Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 4s (2025) 

 

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(𝜶, 𝜷)-Metric on Cartan Space 

 

Robin Kumar1, Mohammad Rafee2, Gaurav Kumar3 
1,2,3 Department of Mathematics, School of Sciences, RIMT University, Punjab, India 

Email Id: robinkumar15101983@gmail.com1 ,Corresponding author: mohd_rafee60@yahoo.com 

  

Article History: 

Received: 10-09-2024 

Revised: 15-11-2024 

Accepted: 25-11-2024 

Abstract:  

In Finsler geometry, a Cartan space or Cartan manifold refers to a special type of geometrical 

structure where there is a preferred connection or curvature, often associated with a 

homogeneous space that carries additional symmetries. In the present research paper, we have 

deduced necessary and sufficient conditions under which an (α,β)-metric, 

K(x,ω)=α(x,ω)+ϵβ(x,ω)+2k (β^2 (x,ω))/(α(x,ω))-k^2/3  (β^4 (x,ω))/(α^3 (x,ω)), on a Cartan 

space admitting h-metrical d-connection becomes a locally Minkowski and conformally flat 

space. 

Keyword: Cartan Space, (α,β)-metric, Minkowski space, Conformally flat space. 

AMS Subject Classification: 53B40, 53C60 

 

1  Introduction 

The term "Cartan space" can refer to spaces that have a particular kind of connection (typically the 

Cartan connection) or spaces that have been generalized to accommodate more complex geometrical 

structures. This space was founded by E. Cartan a French mathematician and geometer [2]. Cartan 

space is the dual of a Finsler space [6] and this dual space was defined using a linear functional named 

as Legender transformation. The relation between Cartan space and Finsler space has been studied by 

F. Brickell [1], H. Rund [10] and others. R. Miron ( [6], [7]) introduced the theory of Hamiltonian 

space, He proved that Cartan space is a particular case of Hamilton space. The notion of (𝛼, 𝛽)-metric 

in Cartan space was introduced by T. Igrashi ( [4], [3]. He obtained the metric tensors and some 

invariants which characterize the special class of Cartan spaces with (𝛼, 𝛽)-metric. H.G. Nagaraja [8], 

G. Shanker [11], M. Rafee ([9], [15]) and Tripathi [13] have also made significant development in 

the theory of Cartan spaces with (𝛼, 𝛽)-metric. The paper is organized as follows: 

In Section 2, we give basic definitions and results required for subsequent sections. In Section 3, we 

deal with Cartan space with an (𝛼, 𝛽) -metric, 𝐾(𝑥, 𝜔) = 𝛼(𝑥, 𝜔) + 𝜖𝛽(𝑥, 𝜔) + 2𝑘
𝛽2(𝑥,𝜔)

𝛼(𝑥,𝜔)
−

𝑘2

3

𝛽4(𝑥,𝜔)

𝛼3(𝑥,𝜔)
, admitting ℎ-metrical 𝑑-connection. In Section  4, we study the conformal change of Cartan 

space and find some important results. 

2  Preliminaries 

We recall some important definitions like Finsler metric in cotangent bundle, Cartan space etc:  

Definition 2.1 (Finsler metric of cotangent bundle )  

Let 𝑀 be a smooth manifold and 𝑇∗𝑀 be its cotangent bundle. A 𝐶∞ function 𝐾: 𝑇∗𝑀\{0} → 𝑅 is 

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Communications on Applied Nonlinear Analysis 

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called Finsler metric or Finsler fundamental function on the cotangent bundle 𝑇∗𝑀 if it satisfies the 

following properties: 

 (1) Positivity: 𝐾(𝑥, 𝜔) ≥ 0 for all 𝜔 ∈ 𝑇𝑝
∗𝑀. 

(2) Positive Homogeneity: 𝐾(𝑥, 𝜔) is +ve 1-homogeneous on the fibers of the cotangent  

bundle 𝑇∗𝑀, i.e.,  𝐾(𝑥, 𝜆𝜔) = 𝜆𝐾(𝑥, 𝜔), ∀ 𝜆 > 0; for any 𝑥 ∈ 𝑀, 𝜔 ∈ 𝑇𝑥
∗𝑀. 

