Let Fⁿ =(Mⁿ, L) be a Finsler space, where M is an n-dimensional differentiable manifold equipped with fundamental metric function L. i.e., a Finsler metric on M is a function L=L(x, y) defined as L:TM→[0, ∞) with the following properties:

i) L=L(x, y) is C^∞ on TM₀,

ii) L_x(y)=L(x, y), is positively 1-homogeneous on the fibres of the tangent bundle TM, 

iii) Strong Convex: The n×n Hessian matrix is strong definite at every point of TM₀. Thus the pair (M, L) is called a Finsler Space.

The Finsler space Fⁿ =(Mⁿ, L) is called Finsler space with (α, β)-metric if there exists a 2-homogeneous function L of two variables and such that the Finsler metric F:TM→R is given by F² (x, y)=L(α(x, y), β(x, y)), where is a Riemannian metric and β=b_i(x) yⁱ, is differential 1-form.

An (α, β)-metric is a Finsler metric expressed in the following form L=αφ(s), s=β/α. The function φ(s) is a C^∞ positive function on an open interval (-b₀, b₀) satisfying

(1)

In this case, the fundamental form of the metric tensor induced by L is positive definite.

The tensor , where is called the Douglas tensor. A Finsler space Fⁿ is a Douglas space if and only if the Douglas tensor vanishes identically.

Let Fⁿ =(Mⁿ, L) be an n-dimensional Finsler space on the differentiable manifold Mⁿ, equipped with the fundamental function L(x, y), then β-change of Finsler metric is given by,

(2)

where f is positively homogeneous function of degree one in L and β and β given by β(x, y)=b_i(x) yⁱ is a one-form on Mⁿ. The Finsler space (Mⁿ, L) obtained from Fⁿ by the β change equation (1) will be denoted by .

Finsler geometry was first introduced locally by Finsler himself (Antonelli, Ingarden, & Matsumoto, 1993), to be studied by many eminent mathematicians for its theoretical importance and applications in the variational calculus, mechanics, and theoretical physics. Moreover, the dependence of the fundamental function L(x, y) on both the positional argument x and directional argument y offers the possibility to use it to describe the anisotropic properties of the physical space.

Bácsó (1997) have introduced the concept of Douglas space, as a generalization of Berwlad space from the view point of geodesic equations. The condition for Finsler space with (α, β)-metric to be of Douglas type was studied by many authors (Li, Shen, & Shen, 2009; Aveesh & Narasimhamurthy, 2013; Park & Choi, 1999; Ramesha & Narasimhamurthy, 2017).

Randers (1941) had considered the simplest possible asymmetrical generalization of a Riemannian metric. Adding a 1-form to the existing Riemannian structure, he was the first to introduce a special Finsler space. This space - which became known in the literature as a Randers space - proved to be mathematically and physically very important. It was one of the first attempts to study a physical theory in the wider context of Finsler geometry, although Randers was not aware that the geometry he used was a special type of Finsler geometry. He has introduced the Finsler change , where L is a Riemannian metric and β is differential 1-form on M. Such a change of Finsler space is termed as Randers change. Randers well-known method for giving examples of Finsler spaces has the form , where a_ij is a Riemannian metric and β=b_i (x) yⁱ is one form with the condition .

And also In 1984, shibata and his co-authors have dealt with a change of Finsler metric, which is called a β-change of metric (Shibata, Shimada, Azuma, & Yasda, 1977). For a β-change of Finsler metric, the differential one-form β play very important roles.

In (Park & Lee, 2001a) the special Randers changes of Finsler spaces with (α, β)-metrics of Douglas type are characterized. And also (Park & Lee, 2001b), studied the Finsler space with certain (α, β)-metric of Douglas type with Randers change. (Matsumoto, 1974, 1986) had worked on Finsler spaces with Randers metric and special forms of important tensors. And also In 1984, Shibata have studied the general case of any β-change, that is, L=f(L, β) which generalizes many changes in Finsler geometry. In this context, they investigated the change of torsion and curvature tensors corresponding to the above transformation. In addition, they also studied some special Finsler spaces corresponding to specific forms of the function f(L, β).

