CUBO, A Mathematical Journal Vol.22, N◦02, (257–271). August 2020 http://dx.doi.org/10.4067/S0719-06462020000200257 Received: 07 November, 2019 | Accepted: 20 July, 2020 Results on para-Sasakian manifold admitting a quarter symmetric metric connection Vishnuvardhana. S.V. 1 and Venkatesha 2 1 Department of Mathematics, GITAM School of Science, GITAM (Deemed to be University) Bengaluru, Karnataka-561 203, INDIA. 2 Department of Mathematics, Kuvempu University, Shankaraghatta - 577 451, Shimoga, Karnataka, INDIA. svvishnuvardhana@gmail.com, vensmath@gmail.com ABSTRACT In this paper we have studied pseudosymmetric, Ricci-pseudosymmetric and projec- tively pseudosymmetric para-Sasakian manifold admitting a quarter-symmetric metric connection and constructed examples of 3-dimensional and 5-dimensional para-Sasakian manifold admitting a quarter-symmetric metric connection to verify our results. RESUMEN En este art́ıculo hemos estudiado variedades para-Sasakianas seudosimétricas, Ricci- seudosimétricas y proyectivamente seudosimétricas que admiten una conexión métrica cuarto-simétrica, y construimos ejemplos de variedades para-Sasakianas 3-dimensional y 5-dimensional que admiten una conexión métrica cuarto-simétrica para verificar nue- stros resultados. Keywords and Phrases: Para-Sasakian manifold, pseudosymmetric, Ricci-pseudosymmetric, projectively pseudosymmetric, quarter-symmetric metric connection. 2020 AMS Mathematics Subject Classification: 53C35, 53D40. c©2020 by the author. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://dx.doi.org/10.4067/S0719-06462020000200257 258 Vishnuvardhana. S.V. & Venkatesha CUBO 22, 2 (2020) 1 Introduction One of the most important geometric property of a space is symmetry. Spaces admitting some sense of symmetry play an important role in differential geometry and general relativity. Cartan [5] introduced locally symmetric spaces, i.e., the Riemannian manifold (M, g) for which ∇R = 0, where ∇ denotes the Levi-Civita connection of the metric. The integrability condition of ∇R = 0 is R · R = 0. Thus, every locally symmetric space satisfies R · R = 0, whereby the first R stands for the curvature operator of (M, g), i.e., for tangent vector fields X and Y one has R(X, Y ) = ∇X∇Y −∇Y ∇X −∇[X,Y ], which acts as a derivation on the second R which stands for the Riemann- Christoffel curvature tensor. The converse however does not hold in general. The spaces for which R · R = 0 holds at every point were called semi-symmetric spaces and which were classified by Szabo [19]. Semisymmetric manifolds form a subclass of the class of pseudosymmetric manifolds. In some spaces R · R is not identically zero, these turn out to be the pseudo-symmetric spaces of Deszcz [9, 10, 11], which were characterized by the condition R · R = L Q(g, R), where L is a real function on M and Q(g, R) is the Tachibana tensor of M. If at every point of M the curvature tensor satisfies the condition