(3) Strict Convexity of 𝐾(𝑥, 𝜔): The hessian matrix defined by 𝑔𝑖𝑗(𝑥, 𝜔) =
1

2

𝜕2𝐾2

𝜕𝜔𝑖𝜕𝜔𝑗 (𝑥, 𝜔) is 

positive definite for all (𝑥, 𝜔) ∈ 𝑇∗𝑀\{0}.  

Definition 2.2 (Cartan Space) 

A differentiable manifold 𝑀 equipped with a Finsler metric 𝐾(𝑥, 𝜔) defined on the cotangent bundle 

𝑇∗𝑀 is called a Cartan space..  

Cartan space is denoted by 𝐶 = (𝑀, 𝐾(𝑥, 𝜔)), where 𝐾(𝑥, 𝜔) represents norm of the differential one 

form 𝜔 ∈ 𝑇𝑥
∗𝑀 based at any point 𝑥 ∈ 𝑀. The function 𝐾(𝑥, 𝜔) is called the fundamental function 

and 𝑔𝑖𝑗(𝑥, 𝜔) =
1

2

𝜕2𝐾2

𝜕𝜔𝑖𝜕𝜔𝑗
(𝑥, 𝜔)  is called the fundamental metric tensor of the Cartan space 𝐶 . In 

Cartan space the metric 𝐾: 𝑇∗𝑀 → [0, ∞) is defined from cotangent bundle 𝑇∗𝑀 to non-negative real 

numbers, so at a point 𝑥 ∈ 𝑀 , 𝐾(𝑥, −)  eats one-form 𝜔 ∈ 𝑇𝑝
∗𝑀  and spits non-negative reals, 

amounts to saying that Cartan space is constructed on the cotangent bundle 𝑇∗𝑀 in the same way a 

Finsler space (𝑀, 𝐹(𝑥, 𝑦)), where 𝐹: 𝑇𝑀 → [0, ∞), is constructed on the tangent bundle 𝑇𝑀. 

Next we define the norm of a differential one form 𝜔 ∈ 𝑇𝑝
∗𝑀  in local coordinates or in terms of 

fundamental metric tensor 𝑔𝑖𝑗 of the corresponding Cartan space (𝑀, 𝐾(𝑥, 𝜔)).  

Definition 2.3 (Finsler norm of a differential one form) 

Let (𝑀, 𝐾(𝑥, 𝜔)) be a Cartan space , where 𝐾(𝑥, 𝜔) is a Finsler metric on the cotangent bundle 𝑇∗𝑀. 

Then the norm of a differential one form 𝜔 ∈ 𝑇𝑝
∗𝑀 at any fixed point 𝑥 ∈ 𝑀 is denoted by 𝐾𝑥(𝜔) 

and defined by  

 𝐾𝑥
2(𝜔) =

1

2

𝜕2𝐾2

𝜕𝜔𝑖𝜕𝜔𝑗
(𝑥, 𝜔)𝜔𝑖𝜔𝑗 

 = 𝑔𝑖𝑗(𝑥, 𝜔)𝜔𝑖𝜔𝑗, 

where 𝑔𝑖𝑗(𝑥, 𝜔) =
1

2

𝜕2𝐾2

𝜕𝜔𝑖𝜕𝜔𝑗
(𝑥, 𝜔) is the fundamental metric tensor of the Finsler metric 𝐾(𝑥, 𝜔) of 

cotangent bundle.  

Moreover, when referring to a Cartan space with an (𝛼, 𝛽)-metric, this could mean the Cartan space 

is endowed with a metric that is parametrized by two parameters, typically denoted as α and 𝛽. This 

type of metric is used in various contexts, including in the study of spacetimes or in certain models of 

differential geometry and gravity, such as generalized theories of relativity. Let us precisely define 

what is meant by a Cartan space with (𝛼, 𝛽)-metric. 



