Conformal transformation of Douglas space of second kind with special ( α , β )-metric

Purpose – In this paper, the authors prove that the Douglas space of second kind with a generalised form of special ( α , β )-metric F , is conformally invariant. Design/methodology/approach – For, the authors have used the notion of conformal transformation and Douglas space. Findings – The authors found some results to show that the Douglas space of second kind with certain ( α , β )-metrics such as Randers metric, first approximate Matsumoto metric along with some special ( α , β )-metrics, is invariant under a conformal change. Originality/value – The authors introduced Douglas space of second kind and established conditions under which it can be transformed to a Douglas space of second kind. and B.N. developed the conformally invariant tensorial quantities in a Finsler space with ( α , β )-metric under conformal β -change.


Introduction
A number of geometers have been studying Douglas space [1,2] from different point of view. The theory of Finsler spaces more precisely Berwald spaces with an (α, β)-metric [3][4][5] have significant role to develop the Finsler geometry [6]. The concept of Douglas space of second kind with (α, β)-metric was first discussed by I. Y. Lee [7] in Finsler geometry. In [8], S. Bacso and Matsumoto developed the concept of Douglas space as an extension of Berwald space. In [9], S. Bacso and Szilagyi introduced the concept of weakly-Berwald space as another extension of Berwald space. In [10], M. S. Kneblman started working on the concept of conformal Finsler spaces and consequently, this notion was explored by M. Hashiguchi [11]. In [12,13] Y. D. Lee and B.N. Prasad developed the conformally invariant tensorial quantities in a Finsler space with (α, β)-metric under conformal β-change.

Conformal transformation of Douglas space
In this paper, we prove that the Douglas space of second kind with generalised special (α, β)-metric is conformally invariant. In the consequence, we find some results to show that the Douglas space of second kind with certain (α, β)-metric such as Randers metric, first approximate Matsumoto metric and Finsler space with some generalised form of (α, β)-metric remains unchanged geometrically under a confomal transformation.

Preliminaries
A Finsler space F n 5 (M, F(α, β)) is said to be with an (α, β)-metric if F(α, β) is a positively homogeneous function in α and β of degree 1, where α is Riemannian metric given by α 2 5 a ij (x)y i y j and β 5 b i (x)y i is 1-form. The space R n 5 (M, α) is called Riemannian space associated with F n . We shall use the following symbols [6]; of F n plays an important role in this paper. B i jk denotes the difference tensor of G i jk and γ i jk that is Using the subscript 0 and transvecting by y i , we get and then B i A Finsler space F n of dimension n is called a Douglas space [14] if are homogeneous polynomial of (y i ) of degree three. Next, differentiating (3) with respect to y m , we obtain the following definitions; are homogeneous equation in (y i ) of degree two, when B m m is same as given in [14]. Furthermore, differentiating Eqn (4) with respect to y h , y j and y k , we obtain Definition 2. A Finsler space F n with (α, β)-metric is known as Douglas space of second kind if B im m ¼ ðn þ 1ÞB i − B m m y i is a homogeneous polynomial in (y i ) of degree two.

AJMS
3. Douglas space of second kind with (α, β)-metric Under this section, we discuss the criteria for a Finsler space with an (α, β)-metric to be a Douglas space of second kind [2]. The spray coefficient G i (x, y) of F n can be expressed as [4]. where Since γ i 00 ¼ γ i jk ðxÞy j y k is hp(2), Eqn (7) yields By means of (3) and (9), we obtain the following lemma [14]; Differentiating (9) with respect to y h , y k , y p and y q , we can have D ij Hence, a Finsler space F n satisfying the condition D ij hkpq ¼ 0 is called Douglas space. Now, differentiating Eqn (9) with respect to y m and contracting m and j in the resulting equation, we get where Following result is used in the succeeding section [7]:  (10) and (11), provided Ω ≠ 0.

Conformal change of Douglas space of second kind with (α, β)-metric
In this section, we find the criteria for a Douglas space of second kind to be conformally invariant.
Let F n 5 (M, F ) and F n ¼ ðM ; FÞ be two Finsler spaces. Then F n is called conformal to F n if we have a function σ(x) in each coordinate neighbourhood of M n such that Fðx; yÞ ¼ e σ Fðx; yÞ and this transformation F → F is called conformal change.
Therefore, we obtain the following result: Theorem 2. A Douglas space of second kind is conformally invariant if and only if K im m ðxÞ are homogeneous polynomial in (y i ) of degree two.

Conformal change of Douglas space of second kind with special (α, β)-metric
Consider a Finsler manifold with special (α, β)-metric defined as Where, e and k are constant.
Then we obtain

Conformal transformation
of Douglas space Therefore, using Eqn (11), we obtain Hence, using Eqn (26), K im m can be reduced as Where, Now, Eqn (28) can also be written as where, showing that K im m is homogeneous polynomial of degree 2 in y i .
α t , where e and k are constants, is conformally invariant.
With the help of Theorem 3 it can be proved that a Douglas space of second kind with a Finsler space of certain (α, β)-metric is conformally transformed to a Douglas space of second kind. In this way, one can have following possible cases; Case(i). If e 5 1 and k 5 0, we have F 5 α þ β which is Randers metric. In case, 2K im m occupies the form Which shows K im m is homogeneous polynomial in (y i ) of degree two.

Conformal transformation of Douglas space
Corollary 1. A Douglas space of second kind with Randers metric F 5 α þ β, is conformally invariant.
Case(ii). If e 5 0 and k 5 1, we have F ¼ α þ β tþ1 α t . In this case 2K im m obtains the form Where, Showing that K im m is homogeneous polynomial in (y i ) of degree 2. Thus, we can have following; Corollary 2. A Douglas space of second kind with special (α, β)-metric F ¼ α þ β tþ1 α t is conformally transformed to a Douglas space of second kind.
Case(iii). If e 5 1 and k 5 1, we obtain F ¼ α þ β þ β tþ1 α t . In the case, 2K im m occupies the form where, Showing that K im m is a homogeneous polynomial in (y i ) of degree 2. Thus, we obtain the following; Corollary 3. A Douglas space of second kind with special (α, β)-metric F ¼ α þ β þ β tþ1 α t is conformally invariant.