R(X, Y ) · J = LJ [(X ∧g Y ) · J ], (1.1) then a Riemannian manifold M is called pseudosymmetric (resp., Ricci-pseudosymmetric, projec- tively pseudosymmetric) when J = R(resp., S, P) . Here (X ∧g Y ) is an endomorphism and is defined by (X ∧g Y )Z = g(Y, Z)X − g(X, Z)Y and LJ is some function on UJ = {x ∈ M : J 6= 0} at x. A geometric interpretation of the notion of pseudosymmetry is given in [13]. It is also easy to see that every pseudosymmetric manifold is Ricci-pseudosymmetric, but the converse is not true. An analogue to the almost contact structure, the notion of almost paracontact structure was introduced by Sato [18]. An almost contact manifold is always odd-dimensional but an almost paracontact manifold could be of even dimension as well. Kaneyuki and Williams [14] studied the almost paracontact structure on a pseudo-Riemannian manifold. Recently, almost paracontact geometry in particular, para-Sasakian geometry has taking interest, because of its interplay with the theory of para-Kahler manifolds and its role in pseudo-Riemannian geometry and mathematical physics ([4, 7, 8], etc.,). As a generalization of semi-symmetric connection, quarter-symmetric connection was intro- duced. Quarter-symmetric connection on a differentiable manifold with affine connection was defined and studied by Golab [12]. From thereafter many geometers studied this connection on different manifolds. Para-Sasakian manifold with respect to quarter-symmetric metric connection was studied by CUBO 22, 2 (2020) Results on para-Sasakian manifold admitting a quarter symmetric . . . 259 De et.al., [16, 1], Pradeep Kumar et.al., [17] and Bisht and Shanker [15]. Motivated by the above studies in this article we study properties of projective curvature tensor on para-Sasakian manifold admitting a quarter-symmetric metric connection. The organi- zation of the paper is as follows: In Section 2, we present some basic notions of para-Sasakian manifold and quarter-symmetric metric connection on it. Section 3 and 4 are respectively devoted to study the pseudosymmetric and Ricci-pseudosymmetric para-Sasakian manifold admitting a quarter-symmetric metric connection. Here we prove that if a para-Sasakian manifold Mn admit- ting a quarter-symmetric metric connection is Pseudosymmetric (resp., Ricci pseudosymmetric) then Mn is an Einstein manifold with respect to quarter-symmetric metric connection or it satisfies L R̃ = −2 (resp., L S̃ = −2). Section 5 and 6 are concerned with projectively flat and projectively pseudosymmetric para-Sasakian manifold Mn admitting a quarter-symmetric metric connection. Finally, we construct examples of 3-dimensional and 5-dimensional para-Sasakian manifold admit- ting a quarter-symmetric metric connection and we find some of its geometric characteristics. 