Communications on Applied Nonlinear Analysis 

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Definition 2.4 If the fundamental function 𝐾(𝑥, 𝜔) of a Cartan space 𝐶 = (𝑀, 𝐾(𝑥, 𝜔)) is a function 

of variables 𝛽(𝑥, 𝜔) = 𝜔𝑖𝑏
𝑖(𝑥), where 𝑎𝑖𝑗(𝑥) is a Riemannian metric and 𝑏𝑖(𝑥) is a vector field 

depending only on 𝑥, then 𝐶 is called Cartan space with (𝛼, 𝛽)-metric. Here it is to be remarked that 

𝐾(𝑥, 𝜔) must satisfy all the conditions imposed on the fundamental function of a Cartan space.  

Let us consider a Cartan space 𝐶 = (𝑀, 𝐾(𝑥, 𝜔))  with an (𝛼, 𝛽) -metric, 𝐾(𝑥, 𝜔) = 𝛼(𝑥, 𝜔) +

𝜖𝛽(𝑥, 𝜔) + 2𝑘
𝛽2(𝑥,𝜔)

𝛼(𝑥,𝜔)
−

𝑘2

3

𝛽4(𝑥,𝜔)

𝛼3(𝑥,𝜔)
, where 𝛼 = (𝑎𝑖𝑗(𝑥, 𝜔)𝜔𝑖𝜔𝑗)

1

2 and 𝛽 = 𝜔𝑖𝑏
𝑖(𝑥). 

The fundamental tensor 𝑔𝑖𝑗(𝑥, 𝜔)  and its reciprocal tensor 𝑔𝑖𝑗(𝑥, 𝜔)  of the Cartan space 𝐶 =

(𝑀, 𝐾(𝛼, 𝛽)) are given by [3]  

 𝑔𝑖𝑗 = 𝜌𝑎𝑖𝑗 + 𝜌0𝑏𝑖𝑏𝑗 + 𝜌−1(𝑏𝑖𝜔𝑗 + 𝑏𝑗𝜔𝑖) + 𝜌−2𝜔𝑖𝜔𝑗, (1) 

where 𝜌, 𝜌0, 𝜌−1 and 𝜌−2 are invariants which are defined and calculated as follows: 

 𝜌 =
1

2𝛼
𝐾𝛼 

 =
𝛼5−2𝑘𝛼2𝛽2+𝑘2𝛽4

2𝛼5
 

 𝜌0 =
1

2
𝐾𝛽𝛽 

 =
2𝑘𝛼2−2𝑘2𝛽2

𝛼3
 

 𝜌−1 =
1

2𝛼
𝐾𝛼𝛽 

 = −
2𝑘2𝛽3−2𝑘𝛼2𝛽

𝛼5
 

 𝜌−2 =
1

2𝛼2
(𝐾𝛼𝛼 −

1

𝛼
𝐾𝛼) 

 =
6𝑘𝛼2𝛽2−5𝑘2𝛽4−𝛼4

2𝛼7
 

and  

 𝑔𝑖𝑗 = 𝜎𝑎𝑖𝑗 − 𝜎0𝑏𝑖𝑏𝑗 + 𝜎−1(𝑏𝑖𝜔𝑗 + 𝑏𝑗𝜔𝑖) + 𝜎−2𝜔𝑖𝜔𝑗 , (2) 

where  

 𝜎 =
1

𝜌
 

 =
2𝛼5

𝛼5−2𝑘𝛼2𝛽2+𝑘2𝛽4
 

 𝜎0 =
𝜌0

𝜌𝜏
 

 𝜏 = 𝜎 + 𝜎0𝐵2 + 𝜌−1𝛽 

 𝜎−1 =
𝜌−1

𝜌𝜏
 



Communications on Applied Nonlinear Analysis 

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 𝜎−2 =
𝜌−2

𝜌𝜏
, 

where 𝐵2 = 𝑏𝑖𝑏𝑗 and 𝐵 represents the norm of the differential form 𝛽(𝑥, 𝜔) = 𝜔𝑖𝑏
𝑖(𝑥) ∈ 𝑇𝑝

∗𝑀. 

Morover, the Cartan connection, which is a generalized connection often used to describe 

homogeneous spaces or spaces with symmetries, the torsion tensor is defined in a similar way, but it 

incorporates the structure of the Cartan connection itself. A Cartan connection provides a framework 

to study affine connections on a space that is often non-Riemannian. The Cartan connection is 

particularly useful for describing curved spaces with torsion and other structural properties, especially 

in the study of Lie groups, affine manifolds, and non-metric connections. 

