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

Abstract

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.

Citation

Ranjan, R., Pandey, P.N. and Paul, A. (2021), "Conformal transformation of Douglas space of second kind with special (α, β)-metric", Arab Journal of Mathematical Sciences, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/AJMS-08-2021-0189

1. 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 [3–5] 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.

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.

2. Preliminaries

A Finsler space Fⁿ = (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 α² = a_ij(x)yⁱy^j and β = b_i(x)yⁱ is 1-form. The space Rⁿ = (M, α) is called Riemannian space associated with Fⁿ. We shall use the following symbols [6];

bi=airbr,b2=arsbrbs

2rij=bi|j+bj|i,2sij=bi|j−bj|i

sji=airsrj,sj=brsjr

The Berwald connection

BΓ={Gjki(x,y),Gji}

of Fⁿ plays an important role in this paper. Bjki denotes the difference tensor of Gjki and γjki that is

(1)Gjki(x,y)=γjki(x)+Bjki(x,y).

Using the subscript 0 and transvecting by yⁱ, we get

(2)Gji=γ0ji+Bjiand2Gi=γ00i+2Bi,

and then Bji=∂̇jBi and Bjki=∂̇kBji. A Finsler space Fⁿ of dimension n is called a Douglas space [14] if

(3)Dij=Gi(x,y)Yi−Gj(x,y)yi,

are homogeneous polynomial of (yⁱ) of degree three.

Next, differentiating (3) with respect to y^m, we obtain the following definitions;

Definition 1.

([14]) A Finsler space Fⁿ is a Douglas space of second kind if Dimi=(n+1)Gi−Gmimyi is a two homogeneous polynomial in (yⁱ).

On the other hand, a Finsler space with (α, β)-metric is a Douglas space of second kind if and only if

(4)Bmim=(n+1)Bi−Bmmyi,

are homogeneous equation in (yⁱ) of degree two, when Bmm is same as given in [14].

Furthermore, differentiating Eqn (4) with respect to y^h, y^j and y^k, we obtain

(5)Bhjkmim=Bhjki=0.

Definition 2.

A Finsler space Fⁿ with (α, β)-metric is known as Douglas space of second kind if Bmim=(n+1)Bi−Bmmyi is a homogeneous polynomial in (yⁱ) of degree two.

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 Gⁱ(x, y) of Fⁿ can be expressed as [4].

(6)2Gi=γ00i+2Bi

(7)Bi=αFβFαs0i+C*βFβαFyi−αFααFαyiα−αbiβ,

where

C*=αβr00Fα−2αs0Fβ2β2Fα+αγ2Fαα,

(8)γ2=b2α2−β2.

Since γ00i=γjki(x)yjyk is hp(2), Eqn (7) yields

(9)Bij=αFβFαs0iyj−s0jyi+α2FααβFαC*biyj−bjyi.

By means of (3) and (9), we obtain the following lemma [14];

Lemma 1.

A Finsler space Fⁿ with an (α, β)-metric is a Douglas space if and only if B^ij = Bⁱy^j − B^jyⁱ are hp(3).

Differentiating (9) with respect to y^h, y^k, y^p and y^q, we can have Dhkpqij=0 which are equivalent to Dhkpmim=(n+1)Dhkpi=0. Hence, a Finsler space Fⁿ satisfying the condition Dhkpqij=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

(10)Bim=n+1αFβs0iFα+αn+1α2ΩFααbi+βγ2Ayir002Ω2−α2n+1α2ΩFβFααbi+Byis0FαΩ2−α3Fααyir0Ω

where Ω=β2Fα+αγ2Fαα, provided that Ω ≠ 0, A=αFαFααα+3FαFαα−3αFαα2 and

(11)B=αβγ2FαFβFααα+β3γ2−β2Fα−4αγ2FααFβFαα+ΩFFαα

Following result is used in the succeeding section [7]:

Theorem 1.

A Finsler space Fⁿ is a Douglas space if second kind if and only if Bmim are homogeneous polynomials in (y^m) of degree two, where Bmim is given by Eqs (10) and (11), provided Ω ≠ 0.

4. 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 Fⁿ = (M, F) and F¯n=(M,F¯) be two Finsler spaces. Then Fⁿ is called conformal to F¯n if we have a function σ(x) in each coordinate neighbourhood of Mⁿ such that F¯(x,y)=eσF(x,y) and this transformation F→F¯ is called conformal change.