2 Preliminaries A differential manifold Mn is said to admit an almost paracontact Riemannian structure (φ, ξ, η, g), where φ is a tensor field of type (1, 1), ξ is a vector field, η is a 1-form and g is a Riemannian metric on Mn such that φ2X = X − η(X)ξ, η(ξ) = 1, φ(ξ) = 0, η(φX) = 0, (2.1) g(X, ξ) = η(X), g(φX, φY ) = g(X, Y ) − η(X)η(Y ), (2.2) for all vector fields X, Y ∈ χ(Mn). If (φ, ξ, η, g) on Mn satisfies the following equations (∇Xφ)Y = −g(X, Y )ξ − η(Y )X + 2η(X)η(Y )ξ, (2.3) dη = 0 and ∇Xξ = φX, (2.4) then Mn is called para-Sasakian manifold [3]. In a para-Sasakian manifold, the following relations hold [6]: (∇Xη)Y = −g(X, Y ) + η(X)η(Y ), (2.5) η(R(X, Y )Z) = g(X, Z)η(Y ) − g(Y, Z)η(X), (2.6) R(X, Y )ξ = η(X)Y − η(Y )X, R(ξ, X)Y = η(Y )X − g(X, Y )ξ, (2.7) S(X, ξ) = −(n − 1)η(X), (2.8) S(φX, φY ) = S(X, Y ) + (n − 1)η(X)η(Y ), (2.9) for every vector fields X, Y, Z on Mn. Here ∇ denotes the Levi-Civita connection, R denotes the Riemannian curvature tensor and S denotes the Ricci curvature tensor. 260 Vishnuvardhana. S.V. & Venkatesha CUBO 22, 2 (2020) Here we consider a quarter-symmetric metric connection ∇̃ on a para-Sasakian manifold [16] given by ∇̃XY = ∇XY + η(Y )φX − g(φX, Y )ξ. (2.10) The relation between curvature tensor R̃(X, Y )Z of Mn with respect to quarter-symmetric metric connection ∇̃ and the curvature tensor R(X, Y )Z with respect to the Levi-Civita connection ∇ is given by R̃(X, Y )Z = R(X, Y )Z + 3g(φX, Z)φY − 3g(φY, Z)φX +{η(X)Y − η(Y )X}η(Z) − [g(Y, Z)η(X) − η(Y )g(X, Z)]ξ. (2.11) Also from (2.11) we obtain S̃(Y, Z) = S(Y, Z) + 2g(Y, Z) − (n + 1)η(Y )η(Z) − 3traceφ g(φY, Z), (2.12) where S̃ and S are Ricci tensors of connections ∇̃ and ∇ respectively. 3 Pseudosymmetric para-Sasakian manifold admitting a quarter- symmetric metric connection A para-Sasakian manifold Mn admitting a quarter-symmetric metric connection is said to be pseudosymmetric if R̃(X, Y ) · R̃ = L R̃ [(X ∧g Y ) · R̃], (3.1) holds on the set U R̃ = {x ∈ Mn : R̃ 6= 0 at x}, where L R̃ is some function on U R̃ . Suppose that Mn be pseudosymmetric, then in view of (3.1) we have R̃(ξ, Y )R̃(U, V )W − R̃(R̃(ξ, Y )U, V )W − R̃(U, R̃(ξ, Y )V )W −R̃(U, V )R̃(ξ, Y )W = L R̃ [(ξ ∧g Y )R̃(U, V )W − R̃((ξ ∧g Y )U, V )W −R̃(U, (ξ ∧g Y )V )W − R̃(U, V )(ξ ∧g Y )W ]. (3.2) By virtue of (2.7) and (2.11), (3.2) takes the form (L R̃ + 2)[η(R̃(U, V )W)Y − g(Y, R̃(U, V )W)ξ − η(U)R̃(Y, V )W + g(Y, U)R̃(ξ, V )W −η(V )R̃(U, Y )W + g(Y, V )R̃(U, ξ)W − η(W)R̃(U, V )Y + g(Y, W)R̃(U, V )ξ] = 0. (3.3) Taking inner product of (3.3) with ξ and using (2.6) and (2.11), we get (L R̃ + 2)[g(Y, R(U, V )W) + 3g(φU, W)g(φV, Y ) − 3g(φV, W)g(φU, Y ) +η(W){η(U)g(V, Y ) − η(V )g(U, Y )} − {g(V, W)η(U) − η(V )g(U, W)}η(Y ) +2{g(V, W)g(Y, U) − g(V, Y )g(U, W)}] = 0. (3.4) CUBO 22, 2 (2020) Results on para-Sasakian manifold admitting a quarter symmetric . . . 