The Cartan torsion tensor can be thought of as a measure of the failure of the Cartan connection to 

satisfy the torsion-free condition in a way similar to the standard torsion tensor in Riemannian 

geometry. 

The Cartan torsion tensor 𝐶𝑖𝑗𝑘 [5] in the Cartan space with an (𝛼, 𝛽)-metric is given by  

 𝐶𝑖𝑗𝑘 = −
1

2
[𝑟−1𝑏𝑖𝑏𝑗𝑏𝑘 + {𝜌−1𝑎𝑖𝑗𝑏𝑘 + 𝜌−2𝑎𝑖𝑗𝜔𝑘 + 𝑟−2𝑏𝑖𝑏𝑗𝜔𝑘 + 

     𝑟−3𝑏𝑖𝜔𝑗𝜔𝑘 + 𝑖|𝑗|𝑘} + 𝑟−4𝜔𝑖𝜔𝑗𝜔𝑘], (3) 

where its coefficients 𝑟−1, 𝑟−2, 𝑟−3 and 𝑟−4 are defined and calculated as follows:  

 𝑟−1 =
1

2
𝐾𝛽𝛽𝛽 

 =
−4𝑘2𝛽2

𝛼3
 

 𝑟−2 =
1

2𝛼
𝐾𝛼𝛽𝛽 

 =
6𝑘2𝛽2−2𝑘𝛼2

𝛼5
 

 𝑟−3 =
1

2𝛼2
(𝐾𝛼𝛼𝛽 −

1

𝛼
𝐾𝛼𝛽) 

 =
6𝑘𝛼2𝛽−10𝑘2𝛽3

𝛼7
 

 𝑟−4 =
1

2𝛼3
(𝐾𝛼𝛼𝛼 −

3

𝛼
𝐾𝛼𝛼 +

3

𝛼2
𝐾𝛼) 

 =
35𝑘2𝛽4−30𝑘𝛼2𝛽2+3𝛼4

2𝛼9
. 

We use the symole ’:’ to denote the covariant differentiation with respect to Christoffel symbols 𝛾𝑗𝑘
𝑖  

constructed from 𝑎𝑖𝑗 . Whenever we talk about Christoffel symbols 𝛾𝑗𝑘
𝑖   constructed from 𝑎𝑖𝑗 , we 

mean 𝛾𝑗𝑘
𝑖 =

1

2
𝑎𝑙𝑖 (

𝜕𝑎𝑘𝑙

𝜕𝑥𝑗
+

𝜕𝑎𝑙𝑗

𝜕𝑥𝑘
−

𝜕𝑎𝑗𝑘

𝜕𝑥𝑙
) . Since 𝜔𝑖:𝑘 = 0  and from Ricci’s theorem of tensor calculus  

[14] we have 𝑎:𝑘
𝑖𝑗

= 0 , if 𝑏:𝑘
𝑖 = 0, then 𝑔:𝑘

𝑖𝑗
= 0. Also, let Γ𝑗𝑘

𝑖 (𝑝) =
1

2
𝑔𝑖𝑟(𝜕𝑗𝑔𝑟𝑘 + 𝜕𝑘𝑔𝑗𝑟 − 𝜕𝑟𝑔𝑗𝑘) be 

the Christoffel symbols constructed from fundamental metric tensor 𝑔𝑖𝑗(𝑥, 𝜔) of the Cartan space 

(𝑀, 𝐾(𝑥, 𝜔)). Now, for the Cartan space (𝑀, 𝐾(𝑥, 𝜔)), we state canonical 𝑑-connection is a triplet 



Communications on Applied Nonlinear Analysis 

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given by  

 𝐷Γ = (𝑁𝑗𝑘, 𝐻𝑗𝑘
𝑖 , 𝐶𝑖

𝑗𝑘
), 

where  

 𝑁𝑖𝑗 = Γ𝑖𝑗
𝑘 𝜔𝑘 −

1

2
Γℎ𝑟

𝑘 𝜔𝑘𝜔𝑟�̇�ℎ𝑔𝑖𝑗 (4) 

