If the space F^{n} admits a neo-pseudo projective curvature tensor

If the neo-pseudo projective deviation tensor

If F^{n} admits the projectively flat Q-recurrent space then the relation

If a Finsler space F^{n} admits projectively flat Q-birecurrent space then the relation

If the space is Q-birecurrent then the generalised Q-recurrent space is Q-symmetric.

For the projective flat generalised Q-recurrent space the relation

Let F^{n} be an n-dimensional Finsler space with a positive definite metric g_{αβ}, which admit a projective deviation tensor field

where in p and q are scalars which are positively homogenous of degree zero in

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

With the help of tensor

and

It is easy to verify that the neo-pseudo projective curvature tensor satisfies the following relations

and

Moreover, these curvature tensor also satisfy the following identities

and

As it is well known, in the Finsler space a scalar function

Let us consider a curvature tensor

Wherein

In analogy with the relation (1.1) the projective curvature tensors

and

In view of above discussions, we have the following theorems:

For the neo-pseudo projective curvature tensor the relation

holds good.

Interchanging β and γ in equation (1.2) and adding this with new equation, we get the desired result.

If the space F^{n} admits a neo-pseudo projective curvature tensor

Interchanging γ and δ in equation (1.3) and adding the new equation to the equation (1.3), we obtain

From equations (1.14) and (1.15), we get

If the neo-pseudo projective deviation tensor ^{n} then projective deviation tensor field

If the neo-pseudo projective deviation tensor

Hence projective deviation tensor field

If the neo-pseudo projective deviation tensor

Inseting q = 1 in equation (1.1), we obtain

If the neo-pseudo projective deviation tensor

This manifests that the space is W-flat.

If the neo-pseudo projective deviation tensor

If the neo-pseudo projective deviation tensor

Consequently, the space is H-flat.

If the projective deviation tensor field

If the projective deviation tensor field

Therefore the space is Q-flat.

In view of the investigation of Prof. U.P. Singh and Prof. A.K. Singh

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 in

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

An n-dimensional Finsler space F^{n} is called Q-symmetric when the covariant derivative of curvature tensor is everywhere zero i.e.

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:

Contracting (2.3) with

Again contracting (2.6) with

Thus, we have now the following theorem:

For the recurrence vector space ^{n} there exists the relation

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) with

Contracting α and δ in equation (2.10) and use of equation (1.9), we obtain

If the neo-pseudo projective curvature tensor ^{n} satisfies the relation

then F^{n} is termed as Q-recurrent with recurrence vector field

Consequently, we have a theorem:

If F^{n} admits the projectively flat Q-recurrent space then the relation

If the space is projectively flat then from equation (1.12), we have

Differentiating (2.13) covariantly with respect to x^{ϵ}, we get

Taking the cyclic permutation in β, γ, ϵ and adding, we have

The first part of equation (2.15) vanishes due to commutation formula (

Neo-pseudo projective curvature tensor

wherein

If the covariant derivative of neo-pseudo projective curvature tensor

As a consequence of above definition follows the result

In this regard we shall now establish the following theorem:

The necessary and sufficient condition for Finsler space to be Q-bisymmetric space is that the neo-pseudo projective tensor vanishes identically.

Since neo-pseudo projective tensor vanishes i.e.

Conversely, if the space to be Q-bisymmetric then the converse of theorem is immediately proof.

It is noteworthy that every Q-recurrent is necessarily Q-birecurrent.

In a Finsler space F^{n}, the recurrent tensor field

Contracting α and δ in equation (3.1) yields

Interchanging ϵ and ρ in equation (3.3) and subtracting the new equation from equation (3.3), we obtain

Contracting (3.4) with

Using commutation formula (

From equations (1.9) and (3.6), consequently, follows

Yields the result

If a Finsler space F^{n} admits projectively flat Q-birecurrent space then the relation

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

Let us consider the relation

wherein

The neo-pseudo projective curvature tensor ^{n} satisfying the condition (4.1) is called generalised neo-pseudo projective recurrent curvature tensor

Finsler space F^{n} equipped with the generalised neo-pseudo projective recurrent curvature tensor

In this regard, we have the following theorems:

The necessary and sufficient condition for Finsler space F^{n} to be Q-symmetric is that the space has to be Q-birecurrent.

If the space is to be Q-symmetric i.e.

Consequently, from equation (4.1) follows

which is the condition of Q-birecurrent.

Conversely, let us assume that the space be Q-birecurrent, follows the condition (4.2). Inserting equation (4.2) into equation (4.1), we obtain

If the space F^{n} is Q-symmetric and Q-flat then its generalised neo-pseudo projective recurrent space vanishes identically.

If the space be Q-symmetric i.e.

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.

If space F^{n} admits Q-symmetric and Q-flat then the space is a generalised Q-symmetric one.

It follows immediately from theorem 4.2.

In Finsler space F^{n}, if the space is Q-birecurrent then the generalised Q-recurrent space is Q-symmetric.

It is obvious from equations (2.4), (3.1) and (4.1).

For the recurrence vector

holds good.

Contracting (4.1) by α and δ, we obtain

Interchanging ϵ and ρ in equation (4.4) and subtracting the new equation from equation (4.4), we get

Contracting (4.5) with

By virtue of equations (3.5) and (4.6), we get

Inserting equation (1.9) in equation (4.7), we get the desired result.

For the projective flat generalised Q-recurrent space the relation

Contracting (4.1) with

Taking the cyclic permutation in ϵ, γ, δ and on making use of equations (2.16) and (3.10), we observe that