261 Assuming that L R̃ + 2 6= 0, the above equation becomes g(Y, R(U, V )W) + 3g(φU, W)g(φV, Y ) − 3g(φV, W)g(φU, Y ) +η(W){η(U)g(V, Y ) − η(V )g(U, Y )} − [g(V, W)η(U) − η(V )g(U, W)]η(Y ) +2[g(V, W)g(Y, U) − g((V, Y )g(U, W)] = 0. (3.5) Putting V = W = ei, where {ei} is an orthonormal basis of the tangent space at each point of the manifold and taking summation over i, i = 1, 2, 3, · · · , n, we get S̃(Y, U) = −2(n − 1)g(Y, U). (3.6) Hence, we can state the following: Theorem 1. If a para-Sasakian manifold Mn admitting a quarter-symmetric metric connection is pseudosymmetric then Mn is an Einstein manifold with respect to quarter-symmetric metric connection or it satisfies L R̃ = −2. 4 Ricci-pseudosymmetric para-Sasakian manifold admitting a quarter-symmetric metric connection A para-Sasakian manifold Mn admitting a quarter-symmetric metric connection is said to be Ricci-pseudosymmetric if the following condition is satisfied R̃(X, Y ) · S̃ = L S̃ [(X ∧g Y ) · S̃], (4.1) on U S̃ . Let para-Sasakian manifold Mn admitting a quarter-symmetric metric connection be Ricci- pseudosymmetric. Then we have S̃(R̃(X, Y )Z, W) + S̃(Z, R̃(X, Y )W) = L S̃ [S̃((X ∧g Y )Z, W) + S̃(Z, (X ∧g Y )W)]. (4.2) By taking Y = W = ξ and making use of (2.7), (2.8) and (2.11), the above equation turns into (L S̃ + 2)[S̃(X, Z) + 2(n − 1)g(X, Z)] = 0 (4.3) Thus, we have the following assertion: Theorem 2. If a para-Sasakian manifold Mn admitting a quarter-symmetric metric connection is Ricci-pseudosymmetric then Mn is an Einstein manifold with respect to quarter-symmetric metric connection or it satisfies L S̃ = −2. 262 Vishnuvardhana. S.V. & Venkatesha CUBO 22, 2 (2020) 5 Projectively flat para-Sasakian manifold admitting a quarter- symmetric metric connection The projective curvature tensor on a Riemannian manifold is defined by [2] P(X, Y )Z = R(X, Y )Z − 1 (n − 1) [S(Y, Z)X − S(X, Z)Y ]. (5.1) For an n-dimensional para-Sasakian manifold Mn admitting a quarter-symmetric metric con- nection, the projective curvature tensor is given by P̃(X, Y )Z = R̃(X, Y )Z − 1 (n − 1) [S̃(Y, Z)X − S̃(X, Z)Y ]. (5.2) Theorem 3. A projectively flat para-Sasakian manifold Mn admitting a quarter-symmetric metric connection is an Einstein manifold with respect to quarter-symmetric metric connection. Proof. Consider a projectively flat para-Sasakian manifold admitting a quarter-symmetric metric connection. Then from (5.2) we have g(R̃(X, Y )Z, W) = 1 (n − 1) [S̃(Y, Z)g(X, W) − S̃(X, Z)g(Y, W)]. (5.3) Setting X = W = ξ in (5.3) and using (2.7), (2.8), (2.11) and (2.12), we get S̃(X, Z) = −2(n − 1)g(X, Z). (5.4) Hence, the proof is completed. 6 Projectively pseudosymmetric para-Sasakian manifold ad- mitting a quarter-symmetric metric connection A para-Sasakian manifold admitting a quarter-symmetric metric connection is said to be projec- tively pseudosymmetric if R̃(X, Y ) · P̃ = L P̃ [(X ∧g Y ) · P̃], (6.1) holds on the set U P̃ = {x ∈ Mn : P̃ 6= 0 at x}, where L P̃ is some function on U P̃ . Let Mn be projectively pseudosymmetric, then we have R̃(X, ξ)P̃(U, V )ξ − P̃(R̃(X, ξ)U, V )ξ − P̃(U, R̃(X, ξ)V )ξ −P̃(U, V )R̃(X, ξ)ξ = L P̃ [(X ∧g ξ)P̃(U, V )ξ − P̃((X ∧g ξ)U, V )ξ −P̃(U, (X ∧g ξ)V )ξ − P̃(U, V )(X ∧g ξ)ξ]. (6.2) CUBO 22, 2 (2020) Results on para-Sasakian manifold admitting a quarter symmetric . . . 