Generalised neo-pseudo projective recurrent Finsler space

Objectives: The purpose of this paper is to obtain several results in the field of generalised neo-pseudo projective recurrent Finsler space. Methods: A generalization technique is employed to solve the resulting problem. We provide its application in the study of space-time. Findings: In section 1, we have defined and studied some of the basic and useful results for later work. Section 2 deals for the neo-pseudo projective recurrent curvature tensor. The notion of neo-pseudo projective recurrent space of second order has been delineated in the section 3. In the section 4 we have studied the generalised neo-pseudo projective recurrent space and established several new results. Novelty/Conclusion: In this paper we have studied some recurrent properties of neo-pseudo projective curvature tensor in a Finsler space. We have obtained several new results which are as follows: • If the space Fn admits a neo-pseudo projective curvature tensor Qα βγδ then Qα βγδ is skew-symmetric with regard to last two indices. • If the neo-pseudo projective deviation tensor Qα β and pseudo deviation tensor field T α β coincides to each other for q = 1 then space is W-flat. • If Fn admits the projectively flat Q-recurrent space then the relation ∇ε Qα βγ +∇β Q α γε +∇γ Qα εβ = 0 holds good. • If a Finsler space Fn admits projectively flat Q-birecurrent space then the relation Kερ Qα βγ +Kβρ Q α γε +Kγρ Q α εβ = 0 holds good. • If the space is Q-birecurrent then the generalised Q-recurrent space is Qsymmetric. • For the projective flat generalised Q-recurrent space the relation ∇ρ ∇ε Qα γδ +∇ρ ∇γ Q α δε +∇ρ ∇δ Q α εγ = 0 holds good. AMS Subject Classification: 58B20, 53C20, 53C60.


Introduction
Let F n be an n-dimensional Finsler space with a positive definite metric g α β , which admit a projective deviation tensor field W α β and pseudo deviation tensor field T α β satisfing where in p and q are scalars which are positively homogenous of degree zero inẋ α . Prof. U.P. Singh and Prof. A.K. Singh while developing the theory of neo-pseudo projective curvature tensor, obtain two kinds of curvature tensor Q α β γ and Q α β γδ (1) . With a view to defining the projective deviation tensor field and pseudo deviation tensor field, he constructed the quantities Q α β (x,ẋ) which behave like neo-pseudo projective deviation tensor. With the help of tensor Q α β (x,ẋ) the absolute differential of concerning vector referred to the scalar function Q(x,ẋ) is defined as follows (1)(2)(3) : It is easy to verify that the neo-pseudo projective curvature tensor satisfies the following relations (1,4) : Moreover, these curvature tensor also satisfy the following identities and Q α βẋ β = 0 (1.8) As it is well known, in the Finsler space a scalar function Q(x,ẋ) is given by Let us consider a curvature tensor W α β in Finsler space, is termed as projective curvature tensor in the Finsler space and is defined as follows (2,5,6) : Wherein H α β is positively homogeneous of degree one inẋ α . In analogy with the relation (1.1) the projective curvature tensors W α β γ and W α β γδ in the Finsler space with the condition p = q = 1 may be defined as follows (2,7) : From equations (1.14) and (1.15), we get

Theorem 1.3:
If the neo-pseudo projective deviation tensor Q α β coincides with geodesic deviation tensor field H α β in the Finsler space F n then projective deviation tensor field W α β and the neo-pseudo projective deviation tensor Q α β are identically equal to each other.

Proof:
If the neo-pseudo projective deviation tensor Q α β coincides with geodesic deviation tensor field H α β . Consequently, from equation (1.11) follows Hence projective deviation tensor field W α β and the neo-pseudo projective deviation tensor Q α β are identically equal to each other.

Theorem 1.4:
If the neo-pseudo projective deviation tensor Q α β and pseudo deviation tensor field T α β coincides to each other for q = 1 then Finsler space admits the condition W α β = 0 i.e. W-flat. Proof: If the neo-pseudo projective deviation tensor Q α β and pseudo deviation tensor field T α β coincides to each other then from equation (1.18) we observe that This manifests that the space is W-flat.

Theorem 1.5:
If the neo-pseudo projective deviation tensor Q α β and projective deviation tensor field W α β coincides to each other then the geodesic deviation tensor field H α β vanish identically i.e. H-flat. Proof: https://www.indjst.org/ If the neo-pseudo projective deviation tensor Q α β and projective deviation tensor field W α β coincides to each other then from equation (1.11) follows the result Consequently, the space is H-flat. Theorem 1.6: If the projective deviation tensor field W α β and geodesic deviation tensor field H α β coincides to each other then the neo-pseudo projective deviation tensor Q α β vanish identically i.e. Q-flat.