A conformal change of (α, β)-metric is given as α,β→α¯,β¯, where α¯=eσα, β¯=eσβ that is,

(12)a¯ij=e2σaij,b¯i=eσbi

(13)a¯ij=e−2σaij,b¯i=e−σbi

and b2=aijbibj=a¯ijb¯ib¯j

From Eqn (13), the Christoffel symbols are given by:

(14)γ¯jki=γjki+δjiσk+δkiσj−σiajk,

Where, σ_j = ∂_jσ and σⁱ = a^ijσ_j.

Using (13) and (14), we obtain the following identities:

∇¯jb¯i=eσ∇jbi+ρaij−σibj,

r¯ij=eσrij+ρaij−12biσj+bjσi,

s¯ij=eσsij+12biσj−bjσi,

s¯ji=e−σsji+12biσj−bjσi,

(15)s¯j=sj+12b2σj−ρbj,

Where, ρ = σ_rb^r.

Using Eqs (14) and (15), we get easily the followings:

(16)γ¯00i=γ00i+2σ0yi−α2σj,

(17)r¯00=eσr00+ρα2−σ0β,

(18)s¯0i=e−σs0i+12σs0bi−βσi,

(19)s¯0=s0+12σ0bi−ρβ.

Now we obtain the conformal transformation of B^ij given by Eqn (9).

Consider F¯(α,β)=eσF(α,β) then

(20)F¯α¯=Fα,F¯α¯α¯=e−σFαα,F¯β¯=Fβ,γ¯2=e2αγ2

From Eqs (8), (19), (20) and using Theorem 3.1, we obtain

(21)C¯*=eσC*+D*,

Where,

(22)D*=αββα2−σ0βFα−αb2σ0−ρβFβ2β2Fα+αγ2Fαα

Hence B^ij can be expressed as:

B¯ij=αFβFαs0iyj−s0jyi+α2FααβFαC*biyj−bjyi

+ασ0FβFα+α2FααβFαD*biyj−bjyi−αβFβ2Fασiyj−σjyi,

=Bij+Cij,

Where,

Cij=ασ0FβFα+α2FααβFαD*biyj−bjyi−αβFβ2Fασiyj−σjyi.

Using Eqn (11), we can have

(23)Ω¯=e2αΩ,A¯=e−σA,B¯=e2αB.

Now, we use conformal transformation on Bmim and obtain

(24)B¯mim=Bmim+Kmim

Where, Kmim is given by [15, 16].

(25)2Kmim=n+1αFβFασ0bi−βσi+αn+1α2ΩFααbi+βγ2AyiΩ2ρα2−σ0β−α2n+1α2ΩFβFααbi+ByiFαΩ2b2σ0−ρβ.

Therefore, we obtain the following result:

Theorem 2.

A Douglas space of second kind is conformally invariant if and only if Kmim(x) are homogeneous polynomial in (yⁱ) of degree two.

5. Conformal change of Douglas space of second kind with special (α, β)-metric F=α+ϵβ+kβt+1αt

Consider a Finsler manifold with special (α, β)-metric defined as

F=α+ϵβ+kβt+1αt,

Where, ϵ and k are constant.

Then we obtain

Fα=1−tkβt+1αt+1,

(26)Fβ=ϵ+k(t+1)βtαt,

Fαα=t(t+1)kβt+1αt+2

Fααα=−6kβ2α4.

Therefore, using Eqn (11), we obtain

Ω=−t(t+2)kβt+3+αtβ+b2t(t+1)αβtαβαt+1

(27)A=t(t+1)kβt+1αt+2(1−t)−2t(t+2)kβt+1αt+1

B=∏1+∏2+∏3

Where,

∏1=−t(t+1)(t+2)kβt+2αt+2ϵ+k(t+1)βtαt−ϵnkβt+1αt+1−t(t+1)k2β2t+1α2t+1b2α2−β2,

∏2=t(t+1)kβt+2αt+2ϵ+k(t+1)βtαt3−t(4t+7)βt+1αt+1b2α2+t(t+2)kβt+1αt+1−14β2,

∏3=t(t+1)kβt+2αt+2(αβ+εβ2)+t(t+1)βtαt(b2α2+εb2αβ−kβ2−εkβ3α−1)+βt+1αt+1(b2α2−k2β2),

Hence, using Eqn (26), Kmim can be reduced as

(28)2Kmim=(n+1)αϵ+k(t+1)βtαt(σ0bi−βσi)+(αA1+αA2)(ρα2−σ0β)−B0+(B1+B2+B3)yi−C1(b2σ0−ρβ).