 𝐻𝑗𝑘
𝑖 =

1

2
𝑔𝑖𝑟(𝜕𝑗𝑔𝑟𝑘 + 𝜕𝑘𝑔𝑗𝑟 − 𝜕𝑟𝑔𝑗𝑘) (5) 

 𝐶𝑖
𝑗𝑘

(𝑥, 𝜔) = −
1

2
𝑔𝑖𝑟(𝑥, 𝜔)

𝜕𝑔𝑗𝑘(𝑥,𝜔)

𝜕𝜔𝑟 = 𝑔𝑖𝑟(𝑥, 𝜔)𝐶𝑟𝑗𝑘(𝑥, 𝜔). (6) 

are respectively called canonical 𝑁-connection, Christoffel symbols and 𝑑-tensor field of type (2,1).  

Let ℎ -covariant derivative with respect to 𝐷Γ  be denoted by the symbol ′|ℎ′ . Then, we have the 

following definition for later use.  

Definition 2.5 [9]  An ℎ-metrical 𝑑-connection on a Cartan space 𝐶 = (𝑀, 𝐾(𝛼(𝑥, 𝜔), 𝛽(𝜔)) with 

(𝛼, 𝛽)-metric is a 𝑑-connection, 𝐷𝛤 on 𝐶, satisfying the following properties:  

(1)  ℎ-deflection tensor 𝐷𝑖𝑗(= 𝜔𝑖|𝑗) = 0 

(2)  𝑎|ℎ
𝑖𝑗

= 0 

(3)  𝑔|ℎ
𝑖𝑗

= 0. 

3 Cartan spaces with an (𝜶, 𝜷)-metric with 𝒉-metrical 𝒅-connection 

In this section we impose the condition of 𝑑 -connection 𝐷Γ  on the Cartan space with an (𝛼, 𝛽) -

metric to be ℎ-metrical and in consequence we analyse what shapes the corresponding Cartan space 

assumes. 

First we take the ℎ-covariant derivative of the given (𝛼, 𝛽)-metric as follows:  

 𝐾(𝑥, 𝜔) = 𝛼(𝑥, 𝜔) + 𝜖𝛽(𝑥, 𝜔) + 2𝑘
𝛽2(𝑥,𝜔)

𝛼(𝑥,𝜔)
−

𝑘2

3

𝛽4(𝑥,𝜔)

𝛼3(𝑥,𝜔)
 

𝑔𝑖𝑗(𝜔𝑖𝜔𝑗|ℎ + 𝜔𝑗𝜔𝑖|ℎ) + 𝜔𝑖𝜔𝑗𝑔|ℎ
𝑖𝑗

= 𝛼|ℎ + 𝜖𝛽|ℎ + 2𝑘 (
2𝛼𝛽𝛽|ℎ − 𝛽2𝛼|ℎ

𝛼2
) −

𝑘3

3
(

4𝛼3𝛽3𝛽|ℎ − 3𝛼2𝛽4𝛼|ℎ

𝛼6
) 

As we have stipulated the 𝑑-connection 𝐷Γ of the Cartan space is ℎ-metrical, therefore by definition 

2.5, we have  

 𝜔𝑗|ℎ = 0, 𝜔𝑖|ℎ = 0, 𝛼|ℎ = 0, 𝑔|ℎ
𝑖𝑗

= 0 

Using these values in above expression, we get  

 𝑔𝑖𝑗(𝜔𝑖 × 0 + 𝜔𝑗 × 0) + 𝜔𝑖𝜔𝑗 × 0 = 0 + 𝜖𝛽|ℎ + 2𝑘 (
2𝛼𝛽𝛽|ℎ−𝛽2×0

𝛼2
) −

𝑘3

3
(

4𝛼3𝛽3𝛽|ℎ−3𝛼2𝛽4×0

𝛼6
) 



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 𝛽|ℎ = 0    (∵ α ≠ 0, β ≠ 0) (7) 

 (𝜔𝑖𝑏
𝑖(𝑥))|ℎ = 0    (∵ β(x, ω) = ωib

i(x)) 

 𝜔𝑖𝑏
𝑖(𝑥)|ℎ + 𝑏𝑖(𝑥)𝜔𝑖|ℎ = 0 

As we have stipulated the 𝑑-connection 𝐷Γ of the Cartan space is ℎ-metrical, therefore by definition 