263 By virtue of (2.11), (2.12) and (5.2), (6.2) becomes (L P̃ + 2)P̃(U, V )X = 0. (6.3) So, one can state that: Theorem 4. If a para-Sasakian manifold Mn admitting a quarter-symmetric metric connection is projectively pseudosymmetric then Mn is projectively flat with respect to quarter-symmetric metric connection or L P̃ = −2. In view of theorem 3, one can state the above theorem as Theorem 5. If a para-Sasakian manifold Mn admitting a quarter-symmetric metric connection is projectively pseudosymmetric then Mn is an Einstein manifold with respect to quarter-symmetric metric connection or L P̃ = −2. 7 Examples 7.1 Example We consider a 3-dimensional manifold M = {(x, y, z) ∈ R3 : z 6= 0}, where (x, y, z) are standard coordinates in R3. Let {E1, E2, E3} be a linearly independent global frame field on M given by E1 = e z ∂ ∂y , E2 = e z( ∂ ∂y − ∂ ∂x ), E3 = ∂ ∂z , If g is a Riemannian metric defined by g(Ei, Ej) =      1, i = j 0, i 6= j for 1 ≤ i, j ≤ 3, and if η is the 1-form defined by η(Z) = g(Z, E3) for any vector field Z ∈ χ(M). We define the (1, 1)-tensor field φ as φ(E1) = E1, φ(E2) = −E2, φ(E3) = 0. The linearity property of φ and g yields that η(E3) = 1, φ2U = U − η(U)E3, g(φU, φV ) = g(U, V ) − η(U)η(V ), 264 Vishnuvardhana. S.V. & Venkatesha CUBO 22, 2 (2020) for any U, V ∈ χ(M). Now we have [E1, E2] = 0, [E1, E3] = E1, [E2, E3] = E2. The Riemannian connection ∇ of the metric g known as Koszul’s formula and is given by 2g(∇XY, Z) = Xg(Y, Z) + Y g(Z, X) − Zg(X, Y ) − g(X, [Y, Z]) −g(Y, [X, Z]) + g(Z, [X, Y ]). Using Koszul’s formula we get the followings in matrix form     ∇E1E1 ∇E1E2 ∇E1E3 ∇E2E1 ∇E2E2 ∇E2E3 ∇E3E1 ∇E3E2 ∇E3E3     =     −E3 0 E1 0 −E3 E2 0 0 0     . Clearly (φ, ξ, η, g) is a para-Sasakian structure on M. Thus M(φ, ξ, η, g) is a 3-dimensional para-Sasakian manifold. Using (2.10) and the above equation, one can easily obtain the following:     ∇̃E1E1 ∇̃E1E2 ∇̃E1E3 ∇̃E2E1 ∇̃E2E2 ∇̃E2E3 ∇̃E3E1 ∇̃E3E2 ∇̃E3E3     =     −2E3 0 2E1 0 −2E3 2E2 0 0 0     . With the help of the above results it can be easily verified that R(E1, E2)E3 = 0, R(E2, E3)E3 = −E2, R(E1, E3)E3 = −E1, R(E1, E2)E2 = −E1, R(E2, E3)E2 = E3, R(E1, E3)E2 = 0, R(E1, E2)E1 = E2, R(E2, E3)E1 = 0, R(E1, E3)E1 = E3. and R̃(E1, E2)E3 = 0, R̃(E2, E3)E3 = −2E2, R̃(E1, E3)E3 = −2E1, R̃(E1, E2)E2 = −4E1, R̃(E2, E3)E2 = 2E3, R̃(E1, E3)E2 = 0, R̃(E1, E2)E1 = 4E2, R̃(E2, E3)E1 = 0, R̃(E1, E3)E1 = 2E3. (7.1) Since E1, E2, E3 forms a basis, any vector field X, Y, Z ∈ χ(M) can be written as X = a1E1 + b1E2 + c1E3, Y = a2E1 + b2E2 + c2E3, Z = a3E1 + b3E2 + c3E3, where ai, bi, ci ∈ R, i = 1, 2, 3. Using the expressions of the curvature tensors, we find values of Riemannian curvature CUBO 22, 2 (2020) Results on para-Sasakian manifold admitting a quarter symmetric . . . 