Some geometric properties on Lorentzian Sasakian manifolds

The objective of the present paper is to study and investigate the geometric properties of Concircular curvature tensor on a Lorentzian Sasakian manifold (in short LS-manifold) endowed with the quarter-symmetric non metric connection. This research is also supported with an example that satisfies the conditions of (cid:31)(cid:30) Concircularly flat and ϒ -Concircularly flat Lorentzian Sasakian manifold endowed with the quarter-symmetric non metric connection.


Introduction
In 1924, Friedmann and Schouten [10] introduced the idea of semi-symmetric connection on a differentiable manifold.Later Matsumoto [14] and Sato [19], introduced the notion of Lorentzian Sasakian manifolds and an almost paracontact manifold respectively.Mihai [15] introduced the same notion independently and obtained several results.The Lorentzian Sasakian manifolds has also been studied in detail by Sato [19], Matsumoto, Mihai [15], De & Shaikh [7] and many others in [3], [17] and [18].Some interesting results were obtained for conformally recurrent and conformally symmetric P-Sasakian manifold in [1].
A linear connection ∇ on a differentiable manifold M is said to be a semi-symmetric connection if the torsion tensor T of the connection satisfies where Λ is a 1-form and ϑ is a vector field defined by /( ) = ( , ) X g X -, for all vector fields X on Γ( ) TM .

Γ(
) TM is the set of all differentiable vector fields on M. Semi-symmetric metric and non metric connection on para-Sasakian manifold was studied by [4] and [5].In 1975, Golab [11] defined and studied quarter-symmetric connection in differentiable manifolds with affine connections.In 2020, Khan [13] studied properties of tangent bundle endowed with quarter-symmetric non-metric connection on an almost Hermitian manifold.Later in 2023, Khan et al. [12] studied properties on lifts of a quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle.
A linear connection ∇ on an n-dimensional Riemannian manifold (M, g) is called a quartersymmetric connection [11], if its torsion tensor T satisfies where ϒ is a (1,1) tensor field.
In particular, if ϒX X = , then the quarter-symmetric connection reduces to the semi-symmetric connection [10].
Thus the notion of the quarter-symmetric connection generalizes the notion of the semi-symmetric connection.If moreover, a quarter-symmetric connection ∇ satisfies the condition for all X Y Z , , on Γ( ) TM , then ∇ is said to be a quarter-symmetric non metric connection (in short qsnmc).
A relation between the quarter-symmetric non metric connection ∇ and the Levi-Civita connection ∇ in an n-dimensional Lorentzian Sasakian manifold M is given by [8] The 1-form Λ is defined by /( ) = ( , ) X g Xand ϑ is the corresponding vector field.Bagewadi and Venkatesha [2] studied Concircular ϒ -recurrent Lorentzian Sasakian manifolds which generalized the notion of locally Concircular ϒ -symmetric Lorentzian Sasakian manifolds and obtained some interesting results.
Let R and R be the curvature tensors with respect to the quarter-symmetric non metric connection ∇ and the Levi-Civita connection ∇ respectively.Then, we have for all vector fields X Y Z TM , , ( ) * , where S and S be the Ricci tensors with respect to the quartersymmetric non metric connection ∇ and the Levi-Civita connection ∇ respectively.
A Riemannian manifold M is locally symmetric if its curvature tensor R satisfies ∇R = 0.As a generalization of locally symmetric spaces, many geometers have considered semi-symmetric spaces and studied their generalizations.A Riemannian manifold M is said to be semi-symmetric if its curvature tensor R satisfies R X Y R ( , ). = 0, where R(X, Y) acts on R as a derivation.A Riemannian manifold M is said to be Ricci-semi symmetric manifold if it satisfies the relation R X Y S ( , ) = 0, where R X Y ( , ) the curvature operator.A transformation of an n-dimensional Riemannian manifold M, which transforms every geodesic circle of M into a geodesic circle, is called a Concircular transformation.A Concircular transformation is always a conformal transformation.We know, geodesic circle means a curve in M whose first curvature is constant and whose second curvature is identically zero.Thus the geometry of Concircular transformations, i.e., the Concircular geometry, is a generalization of inversive geometry in the sense that the change of metric is more general than that induced by a circle preserving diffeomorphism (see also [6]).An interesting invariant of a transformation is the Concircular curvature tensor C .Some useful properties are given below: Using ( 8), we obtain where and C is the Concircular curvature tensor and r is the scalar curvature with respect to the quarter-symmetric non metric connection respectively.Riemannian manifolds with vanishing Concircular curvature tensor are of constant curvature.Thus the Concircular curvature tensor is a measure of the failure of a Riemannian manifold to be of constant curvature.