2.5, we have  

 𝜔𝑖|ℎ = 0, 

Using these values in above expression, we get 

 𝜔𝑖𝑏
𝑖(𝑥)|ℎ + 𝑏𝑖(𝑥) × 0 = 0 

 𝜔𝑖𝑏
𝑖(𝑥)|ℎ = 0 

 𝑏𝑖(𝑥)|ℎ = 0 (8) 

Now, we find ℎ-covariant derivatives of the coefficients of metric tensor 𝑔𝑖𝑗 and then use conditions 

of ℎ-metrical 𝑑-connection 𝐷Γ of Cartan space as follows, we get  

 ∵ 𝜌 =
𝛼5−2𝑘𝛼2𝛽2+𝑘2𝛽4

2𝛼5
 

 ∴ 𝜌|ℎ = 0. (9) 

 ∵ 𝜌0 =
2𝑘𝛼2−2𝑘2𝛽2

𝛼3
 

 ∴ 𝜌0|ℎ = 0. (10) 

 ∵ 𝜌−1 =
−4𝑘2𝛽2

𝛼3
 

 ∴ 𝜌−1|ℎ = 0. (11) 

 ∵ 𝜌−2 =
6𝑘2𝛽2−2𝑘𝛼2

𝛼5
 

 ∴ 𝜌−2|ℎ = 0. (12) 

The ℎ-covariant differentiation of the equation (1) gives  

 𝑔|ℎ
𝑖𝑗

= 𝜌𝑎|ℎ
𝑖𝑗

+ 𝑎𝑖𝑗𝜌|ℎ + 𝜌0(𝑏𝑖𝑏𝑗)|ℎ + 𝑏𝑖𝑏𝑗𝜌0 + 𝜌−1(𝑏𝑖𝜔𝑗 + 𝑏𝑗𝜔𝑖)|ℎ + 

 (𝑏𝑖𝜔𝑗 + 𝑏𝑗𝜔𝑖)𝜌−1|ℎ + 𝜌−2(𝜔𝑖𝜔𝑗)|ℎ + 𝜔𝑖𝜔𝑗𝜌−2|ℎ 

 𝑔|ℎ
𝑖𝑗

= 𝜌𝑎|ℎ
𝑖𝑗

+ 𝑎𝑖𝑗𝜌|ℎ + 𝜌0(𝑏𝑖𝑏|ℎ
𝑗

+ 𝑏𝑗𝑏|ℎ
𝑖 ) + 𝑏𝑖𝑏𝑗𝜌0|ℎ + 𝜌−1(𝑏𝑖𝜔|ℎ

𝑗
+ 𝜔𝑖𝑏|ℎ

𝑖 + 𝑏𝑗𝜔|ℎ
𝑖 +

𝜔𝑖𝑏ℎ

𝑗
)𝜌−1|ℎ(𝑏𝑖𝜔𝑗 + 𝑏𝑗𝜔𝑖) + 𝜌−2|ℎ(𝜔𝑖𝜔|ℎ

𝑗
+ 𝜔𝑗𝜔|ℎ

𝑖 ) + 𝜔𝑖𝜔𝑗𝜌−2|ℎ. 

Using the conditions of ℎ-metrical 𝑑-connection 𝐷Γ of Cartan space and equations (8), (9), (10), (11) 

and (12), above equation reduces to  



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                   𝑔|ℎ
𝑖𝑗

= 0. 

Thus, allowing 𝑑 -connection 𝐷Γ  of the Cartan space to be ℎ -metrical, it gives two important 

quantities namely 𝑎|ℎ
𝑖𝑗

= 0 (by definition of ℎ-metrical 𝑑-connection) and 𝑔|ℎ
𝑖𝑗

= 0, i.e., ℎ-covariant 

derivatives of fundamental metric tensors of associated Riemannian space and Cartan space vanishes. 