265 and Ricci curvature with respect to quarter-symmetric metric connection as; R̃(X, Y )Z = [−4{a1b2 − b1a2}b3 + 2{c1a2 − a1c2}c3]E1 + [−4{b1a2 − a1b2}a3 + 2{c1b2 − b1c2}c3]E2 + [−2{c1a2 − a1c2}a3 − 2{c1b2 − b1c2}b3]E3, (7.2) S̃(E1, E1) = S̃(E2, E2) = −6, S̃(E3, E3) = −4. (7.3) Using (7.1), (7.3) and the expression of the endomorphism (X ∧g Y )Z = g(Y, Z)X −g(X, Z)Y , one can easily verify that S̃(R̃(X, E3)Y, E3) + S̃(Y, R̃(X, E3)E3) = −2[S̃((X ∧g E3)Y, E3) + S̃(Y, (X ∧g E3)E3)], (7.4) here L S̃ = −2. Thus, the above equation verify one part of the Theorem 2 of section 4. Moreover, the manifold under consideration satisfies R̃(X, Y )Z = −R̃(Y, X)Z, R̃(X, Y )Z + R̃(Y, Z)X + R̃(Z, X)Y = 0. Hence, from the above equations one can say that this example verifies the condition (c) of Theorem 3.1 in [1] and first Bianchi identity. 7.2 Example We consider a 5-dimensional manifold M = {(x1, x2, x3, x4, x5) ∈ R 5}, where (x1, x2, x3, x4, x5) are standard coordinates in R5. We choose the vector fields E1 = ∂ ∂x1 , E2 = ∂ ∂x2 , E3 = ∂ ∂x3 , E4 = ∂ ∂x4 , E5 = x1 ∂ ∂x1 + x2 ∂ ∂x2 + x3 ∂ ∂x3 + x4 ∂ ∂x4 + ∂ ∂x5 , which are linearly independent at each point of M. Let g be a Riemannian metric defined by g(Ei, Ej) =      1, i = j 0, i 6= j for 1 ≤ i, j ≤ 5, and if η is the 1-form defined by η(Z) = g(Z, E5) for any vector field Z ∈ χ(M). Let φ be the (1, 1)-tensor field defined by φ(E1) = E1, φ(E2) = E2, φ(E3) = E3, φ(E4) = E4, φ(E5) = 0. 266 Vishnuvardhana. S.V. & Venkatesha CUBO 22, 2 (2020) The linearity property of φ and g yields that η(E5) = 1, φ 2 U = U − η(U)E5, g(φU, φV ) = g(U, V ) − η(U)η(V ), for any U, V ∈ χ(M). Now we have [E1, E2] = 0, [E1, E3] = 0, [E1, E4] = 0, [E1, E5] = E1, [E2, E3] = 0, [E2, E4] = 0, [E2, E5] = E2, [E3, E4] = 0, [E3, E5] = E3, [E4, E5] = E4. By virtue of Koszul’s formula we get the followings in matrix form           ∇E1E1 ∇E1E2 ∇E1E3 ∇E1E4 ∇E1E5 ∇E2E1 ∇E2E2 ∇E2E3 ∇E2E4 ∇E2E5 ∇E3E1 ∇E3E2 ∇E3E3 ∇E3E4 ∇E3E5 ∇E4E1 ∇E4E2 ∇E4E3 ∇E4E4 ∇E4E5 ∇E5E1 ∇E5E2 ∇E5E3 ∇E5E4 ∇E5E5           =           −E5 0 0 0 E1 0 −E5 0 0 E2 0 0 −E5 0 E3 0 0 0 −E5 E4 0 0 0 0 0           . Above expressions satisfies all the properties of para-Sasakian manifold. Thus M(φ, ξ, η, g) is a 5-dimensional para-Sasakian manifold. From the above expressions and the relation of quarter symmetric metric connection and Riemannian connection, one can easily obtain the following:           ∇̃E1E1 ∇̃E1E2 ∇̃E1E3 ∇̃E1E4 ∇̃E1E5 ∇̃E2E1 ∇̃E2E2 ∇̃E2E3 ∇̃E2E4 ∇̃E2E5 ∇̃E3E1 ∇̃E3E2 ∇̃E3E3 ∇̃E3E4 ∇̃E3E5 ∇̃E4E1 ∇̃E4E2 ∇̃E4E3 ∇̃E4E4 ∇̃E4E5 ∇̃E5E1 ∇̃E5E2 ∇̃E5E3 ∇̃E5E4 ∇̃E5E5           =           −2E5 0 0 0 2E1 0 −2E5 0 0 2E2 0 0 −2E5 0 2E3 0 0 0 −2E5 2E4 0 0 0 0 0           . With the help of the above results it can be easily obtain the non-zero components of curvature tensors as R(E1, E2)E1 = E2, R(E1, E2)E2 = −E1, R(E1, E3)E1 = E3, R(E1, E3)E3 = −E1, R(E1, E4)E1 = E4, R(E1, E4)E4 = −E1, R(E1, E5)E1 = E5, R(E1, E5)E5 = −E1, R(E2, E3)E2 = E3, R(E2, E3)E3 = −E2, R(E2, E4)E2 = E4, R(E2, E4)E4 = −E2, R(E2, E5)E2 = E5, R(E2, E5)E5 = −E2, R(E3, E4)E3 = E4, R(E3, E4)E4 = −E3, R(E3, E5)E3 = E5, R(E3, E5)E5 = −E3, R(E4, E5)E4 = E5, R(E4, E5)E5 = −E4, CUBO 22, 2 (2020) Results on para-Sasakian manifold admitting a quarter symmetric . . . 