P-Sasakian Manifold with Quarter-Symmetric Non-Metric Connection

The object of the present paper is to study on a P-Sasakian manifold with quarter symmetric non-metric connection. In this paper, we consider some properties of the curvature tensor, projective curvature tensor, concircular curvature tensor, conformal curvature tensor with respect to quarter symmetric non-metric connection in a P-Sasakian manifolds. Finally, we give an example.

A linear connection is said to be a quarter-symmetric connection if its torsion tensor T is of the form for any vector fields X, Y on a manifold, where u is a 1−form and ϕ is a tensor of type (1,1). If ϕ = I, then the quartersymmetric connection is reduced to a semi-symmetric connection. Hence quarter-symmetric connection can be viewed as a generalization of semi-symmetric connection. The connection ∇ is said to be a metric connection if there is a Riemannian metric g in M such that ∇g = 0, otherwise it is non-metric. In [12], Sharfuddin and Hussian defined a semi-symmetric metric connection in an almost contact manifold, by setting Ajit Barman and Gopal Ghosh studied P-Sasakian manifolds admittting a semi-symmetric non-metric connection whose concircular curvature tensor satisfies certain curvature conditions. Moreover, some properties of a quarter-symmetric non-metric connection on P-sasakian manifolds are investigated in [11].
124 P-Sasakian Manifold with Quarter-Symmetric Non-Metric Connection In the present paper, we will study P-Sasakian manifold with quarter symmetric non-metric connection. Section 2 is devoted to preliminaries. In section 3, we introduce quarter symmetric non-metric connection on a Para-Sasakian manifold. We calculate curvature tensor and Ricci tensor and scalar curvature of a P-Sasakian manifold with respect to quarter symmetric non-metric connection, respectively. Moreover we show that if a Para-Sasakian manifold with quarter symmetric non-metric connection is Ricci semi-symmetric, then the manifold is η− Einstein manifold with respect to quarter symmetric non-metric connection.
In section 4, we find some results for concircular curvature tensor with respect to quarter symmetric non-metric connection.
In section 5, it is shown that if a Para-Sasakian manifold is φ− projectively flat with respect to quarter symmetric non-metric connection, then the manifold is an η− Einstein manifold with respect to quarter symmetric non-metric connection. In section 6, we have proved that if a Para-Sasakian manifold is conformally flat with respect to quarter symmetric non-metric connection, then the manifold is an Einstein manifold with respect to quarter symmetric non-metric connection. In section 7, we give an example which verify the results of Section 3, Section 4 and Section 5.
A P-Sasakian manifold M is said to be η Einstein if its Ricci tensor S is of the form

Quarter-Symmetric Non-Metric Connection
Let M be an n− dimensional P-Sasakian manifold with Levi-Civita connection ∇. If we set for any vector field X and Y , then ∇ is a linear connection on M . We know that the torsion tensor T with respect to connection ∇ is given From (18) we get Furthermore (18) we have for any vector field X and Y , which implies that ∇ is a quarter symmetric non-metric connection on M . Also by using (4), and (20) we get which means that the metric g is ξ− parallel with respect to quarter symmetric non-metric connection.
From (5),(9),(10), (4) and (18) we have the following proposition: Proposition 1. Let M be a P-Sasakian manifold. Then we have the following equations: The curvature tensor R of the quarter symmetric non-metric connection ∇ on M is defined by From (8), (18) and (23) we have where is the curvature tensor with respect to the Levi-Civita connection ∇. Using (24) and the first Bianchi identity we have the following proposition Proposition 2. Let M be an n− dimensional P −Sasakian manifold with quarter symmetric non-metric connection. Then the first Bianchi identity of the quarter-symmetric nonmetric connection ∇ on M is provided.
From (14) and (24) we have where K and K are given by and Theorem 3. Let M be an n− dimensional P −Sasakian manifold with quarter symmetric non-metric connection. Then we have the following equations: for any X, Y, Z, U ∈ Γ(T M ).
Proposition 4. Let M be an n− dimensional P −Sasakian manifold with quarter symmetric non-metric connection. Then we have the following equations: for any X, Y, Z ∈ Γ(T M ).
The Ricci tensor S of an P-Sasakian manifold M with respect to quarter symmetric non-metric connection ∇ is given by The scalar curvature of M with respect to quarter symmetric non-metric connection ∇ is defined by where X, Y ∈ Γ(T M ), {e 1 , e 2 , ..., e n } is an orthonormal frame.
Theorem 5. Let M be an n− dimensional P −Sasakian manifold. Then we have the following equations: where r is scalar curvature of Levi-Civita connection and β = trace(Φ).
From (34) we have the following corollary Theorem 7. Let M be an n− dimensional P −Sasakian manifold. If M is Ricci semi-symmetric with respect to quarter symmetric non-metric connection, then M is an η Einstein manifold with respect to quarter symmetric non-metric connection.
Proof. Let R(X, Y )S = 0 be on M for any X, Y, Z, U ∈ Γ(T M ), then we have If we choose Z = ξ and X = ξ in (39), we get Using (30), (31) and (36) in (40), we obtain This equation tell us M is an η Einstein manifold with respect to quarter symmetric non-metric connection.
Then contracting the equation (53) over Y and Z and from (3), (5), (7), (4) we get Then we have the following theorem: Theorem 12. Let M be an n− dimensional P −Sasakian manifold. If M is φ− concirculary flat with respect to quarter symmetric non-metric connection, then the scalar curvature tensor with respect to Levi-Civita connection M is equal to β 2 − n(n − 1).
Thus from (34), (38), (62), the equation (61) becomes Moreover we have Using (12)   Let M be an n− dimensional P-Sasakian manifold. The conformal curvature tensor of M with respect to quarter symmetric non-metric connection ∇ is given Suppose that P-Sasakian manifold is conformally flat with respect to quarter symmetric non-metric connection, that is C(X, Y, Z, U ) = 0. By using (67) we get Putting Y = Z = ξ in (68) and using (31), (35) and (36) we obtain from (69) we have the following theorem Theorem 17. If a P-Sasakian manifold is conformally flat with respect to quarter symmetric non-metric connection, then the manifold is Einstein manifold with respect to quarter symmetric non-metric connection.
Let ∇ be the Levi-Civita connection with respect to the Riemannian metric g. Then, we have Using Koszul formula for the Riemannian metric g, we can easily calculate

P-Sasakian Manifold with Quarter-Symmetric Non-Metric Connection
From the above relations, it can be easily seen that for all E 1 = ξ Thus the manifold M is an P-Sasakian with the structure (φ, ξ, η, g). Using (18) in the above equations, we get Using the above relations, we can calculate the non-vanishing components of the curvature tensor as follows: and From the equations (74) and (75), we can easily calculate the non-vanishing components of the Ricci tensor as follows: Let X, Y , Z and U be any four vector fields given by where A i , B i , C i , D i , for all i = 1, 2, 3, are the non-zero real numbers.
Then we see that P − Sasakian manifold M will be ξ− concirculary flat with respect to quarter symmetric non-metric connection if A1 B1 = A2 B2 = A3 B3 . We also see the that the manifold is φ− concirculary flat with respect to quarter symmetric non-metric connection if A2 B2 = A3 B3 or C2 D2 = C3 D3 . Moreover we see that P − Sasakian manifold M is φ− projectively flat with respect to quarter symmetric non-metric connection if A2 B2 = A3 B3 or C2 D2 = C3 D3 . This is verifying some results of section 4 and section 5.