Where,

αA1=(n+1)t(t+1)kα2βt+1{αt+1β2+b2t(t+1)α2βt+1}−t(t+2)kβt+3bi,

αA2=t(t+1)k(1−t)αt+1−2t(t+2)kβt+2βtγ2{αt+1β+b2t(t+1)α2βt}−t(t+2)kβt+22yi

B0=(n+1)t(t+1)kα4βt(εαt+k(t+1)βt)(αt+1−tkβt+1)αt+1β+b2t(t+1)α2βt−t(t+2)kβt+2bi

B1=−t(t+1)(t+2)kα2βt+2ϵα2t+1+k(t+1)αt+1βt−εtkαtβt+1−t(t+1)k2β2t+1(αt+1−tkβt+1)αt+1β2+b2t(t+1)α2βt+1−t(t+2)kβt+32γ2,

B2=t(t+1)(t+2)kα2βt+2(εαt+k(t+1)βt)(αt+1−tkβt+1)αt+1β2+b2t(t+1)α2βt+1−t(t+2)kβt+323b2αt+3−t(4t+7)kb2α2βt+1−4αt+1β2+4t(t+2)kβt+3.

B3=kt(t+1)(αβ)t+2(αt+1−tkβt+1)αt+1β2+b2t(t+1)α2βt+1−t(t+2)kβt+32αt+2β+ϵαt+1β2t(t+1)(b2α3βt−kαβt+2)+(t(t+1)+ϵb2)α2βt+2−(t(t+1)kϵ+k2)βt+3

C=−t(t+1)kα2βt+1αt+1β2+b2t(t+1)α2βt+1−t(t+2)kβt+3yi.

Now, Eqn (28) can also be written as

(29)2Kmim=(n+1)αε+k(t+1)(α−1β)(σ0bi−βσi)+p1+p2+p3+p4+p5+p6+p7.

where,

p1=αA1(ρα2−σ0β)

p2=αA2(ρα2−σ0β)

p3=−B0(b2σ0−ρβ)

p4=−B1yi(b2σ0−ρβ)

p5=−B2yi(b2σ0−ρβ)

p6=−B3yi(b2σ0−ρβ)

p7=C(b2σ0−ρβ)

showing that Kmim is homogeneous polynomial of degree 2 in yⁱ.

Theorem 3.

A Douglas space of second kind with special (α, β)-metric F=α+ϵβ+kβt+1αt, where ϵ 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 ϵ = 1 and k = 0, we have F = α + β which is Randers metric. In case, 2Kmim occupies the form

  (30)2Kmim=n+1ασ0bi−βσi,

  Which shows Kmim is homogeneous polynomial in (yⁱ) of degree two.

Note that in this case, p₁ = p₂ = p₃ = p₄ = p₅ = p₆ = p₇ = 0.

Corollary 1.

A Douglas space of second kind with Randers metric F = α + β, is conformally invariant.

• Case(ii). If ϵ = 0 and k = 1, we have F=α+βt+1αt. In this case 2Kmim obtains the form

(31)2Kmim=(n+1)(t+1)(α−1β)α(σ0bi−βσi)+q1+q2+q3+q4+q5+q6+q7,

Where,

q1=(n+1)t(t+1)α2βt+1αt+1β2+b2t(t+1)α2βt+1−t(t+2)kβt+3bi(σ0bi−βσi),

q2=t(t+1)(1−t)αt+1−2t(t+2)βt+1βtγ2αt+1β+b2t(t+1)α2βt−t(t+2)βt+22(ρα2−σ0β),

q3=(n+1)t(t+1)2α4β2tbi(αt+1−tβt+1)αt+1β+b2t(t+1)α2βt−t(t+2)βt+2(b2σ0−ρβ),

q4=t(t+1)2(t+2)α2β2t+2γ2αt+1β2+b2t(t+1)α2βt+1−t(t+2)βt+32(b2σ0−ρβ),

q5=−t(t+1)2α2β2t+23b2αt+3−t(4t+7)b2α2βt+1−4α2βt+1+4t(t+2)βt+3(αt+1−tβt+1)αt+1β2+b2t(t+1)α2βt+1−t(t+2)kβt+32yi(b2σ0−ρβ),

q6=−t(t+1)(αβ)t+2αt+2β+t(t+1){b2α3βt+α2βt+1−αβt+2}−βt+3(αt+1−tβt+1)αt+1β2+b2t(t+1)α2βt+1−t(t+2)kβt+32yi(b2σ0−ρβ),

q7=t(t+1)α2βt+1αt+1β2+b2t(t+1)α2βt+1−t(t+2)kβt+3yi(b2σ0−ρβ),

Showing that Kmim is homogeneous polynomial in (yⁱ) of degree 2.