Now, since 𝑎|ℎ
𝑖𝑗

= 0 and 𝑔|ℎ
𝑖𝑗

= 0, therefore there corresponding Chritoffel symbols will also be same, 

i.e., 𝐻𝑗ℎ
𝑖 = 𝛾𝑗ℎ

𝑖  and its equivalent condition is given by  

 𝑏:𝑘
𝑖 = 0 (13) 

Now, since 𝐻𝑗ℎ
𝑖 = 𝛾𝑗ℎ

𝑖  therefore the curvature tensor 𝐷ℎ𝑗𝑘
𝑖  of 𝐷Γ coincides with the curvature tensor 

𝑅ℎ𝑗𝑘
𝑖  of Riemannian connection 𝑅Γ = (𝛾𝑗𝑘

𝑖 , 𝛾𝑗𝑘
𝑖 𝑦𝑖 , 0), i.e.,  

 𝐷ℎ𝑗𝑘
𝑖 = 𝑅ℎ𝑗𝑘

𝑖  

If the Riemannian curvature tensor vanishes, i.e., 𝑅ℎ𝑗𝑘
𝑖 = 0, the curvature tensor of 𝑑-connection also 

vanishes, i.e., 𝐷ℎ𝑗𝑘
𝑖 = 0. This discussion can be summarized as follows:  

Proposition 3.1 A Cartan space 𝐶  with the (𝛼, 𝛽) -metric 𝐾(𝑥, 𝜔) = 𝛼(𝑥, 𝜔) + 𝜖𝛽(𝑥, 𝜔) +

2𝑘
𝛽2(𝑥,𝜔)

𝛼(𝑥,𝜔)
−

𝑘2

3

𝛽4(𝑥,𝜔)

𝛼3(𝑥,𝜔)
 admitting a h-metrical d-connection is locally flat if and only if the associated 

Riemannian space is locally flat.  

Now, we find ℎ-covariant derivatives of the coefficients of Cartan torsion tensor 𝐶𝑖𝑗𝑘 and then use 

conditions of ℎ-metrical d-connection 𝐷Γ of Cartan space and equation (7) as follows: 

 ∵ 𝑟−1 =
−4𝑘2𝛽2

𝛼3
 

 ∴ 𝑟−1|ℎ = 0 (14) 

 ∵ 𝑟−2 =
6𝑘2𝛽2−2𝑘𝛼2

𝛼5
 

 ∴ 𝑟−2|ℎ = 0 (15) 

 ∵ 𝑟−3 =
6𝑘𝛼2𝛽−10𝑘2𝛽3

𝛼7
 

 ∴ 𝑟−3|ℎ = 0 (16) 

 ∵ 𝑟−4 =
35𝑘2𝛽4−30𝑘𝛼2𝛽2+3𝛼4

2𝛼9
 

 ∴ 𝑟−4|ℎ = 0 (17) 

Now we calculate the value of ℎ-covariant derivative of 𝑑-tensor field 𝐶𝑖
𝑗𝑘

 of type (2,1) under the 

assumption of ℎ-metrical 𝑑-connection as follows:  

 ∵ 𝐶𝑘
𝑖𝑗

= 𝑔𝑘𝑟𝐶𝑟𝑖𝑗 



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 ∴ 𝐶𝑘|ℎ
𝑖𝑗

= (𝑔𝑘𝑟𝐶𝑟𝑖𝑗)
|ℎ

 

 = 𝑔𝑘𝑟 × 𝐶|ℎ
𝑟𝑖𝑗

+ 𝐶𝑟𝑖𝑗 × 0𝑔𝑘𝑟|ℎ 

 = 𝑔𝑘𝑟𝐶|ℎ
𝑟𝑖𝑗

 

