D-homothetic deformation of an η-Einstein Para-Sasakian manifold

Objectives: The main purpose of this paper is to study the theory of Dhomothetic deformation of an η-Einstein Para-Sasakian manifold. Methods: A deformation technique is employed to solve the resulting problem. We provide its application in the general theory of relativity. Findings: Section 2 deals with recurrent and symmetric Para-Sasakian Manifolds. In section 3 we have defined and studied projectively symmetric Para-Sasakianmanifold. The notion of φ -holomorphic sectional curvature in an η-Einstein Para-Sasakian manifold has been delineated in the section 4. In this very section, we shall study the relevant commutation formulae which give rise to the required curvature tensor. The obtained results were compared with the previous works (1--3) in this field andwere found to be in good agreement.Novelty/Conclusion: The study concludes the following results: 1. If n-dimensional Para-Sasakian manifold is to be recurrent projective symmetric then the recurrent projective curvature tensor vanishes identically. 2. If n-dimensional Para-Sasakian manifold is to be projective semi-symmetric then the semi-recurrent projective curvature tensor vanishes identically. 3. If n-dimensional Para-Sasakian manifold is to be projective scalar-symmetric then the scalar projective curvature vanishes identically. 4. If D-homothetic deformation of η-Einstein Para-Sasakian manifold is constant φ -holomorphic sectional curvature thenwe shall obtain the value of scalar


Definition 2.1:
If the Para-Sasakian manifold satisfies the condition is called recurrent Para-Sasakian manifold for some non-zero recurrence vector τ ρ and R ε αβ γ is a recurrent curvature tensor (5) . If the covariant derivative of recurrent curvature tensor R ε αβ γ vanishes everywhere then an n-dimensional Para-Sasakian manifold is called recurrent symmetric Para-Sasakian manifold.
As a consequence of this definition follows the result:

Definition 2.2:
If the Para-Sasakian manifold satisfies the condition (2.3) ∇ ρ R αβ = τ ρ R αβ then the manifold is called Ricci-recurrent Para-Sasakian manifold or semi-recurrent Para-Sasakian manifold for some nonzero recurrence vector τ ρ and R αβ is a Ricci-recurrent curvature tensor.
If the covariant derivative of Ricci-recrrent curvature tensor R αβ vanishes everywhere then an n-dimensional Para-Sasakian manifold is called Ricci-symmetric Para-Sasakian manifold or semi-symmetric Para-Sasakian manifold.
As a consequence of this definition follows the result:

Definition 2.3:
If the Para-Sasakian manifold satisfies the condition (2.5) ∇ ρ R = τ ρ R then the manifold is called scalar-recurrent Para-Sasakian manifold for some non-zero recurrence vector τ ρ . If the covariant derivative of scalar curvature R vanishes everywhere then an n-dimensional Para-Sasakian manifold is called scalar-symmetric Para-Sasakian manifold.
As a consequence of this definition follows the result: In this regard, we have the following theorems: Theorem 2.1: The necessary and sufficient condition for the n-dimensional Para-Sasakian manifold to be recurrent symmetric manifold is that the recurrent curvature tensor vanishes.

Proof:
Since the recurrent curvature tensor vanishes i.e. R ε αβ γ = 0, https://www.indjst.org/ Inserting above in equation (2.1), we obtain ∇ ρ R ε αβ γ = 0, This manifests that the Para-Sasakian manifold to be recurrent symmetric space. Conversely, If the space to be recurrent symmetric then in view of the above discussion converse of the theorem is proof from the equations (2.1) and (2.2).

Theorem 2.2:
The necessary and sufficient condition for the n-dimensional Para-Sasakian manifold to be Ricci-symmetric manifold is that the Ricci-recurrent curvature tensor vanishes.

Proof:
The proof is self evident in view of the equations (2.3) and (2.4).

Theorem 2.3:
The necessary and sufficient condition for the n-dimensional Para-Sasakian manifold to be scalar symmetric manifold is that the scalar curvature vanishes.
Proof: It is evident in view of the equations (2.5) and (2.6).

Definition 3.1:
If the Para-Sasakian manifold satisfies the condition then the manifold is called Para-Sasakian manifold with recurrent projective curvature tensor for some non-zero recurrence vector τ ρ .
If the covariant derivative of recurrent projective curvature tensor P ε αβ γ vanishes everywhere then an n-dimensional Para-Sasakian manifold is called recurrent projective symmetric Para-Sasakian manifold.
As a consequence of this definition follows the result (6) :
If the covariant derivative of semi-recurrent projective curvature tensor P αβ vanishes everywhere then an n-dimensional Para-Sasakian manifold is called projective semi-symmetric Para-Sasakian manifold.
As a consequence of this definition follows the result: (3.4) ∇ ρ P αβ = 0.

Definition 3.3:
If the Para-Sasakian manifold satisfies the condition (3.5) ∇ ρ P = τ ρ P then the space is called Para-Sasakian manifold with scalar-recurrent projective curvature for some non-zero recurrence vector τ ρ .
If the covariant derivative of scalar projective curvature P vanishes everywhere then an n-dimensional Para-Sasakian manifold is called projective scalar-symmetric Para-Sasakian manifold.
As a consequence of this definition follows the result: (3.6) ∇ ρ P = 0.
In this regard, we have the following theorems: Theorem 3.1: The necessary and sufficient condition for the n-dimensional Para-Sasakian manifold to be recurrent projective symmetric manifold is that the recurrent projective curvature tensor vanishes.

Proof:
Since the recurrent projective curvature tensor vanishes i.e. P ε αβ γ = 0, Inserting above in equation (3.1), we obtain ∇ ρ P ε αβ γ = 0. This manifests that the Para-Sasakian manifold to be recurrent projective symmetric space. https://www.indjst.org/ Conversely, If the space to be recurrent projective symmetric then in view of above discussion, converse of the theorem is immediately proof from the equations (3.1) and (3.2).

Theorem 3.2:
The necessary and sufficient condition for the n-dimensional Para-Sasakian manifold to be projective semi-symmetric manifold is that the semi-recurrent projective curvature tensor vanishes.

Proof:
The proof is self-evident in view of equations (3.3) and (3.4).

Theorem 3.3:
The necessary and sufficient conditions for the n-dimensional Para-Sasakian manifold to be projective scalar-symmetric manifold is that the scalar projective curvature vanishes.
Proof: It is evident in view of the equations (3.5) and (3.6).