By studying all these results, in this article, first the authors discuss that the condition for β-change of Finsler space is of Douglas type. Further, In the next part, the conditions for the β-change of Finsler space are derived with different classes of (α, β)- metrics to be of Douglas type.

2.1 Definition

Let Fⁿ =(Mⁿ, L) be an n-dimensional Finsler space, where Mⁿ be a n-dimensional differentiable manifold and L(x,y) is the fundamental function defined on the manifold TM₀ of non zero tangent vectors. It is assumed that L(x, y) is positive and the metric tensors is positive definite, where

The geodesics of an n-dimensional Finsler space Fⁿ =(Mⁿ, L), are given by the system of differential equations: (d²xⁱ)/(dt²)+2Gⁱ(x,dx/dt)=0, where   

where γ_jkⁱ(x, y) are Christoffel symbols constructed from the Finsler metric tensor g_ij(x, y) with respect to (xⁱ).

The covariant differentiation with respect to Levi-Civita connection {γ_jkⁱ} of Rⁿ is denoted by (;).

2.2 Definition

A Finsler space Fⁿ is said to be Douglas space, if

(3)

are homogeneous polynomial in (y) of degree 3.

According to Basco (1997), a Finsler space Fⁿ is of Douglas type if and only if the Douglas tensor , vanishes identically, where is the hv-curvature tensor of the Berwald connection,

According to (Matsumoto, 1998a), the function Gⁱ (x, y) of Fⁿ with the (α, β)-metric are written in the form,

(4)

(5)

where the subscript 0 means contraction by yⁱ and we put where

Since are homogeneous polynomial in (yⁱ) of degree two.

From equations (3) and (5), we have

(6)

Thus, the condition for a Finsler space to be of Douglas Space is that are hp(3)

Let =(Mⁿ, ) be a Finsler space, which is obtained by β-change of Fⁿ =(Mⁿ, L).

From equation (6), in are written as

(7)

Suppose Fⁿ is a Douglas space. Since B_ij are hp(3), the necessary and sufficient condition for Fⁿ to be also a Douglas space is that

(8)

are homogeneous polynomial of degree 3.

We use the results proved by Park and Lee (2001b)

2.3 Theorem

Let Fⁿ =(Mⁿ, L), be a Finsler space with an (α, β)-metric of Douglas type. Then Fⁿ =(Mⁿ, L+β) which is obtained by a β-change of Fⁿ is also a Douglas space if and only if W^ij is homogeneous polynomial of degree 3.

i.e., Suppose Fⁿ is a Douglas space. The condition for Fⁿ =(Mⁿ, L+β) to be a Douglas space is that equation (8) is hp (3).

In this section, the condition for a β-change of Finsler space Fⁿ is found to be of Douglas type.

Let us denote the following notations:

And again , where (|) denotes the h-cavarient derivative with respect to the cartan connection

(9)

Then and , where and . The tensors D_i, D_ji,D_jkⁱ and are positively homogeneous in yⁱ of degree two, one, and zero, respectively. In the following, the explicit form of Dⁱ is necessary. To find this, we deal with equation L_ij|k =0, where L_ij|k is the h-covarient derivative of L_ij=hij/L in CΓ. Then

(10)

Transvecting equation (10) by y^k, we have

(11)

Again consider L_i|j =0, then

(12)

Where

Plugging equation (12) in Then we have

(13)

(14)

From equations (11) and (14) , we have

(15)

Contracting equation (13) by y which yields

(16)

Again contracting equations (15) and (16) by yⁱ leads to

(17)

which represents the system for linear equations.

Here we use the following lemma proved by (Matsumoto, 1998b),

3.1 Lemma

A system of linear equations in X_i has the unique solution , where Yⁱ=g^ir Y_r and From the lemma (3.1),

(18)

where F_jⁱ =g^ir F_rj and F_j=b_rF_j^r.

Therefore, the tensor Dⁱ of (9) arising from a β-change are given by (18).

Again from equations (9) and (18), we have

(19)

Suppose Fⁿ is a Douglas space, that is, Gⁱy^j-G^jy^j are hp (3). Thus we have

Proposition 1: Let Fⁿ be a Douglas space and Fⁿ a Finsler space which is obtained by a β-change by β. is also a Douglas space if and only if are hp(3).