267 and R̃(E1, E2)E1 = 4E2, R̃(E1, E2)E2 = −4E1, R̃(E1, E3)E1 = 4E3, R̃(E1, E3)E3 = −4E1, R̃(E1, E4)E1 = 4E4, R̃(E1, E4)E4 = −4E1, R̃(E1, E5)E1 = 2E5, R̃(E1, E5)E5 = −2E1, R̃(E2, E3)E2 = 4E3, R̃(E2, E3)E3 = −4E2, R̃(E2, E4)E2 = 4E4, R̃(E2, E4)E4 = −4E2, R̃(E2, E5)E2 = 2E5, R̃(E2, E5)E5 = −2E2, R̃(E3, E4)E3 = 4E4, R̃(E3, E4)E4 = −4E3, R̃(E3, E5)E3 = 2E5, R̃(E3, E5)E5 = −2E3, R̃(E4, E5)E4 = 2E5, R̃(E4, E5)E5 = −2E4. (7.5) Since E1, E2, E3, E4, E5 forms a basis, any vector field X, Y, Z ∈ χ(M) can be written as X = a1E1 + b1E2 + c1E3 + d1E4 + f1E5, Y = a2E1 + b2E2 + c2E3 + d2E4 + f2E5, Z = a3E1 + b3E2 + c3E3 + d3E4 + f3E5, where ai, bi, ci, di, fi ∈ R, i = 1, 2, 3, 4, 5. Using the expressions of the curvature tensors, we find values of Riemannian curvature and Ricci curvature with respect to quarter-symmetric metric connection as; R̃(X, Y )Z = [−4{a1(b2b3 + c2c3 + d2d3) − a2(b1b3 + c1c3 + d1d3)} − 2(a1f2 − f1a2)f3]E1 + [−4{b1(a2a3 + c2c3 + d2d3) − b2(a1a3 + c1c3 + d1d3)} − 2(b1f2 − f1b2)f3]E2 + [−4{c1(a2a3 + b2b3 + d2d3) − c2(a1a3 + b1b3 + d1d3)} − 2(c1f2 − f1c2)f3]E3 + [−4{d1(a2a3 + b2b3 + c2c3) − d2(a1a3 + b1b3 + c1c3)} − 2(d1f2 − f1d2)f3]E4 + [2{(a1f2 − f1a2)a3 + (b1f2 − f1b2)b3 + (c1f2 − f1c2)c3 + (d1f2 − f1d2)d3}]E5, S̃(E1, E1) = S̃(E2, E2) = S̃(E3, E3) = S̃(E4, E4) = −14, S̃(E5, E5) = −8. (7.6) In view of (7.5), (7.6) and the expression of the endomorphism one can easily verify the equation (7.4) and hence the Theorem 2 of section 4 is verified. This example also verifies the condition (c) of Theorem 3.1 in [1] and first Bianchi identity. Above two examples verifies the one part of the Theorem 2, that is, if a para-Sasakian manifold Mn admitting a quarter-symmetric metric connection is Ricci pseudosymmetric then Mn satisfies L S̃ = −2 (Mn is not Einstein manifold with respect to quarter-symmetric metric connection). Another part of the theorem is that, if a para-Sasakian manifold Mn admitting a quarter-symmetric metric connection is Ricci pseudosymmetric then Mn is an Einstein manifold with respect to quarter-symmetric metric connection (L S̃ 6= −2). Now, the second part of the Theorem 2 can be verified by using the proper example. 7.3 Example We consider a 5-dimensional manifold M = {(x, y, z, u, v) ∈ R5}, where (x, y, z, u, v) are standard coordinates in R5. Let {E1, E2, E3, E4, E5} be a linearly independent global frame field on M given 268 Vishnuvardhana. S.V. & Venkatesha CUBO 22, 2 (2020) by E1 = ∂ ∂x , E2 = e −x ∂ ∂y , E3 = e −x ∂ ∂z , E4 = e −x ∂ ∂u , E5 = e −x ∂ ∂v . Let g be a Riemannian metric defined by g(Ei, Ej) =      1, i = j 0, i 6= j for 1 ≤ i, j ≤ 5, and if η is the 1-form defined