z) −B(U)R(X,Y,Z) − ∇Ur −B(U)r (n− 1) {g(Y,Z)X −g(X,Z)Y}. (4.16) If the tensor α and the curvature tensor of the special projective semi-symmetric connection ∇̄ are recurrent with respect to the Levi-Civita connection ∇, then from the equations (4.6), (4.7) and (4.8), we get (∇UI)(X,Y,Z) = B(U)I(X,Y,Z). Theorem 4.4. Let Mn be a Riemannian manifold admitting a special projec- tive semi-symmetric connection whose Ricci-tensor is recurrent with respect to the Levi-Civita connection. If the manifold is projectively recurrent with respect to Levi-Civita connection, then the curvature tensor of the special projective semi- symmetric connection is recurrent. Proof: Let the manifold Mn be projectively recurrent with respect to Levi Civita connection ∇. Then from the equation (4.12), we have (∇UR)(X,Y,Z) −B(U)R(X,Y,Z) = 1 n− 1 [{(∇URic)(Y,Z) −B(U)Ric(Y,Z)X} −{(∇URic)(X,Z) −B(U)Ric(X,Z)Y}]. (4.17) Now, from equations (3.5) and (4.3), we get (∇UR̄ic)(Y,Z) −B(U)R̄ic(Y,Z) =(∇URic)(Y,Z) −B(U)Ric(Y,Z) −(n− 1){(∇Uα)(Y,Z) −B(U)α(Y,Z)}. (4.18) Since the Ricci tensor of the special projective semi-symmetric connection ∇̄ is recurrent with respect to the Levi-Civita connection ∇, hence the above equation gives (4.19) (∇URic)(Y,Z) −B(U)Ric(Y,Z) = (n− 1){(∇Uα)(Y,Z) −B(U)α(Y,Z)}. Thus, from the equations (4.17) and (4.19), we get (∇UR)(X,Y,Z) −B(U)R(X,Y,Z) ={(∇Uα)(Y,Z) −B(U)α(Y,Z)}X −{(∇Uα)(X,Z) −B(U)α(X,Z)}Y, (4.20) which, on using in the equation (4.5), gives (4.21) (∇UR̄)(X,Y,Z) = B(U)R̄(X,Y,Z). This proves the statement. Theorem 4.5. Let Mn be a Riemannian manifold admitting a special projective semi-symmetric connection whose Ricci-tensor is recurrent with respect to the Levi- Civita connection. If the manifold is of constant curvature, then the curvature tensor of the special projective semi-symmetric connection is recurrent with respect to the Levi-Civita connection. 160 PAL, PANDEY AND SINGH Proof: If the Riemannian manifold Mn is of constant curvature, then we have [9] (4.22) R(X,Y,Z) = 1 n− 1 {Ric(Y,Z)X −Ric(X,Z)Y}. Using the above equation in the equation (3.4), we have (4.23) R̄(X,Y,Z) = 1 n− 1 [{Ric(Y,Z)−(n−1)α(Y,Z)}X−{Ric(X,Z)−(n−1)α(X,Z)}Y ], which, on using the equation (3.5), gives (4.24) R̄(X,Y,Z) = 1 n− 1 {R̄ic(Y,Z)X − R̄ic(X,Z)Y}. Differentiating the above equation covariantly with respect to the Levi-Civita con- nection, we have (∇UR̄)(X,Y,Z) = 1 n− 1 {(∇UR̄ic)(Y,Z)X − (∇UR̄ic)(X,Z)Y}, which can be written as (∇UR̄)(X,Y,Z) −B(U)R̄(X,Y,Z) = 1 n− 1 [{(∇UR̄ic)(Y,Z) −B(U)R̄ic(Y,Z)}X −{(∇UR̄ic)(X,Z) −B(U)R̄ic(X,Z)}Y ]. (4.25) Since the Ricci tensor of special projective semi-symmetric connection is recurrent with respect to the Levi-Civita connection ∇, hence from the above equation, we have (∇UR̄)(X,Y,Z) = B(U)R̄(X,Y,Z), which proves the statement. References [1] De, U.C. and De, B. K., On a type of semi-symmetric connection on a Riemannian manifold, Ganita, 47(2), (1996), 11-24. [2] Friedmann, A. and Schouten, J. A., Uber die geometrie der halbsymmetrischen Ubertragun- gen, Math. Zeitschr., 21(1), (1924), 211-223. [3] Fengyun, Fu. and Zhao. P., A property on geodesic mappings of pseudo-symmetric Riemann- ian manifolds, Bull. Malays. Math. Sci. Soc.(2), 33(2), (2010), 265-272. [4] Hayden, H. 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[11] Zhao, P. and Shangguan L., On semi-symmetric connection, J. of Henan Normal University (Natural Science), 19(4), (1994), 13-16. [12] Zhao, P. and Song H., An invariant of the projective semisymmetric connection, Chinese Quarterly J. of Math., 17(4), (2001), 48-52. [13] Zhao, P., Some properties of projective semi-symmetric connections, Int. Math.Forum, 3(7), (2008), 341-347. ON A TYPE OF PROJECTIVE SEMI-SYMMETRIC CONNECTION 161 1Department of Mathematical Sciences, A. P. S. University, Rewa, (M.P.), India, 486003 2Department of Mathematics, University Institute of Technology, Rajiv Gandhi Proudyogiki Vishwavidyalaya Bhopal, (M.P.), India, 462036 ∗Corresponding author