Preliminaries
An n-dimensional differentiable manifold M is said to be an almost para-contact manifold, if it admits an almost para-contact structure ( , , , ) b / -g consisting of a (1, 1) tensor field ϒ, vector field ϑ, 1-form Λ and Lorentzian metric g for any vector field X, Y on M. Such a manifold is termed as Lorentzian para-contact manifold and the structure ( ) b / , , , -g a Lorentzian para-contact structure [14].
for X, Y tangent to M, then M is called a Lorentzian Sasakian manifold or briefly LS-manifold, where ∇ denotes the covariant differentiation with respect to Lorentzian metric g.
For the curvature tensor R, the Ricci tensor S and the Ricci operator Q in a LS-manifold M with respect to the Levi-Civita connection the following relation hold for all vector fields X Y TM , ( ) * .

Main Results
In this paper, we study a type of quarter symmetric non-metric connection (qsnmc) on LS-manifolds.The paper is organized 1.First section includes introduction.
2. Section two is equipped with some prerequisites of a LS-manifold.3. In section three main results of the paper are discussed.4. In section four we studied ϑ -Concircularly flat LS-manifold with respect to the qsnmc.5. ϒ-Concircularly flat LS-manifolds with respect to the qsnmc have been studied in section five.6.In next section, we investigate Ricci-semi symmetric manifolds with respect to the qsnmc of a LS-manifold.7. At last, we construct an example of a 5-dimensional LS-manifolds endowed with the qsnmc which verify the results of section four and five.

ϑ ϑ -Concircularly
Putting Z = ϑ in (23) and using (11), we have Using ( 20) and (24), we get with respect to a qsnmc, then the manifold is an Λ -Einstein manifold with respect to the qsnmc and the scalar curvature r with respect to the qsnmc is a negative constant.
Proof.In order to prove the Theorem, we first state following Lemma.Lemma 4.4.[9] If a LS-manifold is semi-symmetric with respect to the qsnmc, then the manifold is an Λ -Einstein manifold with respect to the qsnmc and the scalar curvature r with respect to the qsnmc is a negative constant.
From the definition of concircular curvature tensor, it follows that Thus using Lemma 4.4, we obtain Theorem 4.3.

Definition 5.2. A LS-manifold is said to be an / Einstein manifold if its Ricci tensor S of the Levi-Civita connection is of the form
where a and b are smooth functions on the manifold.
Theorem 5.3.If a LS-manifold admitting a qsnmc is ϒ -Concircularly flat, then the manifold with respect to the qsnmc is an / Einstein manifold.
From which it follows that the manifold is an / Einstein manifold with respect to the qsnmc.Hence, proof of the Theorem 5.3 is complete.

Lorentzian Para-Sasakian Manifold Satisfying C S
⋅ ⋅ = 0 with Respect to a Quarter-Symmetric Non Metric Connection Theorem 6.1.If LS-manifolds satisfying C S ⋅ = 0 with respect to a qsnmc, then the manifold is an Λ -Einstein manifold with respect to a qsnmc.
Proof.We consider LS-manifolds with respect to a qsnmc ∇ satisfying the curvature condition C S ⋅ = 0. Then Putting X = ϑ in (31) and using (10), we get where a n r r n n = 2( 1) 2 ( 1) This means that the manifold is an Λ -Einstein manifold with respect to the qsnmc.

Example
In this section, we construct an example on LS-manifold with respect to the qsnmc ∇ , which verify the results of section three and section four.We consider the 5-dimensional manifold ( , , , , ) 5 x y z u v ∈ » , where ( , , , , ) x y z u v are the standard coordinates in » 5 .We choose the vector fields which are linearly independent at each point of M. Let g be the Lorentzian metric defined by Using the linearity of ϒ and g, we obtain Flat Lorentzian Para-Sasakian Manifold with Respect to the Quarter-Symmetric Non Metric Connection Definition 5.1.A LS-manifold is said to be b Concircularly flat with respect to the qsnmc if be a local orthonormal basis of vector fields in M, then { , orthonormal basis.Putting X U e i = = in (27) and summing over i = 1 to n −1, we obtain
Also it follows that the scalar curvature tensor with respect to the quarter-symmetric metric connection is r = 40.Let X, Y, Z and U be any four vector fields given by , for all i = 1,2,3,4,5 are all non-zero real numbers.Using the above curvature tensors and the scalar curvature tensors of the qsnmc, we have result of Section three.Now, we see that the ϒ -Concircularly flat with respect to the qsnmc from the above relations as follow:Hence LS-manifolds will be ϒ -Concircularly flat with respect to the quarter-symmetric metric conarguments tell us that the 5-dimensional LS-manifold with respect to the qsnmc under consideration agrees with the section five.

Flat Lorentzian Para-Sasakian Manifold with Respect to the Quarter-Symmetric Non Metric Connection Definition
[2].A LS-manifold is said to be -Concircularly flat[2]with respect to the qsnmc if