4.1 Finsler Space with (α, β)-Metrics of Douglas Type

Let Fⁿ =(Mⁿ, L), be a Finsler space. And be a Finsler space, which is transformed by a β-change of Finsler space Fⁿ =(Mⁿ, L) is of Douglas type.

Here we are discussing three different classes of (α, β)-metrics, which is transformed by a β-change of Finsler space Fⁿ =(Mⁿ, L),

are of Douglas types

4.2 Finsler Space with Metric is of Douglas Type

In this section, the authors characterize the condition for a Finsler space which is obtained by a β-change of Finsler space is of Douglas type.

For an (a, b)-metric,

(20)

The partial derivatives of equation (18) are as follows:

(21)

For , which is obtained by a b-change of Fⁿ =(Mⁿ, L), Substituting equaition (21) in (7), which gives

(22)

Suppose that be a Douglas space, i.e., be homogeneous polynomial in (yⁱ) of degree 3. Since α is irrational in (yⁱ), the terms of equation (22) can be divided in two equations:

(23)

(24)

Eliminating from equations (23) and (24), which yields,

(25)

where,

Contracting equation (25) by b_iy_i, which yields

(26)

Since the term 12β⁵ (2βs₀ -r₀₀) of (26) seemingly does not contain α², thus we must have hp(5) v₅, such that

(27)

Let us study the following cases.

Now the terms is equation (26) can be written as,

(28)

The terms in equation (28) does not contain α² as a factor are . Therefore, there exist a hp(3) v₃, such that

(29)

From α²(mod β), then v₃ vanishes and

(30)

Contract equation (30) by b_i, which yields k₂b² =0.

If k₂ =0, then b² =2/3 or s_i=0.

Again if b² =2/3, the terms of equation (28) can be written as 3α² (7 α²-33β²) s₀ =0, which yields s₀ =0 and r₀₀ =0. 

Next, if s_i=0, then we have s₀ =0 and r₀₀ =0.

On the other hand, if b² =0, terms of equation (28) can be written as,

(31)

which implies s₀ =0 and k₂ =0.

Therefore, for n>2, in both the cases of v₅ =0 and v₅ ≠0 leads to r₀₀ =0 and s₀ =0. So that equation (25) can be written as,

(32)

Contract equation (31) by y_i gives s₀ⁱ =0. Since r_ij =s_ij =s_jⁱ =0, we conclude that b_ij =0.

Thus, we state that,

Theorem

A Finsler space which is obtained by a β-change of aFinsler space Fⁿ =(Mⁿ, L) with equation (20) is of Douglas type if and only if b_ij =0.

4.3 Finsler SpaceWith Metric is of Douglas Type:

Let us consider an (α, β)-metric

(33)

And partial derivatives of equation (20) are:

(34)

To deduce the condition for Finsler space , which is obtained by a β-change of Fⁿ =(Mⁿ,L), using equation (34) in (7) yields

(35)

Suppose be a Douglas space, that is, be homogeneous polynomial in (y) of degree 3. Since α is irrational in (yⁱ), the terms of equation (35) can be divided in two equations:

(36)

(37)

Transvecting equation (37) by b_iy_i, which yields

(38)

From equation (38), s₀α²=0⟹s_i=0.

∴ equation (37) reduces to

(39)

Transvect equation (39) by y_i yields s0ⁱ=0 immediately after s_ij =0. Again, put equation (37) in (36), we have

(40)

In equation (40), only the term [3kb ] does not contain α² as a factor, thus we must havehp(3) v₃^ij ;

(41)

If suppose α² ≢0(modβ). Then equation (41) is reduced to

" style="vertical-align: bottom; display: inline;" width="" /> where v^ij are hp(1). Thus equation (40) leads to

(42)

Transvecting equation (42) by b_iy_j, we get

(43)

Now we define a function f₁ (x);

(44)

Removing b_i v_ij y_j from equation (44), which yields

(45)

∴from equation (45) and s_ij =0, which yields

(46)

where,

Conversely, if equation (46) is holds good, then s_ij =0 and

(47)

then of equation (35) is hp(3).