by η(Z) = g(Z, E1) for any vector field Z ∈ χ(M). Let the (1, 1)-tensor field φ be defined by φ(E1) = 0, φ(E2) = E2, φ(E3) = E3, φ(E4) = E4, φ(E5) = E5. With the help of linearity property of φ and g, we have η(E1) = 1, φ2V = V − η(V )E1, g(φX, φY ) = g(X, Y ) − η(X)η(Y ), for any X, Y ∈ χ(M). Now we have [E1, E2] = −E2, [E1, E3] = −E3, [E1, E4] = −E4, [E1, E5] = −E5, [E2, E3] = [E2, E4] = [E2, E5] = [E3, E4] = [E3, E5] = E4, E5] = 0. With the help of Koszul’s formula we get the followings in matrix form           ∇E1E1 ∇E1E2 ∇E1E3 ∇E1E4 ∇E1E5 ∇E2E1 ∇E2E2 ∇E2E3 ∇E2E4 ∇E2E5 ∇E3E1 ∇E3E2 ∇E3E3 ∇E3E4 ∇E3E5 ∇E4E1 ∇E4E2 ∇E4E3 ∇E4E4 ∇E4E5 ∇E5E1 ∇E5E2 ∇E5E3 ∇E5E4 ∇E5E5           =           0 0 0 0 0 E2 −E1 0 0 0 E3 0 −E1 0 0 E4 0 0 −E1 0 E5 0 0 0 −E1           . In this case, (φ, ξ, η, g) is a para-Sasakian structure on M and hence M(φ, ξ, η, g) is a 5- dimensional para-Sasakian manifold. Using (2.10) and the above equation, one can easily obtain the following:           ∇̃E1E1 ∇̃E1E2 ∇̃E1E3 ∇̃E1E4 ∇̃E1E5 ∇̃E2E1 ∇̃E2E2 ∇̃E2E3 ∇̃E2E4 ∇̃E2E5 ∇̃E3E1 ∇̃E3E2 ∇̃E3E3 ∇̃E3E4 ∇̃E3E5 ∇̃E4E1 ∇̃E4E2 ∇̃E4E3 ∇̃E4E4 ∇̃E4E5 ∇̃E5E1 ∇̃E5E2 ∇̃E5E3 ∇̃E5E4 ∇̃E5E5           =           0 0 0 0 0 2E2 −2E1 0 0 0 2E3 0 −2E1 0 0 2E4 0 0 −2E1 0 2E5 0 0 0 −2E1           . CUBO 22, 2 (2020) Results on para-Sasakian manifold admitting a quarter symmetric . . . 269 From above results it can be easily obtain the non-zero components of Riemannian curvature and Ricci curvature tensors as R(E1, E2)E1 = E2, R(E1, E2)E2 = −E1, R(E1, E3)E1 = E3, R(E1, E3)E3 = −E1, R(E1, E4)E1 = E4, R(E1, E4)E4 = −E1, R(E1, E5)E1 = E5, R(E1, E5)E5 = −E1, R(E2, E3)E2 = E3, R(E2, E3)E3 = −E2, R(E2, E4)E2 = E4, R(E2, E4)E4 = −E2, R(E2, E5)E2 = E5, R(E2, E5)E5 = −E2, R(E3, E4)E3 = E4, R(E3, E4)E4 = −E3, R(E3, E5)E3 = E5, R(E3, E5)E5 = −E3, R(E4, E5)E4 = E5, R(E4, E5)E5 = −E4, and R̃(E1, E2)E1 = 2E2, R̃(E1, E2)E2 = −2E1, R̃(E1, E3)E1 = 2E3, R̃(E1, E3)E3 = −2E1, R̃(E1, E4)E1 = 2E4, R̃(E1, E4)E4 = −2E1, R̃(E1, E5)E1 = 2E5, R̃(E1, E5)E5 = −2E1, R̃(E2, E3)E2 = 2E3, R̃(E2, E3)E3 = −2E2, R̃(E2, E4)E2 = 2E4, R̃(E2, E4)E4 = −2E2, R̃(E2, E5)E2 = 2E5, R̃(E2, E5)E5 = −2E2, R̃(E3, E4)E3 = 2E4, R̃(E3, E4)E4 = −2E3, R̃(E3, E5)E3 = 2E5, R̃(E3, E5)E5 = −2E3, R̃(E4, E5)E4 = 2E5, R̃(E4, E5)E5 = −2E4, (7.7) S̃(E1, E1) = S̃(E2, E2) = S̃(E3, E3) = S̃(E4, E4) = S̃(E5, E5) = −8, (7.8) S̃(X, Y ) = −2(5 − 1)g(X, Y ) = −8g(X, Y ), where X = a1E1 + b1E2 + c1E3 + d1E4 + f1E5 and Y = a2E1 + b2E2 + c2E3 + d2E4 + f2E5. 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Introduction Preliminaries Pseudosymmetric para-Sasakian manifold admitting a quarter-symmetric metric connection Ricci-pseudosymmetric para-Sasakian manifold admitting a quarter-symmetric metric connection Projectively flat para-Sasakian manifold admitting a quarter-symmetric metric connection Projectively pseudosymmetric para-Sasakian manifold admitting a quarter-symmetric metric connection Examples Example Example Example