Thus, we can have following;

Corollary 2.

A Douglas space of second kind with special (α, β)-metric F=α+βt+1αt is conformally transformed to a Douglas space of second kind.

• Case(iii). If ϵ = 1 and k = 1, we obtain F=α+β+βt+1αt. In the case, 2Kmim occupies the form

(32)2Kmim=(n+1)1+(t+1)(α−1β)α(σ0bi−βσi)+r1+r2+r3+r4+r5+r6+r7,

where,

r1=(n+1)t(t+1)α2βt+1αt+1β+b2t(t+1)α2βt−t(t+2)βt+2bi(ρα2−σ0β),

r2=t(t+1)(1−t)αt+1−2t(t+2)βt+1βtγ2αt+1β+b2t(t+1)α2βt−t(t+2)βt+22yi(ρα2−σ0β),

r3=−(n+1)t(t+1)α4βt(αt+(t+1)βt)bi(αt+1−tβt+1)αt+1β+b2t(t+1)α2βt−t(t+2)βt+2(b2σ0−ρβ),

r4=t(t+1)(t+2)α2βt+2(α2t+1+(t+1)αt+1βt−tαtβt+1−t(t+1)β2t+1)γ2(αt+1−tβt+1)αt+1β2+b2t(t+1)α2βt+1−t(t+2)βt+32yi(b2σ0−ρβ),

r5=−t(t+1)α2βt+2(αt+(t+1)βt)(αt+1−tβt+1)αt+1β2+b2t(t+1)α2βt+1−t(t+2)βt+323b2αt+3−t(4t+7)b2α2βt+1−4αt+1β2+4t(t+2)βt+3yi(b2σ0−ρβ),

r6=t(t+1)(αβ)t+2(αt+1−tβt+1)αt+1β2+b2t(t+1)α2βt+1−t(t+2)βt+32αt+2β+αt+1β2+t(t+1)(b2α3βt−αβt+2)+(t2+1+b2)α2βt+1−(t2+t+1)βt+3yi(b2σ0−ρβ),

r7=−t(t+1)α2βtαt+1β+b2t(t+1)α2βt−t(t+2)βt+2yi(b2σ0−ρβ).

Showing that Kmim is a homogeneous polynomial in (yⁱ) of degree 2.

Thus, we obtain the following;

Corollary 3.

A Douglas space of second kind with special (α, β)-metric F=α+β+βt+1αt is conformally invariant.

• Case(iv). If ϵ = 1, k = 1 and t = 1, we obtain F=α+β+β2α. Then, 2Kmim reduces in the form

(33)2Kmim=(n+1)1+2(α−1β)α(σ0bi−βσi)+u1+u2+u3+u4+u5+u6+u7,

Where,

u1=2(n+1)α2β(1+2b2)α2−3β2biρα2−σ0β,

u2=12βγ2(1+2b2)α2−3β22yiρα2−σ0β,

u3=−2(n+1)α4(α+2β)α2−β2(1+2b2)α2−3β2bib2σ0−ρβ,

u4=6α2(α3+2α2β−αβ2−2β3)γ2βα2−β2(1+2b2)α2−3β22yib2σ0−ρβ,

u5=−2α2(α+2β)βα2−β2(1+2b2)α2−3β223b2α4−(11b2+4)α2β2+12β4yib2σ0−ρβ,

u6=2α3α2−β2(1+2b2)α2−3β22(1+2b2)α3+(3+b2)α2β−αβ2−3β3yib2σ0−ρβ,

u7=−2α2(1+2b2)α2−3β2yib2σ0−ρβ.

Showing that Kmim is a homogeneous polynomial in (yⁱ) of degree 2.

Thus, we can have the following;

Corollary 4.

A Douglas space of second kind with first approximate Matsumoto metric F=α+β+β2α is invariant under conformal change.