Now, we define,

Theorem

Let Fⁿ=(Mⁿ, L) Finsler space with an (α, β)-metric L=α+ϵβ+k β²/α; ϵ, k≠0. A Finsler space (n>2) which is obtained by a β-change of a Finsler space Fⁿ =(Mⁿ,L) is of Douglas type if and only if 

where

4.4 Finsler SpaceWith Metric is of Douglas Type

Let us consider an (α,β)-metric,

(48)

partial derivatives of equation (48) are:

(49)

To prove which is obtained by a β-change of Fⁿ =(Mⁿ, L) is of Douglas space,

Using equation (48) in (7) which yields

(50)

If is a Douglas space, i.e., be homogeneous polynomial in (y) of degree 3. Since α is irrational in (y^j), (4.31) can be rewritten as two equations:

(51)

(52)

Transvecting equation (52) by b_iy_j, which yields,

(53)

From equation (43), we have s₀ =0⟹s_i=0.

∴ equation (52) becomes

(54)

Transvecting equation (50) by y_i yields s₀ⁱ =0⟹s_ij =0.

Again, by using equation (52) and s₀ =0 in (51), we have

(55)

In equation (55), it seems that only the term does not contain α², thus we must have hp(3) v₃^ij, such that

(56)

If α² ≢0(modβ). Then equation (56) is reduced to where v^ij are hp(1). Thus equation (55) leads to

(57)

Transvecting equation (57) by b_iyⁱ, we get

(58)

Now we define a function f₁ (x);

(59)

Removing b_i v_ij y_j from equation (59), which yields

(60)

∴ From equation (60) and s_ij =0, which yields

(61)

where, f2(x)= f1(x)/ b2

Conversely, if equation (61) is holds good, then s_ij =0 and

(62)

then of equation (50) is hp(3).

Thus, we define,

Theorem Let Fⁿ =(Mⁿ, L) Finsler space with an (α, β)-metric L=β² /α. A Finsler space , which is obtained by a β-change of a Finsler space Fⁿ =(Mⁿ, L) is of Douglas type if and only if b_i:j =f₂ (x){(2b²) a_ij -3b_ib_j},

where f₂(x) = (f₁(x))/b².

The theory of Finsler space with (α,β)-metric has been developed into faithful branch of Finsler Geometry. From stand point of Finsler Geometry itself, Randers metric is very interesting because its form of simple and properties of Finsler spaces equipped this metric can be looked asRiemannian spaces equipped with the metric L(α, β) = α + β. 

With this M, Matsumoto introduced the transformation of Finsler metric (x, y)=L(x, y)+β(x, y) where β(x, y)=b_i(x) yⁱ is a one-form.C. Shibata introduced thetransformation of Finsler metric (x, y)=f (L, β),where β(x, y)=b_i(x) yⁱ, b_i(x) are components of a covariant vector in (Mⁿ, L)and f is positively homogenous function of degree one in L and β.

And also G. Randers studied the properties of β-change.

By studying all these concepts, the authors have finally summarized the results as follows:

In the first part, it deals with the β-change of a Finsler space with the (α,β)-metric is of Douglas type. Further, the Finsler space n n is investigated, which is transformed by a β-change of a Finsler space Fⁿ =(Mⁿ,L)with different classes of (α,β)-metrics is of Douglas type and vice versa. In this regard, the following conclusions are arrived.

A Finsler space (n > 2), which is obtained by a β-change of a Finsler space Fⁿ =(Mⁿ,L=((α² +αβ+β²))/(α+β)) is of Douglas type if and only if b_i:j =0.

A Finsler space (n > 2), which is obtained by a β-change of a Finsler space Fⁿ =(Mⁿ,L=((α² +ϵβ+k β²/α;ϵ,k≠0) is of Douglas type if and only if b is in the form,

b_i:j = f₂ (x){(2kb²+1) a_ij -3kb_i b_j},

where f₂(x)=(f₁(x))/(k²b² -k).

A Finsler space (n > 2), which is obtained by a β-change of a Finsler space Fⁿ =(Mⁿ, L=β²/α) is of Douglas type if and only if b can be written as b_i:j=f₂ (x){(2b²) a_ij-3b_ib_j}, where f₂(x)=(f₁(x))/b².

Conflict of Interest

The authors confirm that there is no conflict of interest to declare for this publication.

Acknowledgments

The authors would like to appreciate the effort from editors and reviewers. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.