Proof:
If the projective deviation tensor field W α β and geodesic deviation tensor field H α β coincides to each other. Consequently, from equation (1.11) follows Therefore the space is Q-flat.

Recurrent neo-pseudo projective curvature tensor in Finsler space
In view of the investigation of Prof. U.P. Singh and Prof. A.K. Singh (1) we observe that if the neo-pseudo projective deviation tensor Q α β is necessarily recurrent then projective deviation tensor and pseudo deviation tensor are proportional to each other. As a consequence of this follows the result wherein t is a scalar. As a consequence of equations (1.1) and (2.1), we obtain wherein s = pt + q is any scalar and positively homogeneous of degree zero inẋ α .

Definition 2.1:
A Finsler space whose curvature tensor is recurrent is called Q-recurrent Finsler space.
In view of the definition it follows that for a recurrent space, we have wherein R ε is a non-zero vector termed as the recurrent vector field.

Definition 2.2:
An n-dimensional Finsler space F n is called Q-symmetric when the covariant derivative of curvature tensor is everywhere zero i.e.

Definition 2.3:
A Finsler space Fn is said to be Q-flat when its curvature tensor vanishes identically.
As a consequence of this definition follows the result: Proof: Differentiating (2.3) covariantly with regard to x ρ , we get Interchanging ε and δ in equation (2.8) and subtracting the new equation from equation (2.8), we obtain Contracting (2.9) withẋ βẋγ and use of equation (1.7), we get Contracting α and δ in equation (2.10) and use of equation (1.9), we obtain

Definition 2.4:
If the neo-pseudo projective curvature tensor Q α β γ in the Finsler space F n satisfies the relation then F n is termed as Q-recurrent with recurrence vector field R ε . Consequently, we have a theorem: Theorem 2.2: If F n admits the projectively flat Q-recurrent space then the relation ∇ ε Q α β γ + ∇ β Q α γε + ∇ γ Q α εβ = 0 holds good.

Definition 3.1:
Neo-pseudo projective curvature tensor Q α β yδ of a Finsler space satisfies the relation wherein K ερ is non-zero recurrent tensor, then it is called neo-pseudo projective recurrent space of second order or briefly a Q-birecurrent space (3,4) . https://www.indjst.org/

Definition 3.2:
If the covariant derivative of neo-pseudo projective curvature tensor Q α β γδ vanishes identically then the space is termed as Q-bisymmetric.
As a consequence of above definition follows the result In this regard we shall now establish the following theorem: Theorem 3.1: The necessary and sufficient condition for Finsler space to be Q-bisymmetric space is that the neo-pseudo projective tensor vanishes identically. Proof: Since neo-pseudo projective tensor vanishes i.e. Q α β γδ = 0. Consequently from equation (3.1) it follows that ∇ ρ ∇ ε Q α β γδ = 0. This manifests that the space to be Q-bisymmetric.
Conversely, if the space to be Q-bisymmetric then the converse of theorem is immediately proof.

Remark 3.1:
It is noteworthy that every Q-recurrent is necessarily Q-birecurrent.

Theorem 3.2:
In a Finsler space F n , the recurrent tensor field K ερ is not symmetric in general.
Proof: Differentiating equation (2.16) covariantly with respect to x ρ , we get Since the space is Q-birecurrent then equation (3.2) assumes the form

Generalised Neo-Pseudo Projective Recurrent Space:
Let us consider the relation wherein R ρ and K ερ are recurrence vector and recurrence tensor fields respectively. Definition 4.1: The neo-pseudo projective curvature tensor Q α β γδ of Finsler space F n satisfying the condition (4.1) is called generalised neo-pseudo projective recurrent curvature tensor (3,4) .
In this regard, we have the following theorems: Theorem 4.1: The necessary and sufficient condition for Finsler space F n to be Q-symmetric is that the space has to be Q-birecurrent.

Theorem 4.2:
If the space F n is Q-symmetric and Q-flat then its generalised neo-pseudo projective recurrent space vanishes identically.

Remark 4.1:
It is noteworthy that if F n to be Q-symmetric and Q-flat follows that the generalised neo-pseudo projective recurrent space necessarily vanishes. Consequently, the space is simply generalised Q-symmetric one.

Theorem 4.3:
If space F n admits Q-symmetric and Q-flat then the space is a generalised Q-symmetric one.

Proof:
It follows immediately from theorem 4.2.

Theorem 4.4:
In Finsler space F n , if the space is Q-birecurrent then the generalised Q-recurrent space is Q-symmetric.

Theorem 4.5:
For the recurrence vector R ρ the relation holds good.