 = −𝑔𝑘𝑟
1

2
[𝑟−1𝑏𝑟𝑏𝑖𝑏𝑗 + 𝑟−2𝑏𝑟𝑏𝑖𝜔𝑗 + 𝑟−3𝑏𝑟𝜔𝑖𝜔𝑗 + 𝑟−4𝜔𝑟𝜔𝑖𝜔𝑗 + 𝜌−1𝑎𝑟𝑖𝑏𝑗 + 

 𝜌−2𝑎𝑟𝑖𝜔𝑗 + 𝑟|𝑖|𝑗]|ℎ 

 = −𝑔𝑘𝑟
1

2
[𝑟−1 × (𝑏𝑟𝑏𝑖𝑏𝑗)|ℎ + 𝑏𝑟𝑏𝑖𝑏𝑗 × 0𝑟−1|ℎ + 𝑟−2 × (𝑏𝑟𝑏𝑖𝜔𝑗)|ℎ + 𝑏𝑟𝑏𝑖𝜔𝑗 × 

 0𝑟−2|ℎ + 𝑟−3 × (𝑏𝑟𝜔𝑖𝜔𝑗)|ℎ + 𝑏𝑟𝜔𝑖𝜔𝑗 × 0𝑟−3|ℎ + 𝑟−4 × (𝜔𝑟𝜔𝑖𝜔𝑗)|ℎ + 𝜔𝑟𝜔𝑖𝜔𝑗 × 

 0𝑟−4|ℎ + 𝜌−1 × (𝑎𝑟𝑖𝑏𝑗)|ℎ + 𝑎𝑟𝑖𝑏𝑗 × 0𝜌−1|ℎ + 𝜌−2 × (𝑎𝑟𝑖𝜔𝑗)|ℎ + 𝑎𝑟𝑖𝜔𝑗 × 

 0𝜌−2|ℎ + (𝑟|𝑖|𝑗)|ℎ] 

 = −𝑔𝑘𝑟
1

2
[𝑟−1(𝑏𝑟𝑏𝑖𝑏𝑗)|ℎ + 𝑟−2(𝑏𝑟𝑏𝑖𝜔𝑗)|ℎ + 𝑟−3(𝑏𝑟𝜔𝑖𝜔𝑗)|ℎ + 𝑟−4(𝜔𝑟𝜔𝑖𝜔𝑗)|ℎ + 

 𝜌−1(𝑎𝑟𝑖𝑏𝑗)|ℎ + 𝜌−2(𝑎𝑟𝑖𝜔𝑗)|ℎ + (𝑟|𝑖|𝑗)|ℎ] 

 = −𝑔𝑘𝑟
1

2
[𝑟−1(𝑏𝑟𝑏𝑖0𝑏|ℎ

𝑗
+ 𝑏𝑟𝑏𝑗0𝑏|ℎ

𝑖 + 𝑏𝑖𝑏𝑗0𝑏|ℎ
𝑟 ) + 𝑟−2(𝑏𝑟𝑏𝑖0𝜔|ℎ

𝑗
+ 𝑏𝑟𝜔𝑗0𝑏|ℎ

𝑖 +

𝑏𝑖𝜔𝑗0𝑏|ℎ
𝑟 ) + 

 𝑟−3(𝑏𝑟𝜔𝑖0𝜔|ℎ
𝑗

+ 𝑏𝑟𝜔𝑗0𝜔|ℎ
𝑖 + 𝜔𝑖𝜔𝑗0𝑏|ℎ

𝑟 ) + 𝑟−4(𝜔𝑟𝜔𝑖0𝜔|ℎ
𝑗

+ 𝜔𝑟𝜔𝑗0𝜔|ℎ
𝑖 +

𝜔𝑖𝜔𝑗0𝜔|ℎ
𝑟 ) + 

 𝜌−1(𝑎𝑟𝑖0𝑏|ℎ
𝑗

+ 𝑏𝑗0𝑎|ℎ
𝑟𝑖) + 𝜌−2(𝑎𝑟𝑖0𝜔|ℎ

𝑗
+ 𝜔𝑗0𝑎|ℎ

𝑟𝑖) + 0(𝑟|𝑖|𝑗)|ℎ] 

 𝐶𝑘|ℎ
𝑖𝑗

= 0 (18) 

Thus we shown that Cartan torsion tensor vanishes under the assumption of ℎ-metrical 𝑑-connection.  

One knows that a Cartan space 𝐶 is Berwald space if and only if 𝐶𝑘|ℎ
𝑖𝑗

= 0 [12]. Hence from equation 

(18), we have the following proposition:  

Proposition 3.2  A Cartan space 𝐶  with the (𝛼, 𝛽) -metric 𝐾(𝑥, 𝜔) = 𝛼(𝑥, 𝜔) + 𝜖𝛽(𝑥, 𝜔) +

2𝑘
𝛽2(𝑥,𝜔)

𝛼(𝑥,𝜔)
−

𝑘2

3

𝛽4(𝑥,𝜔)

𝛼3(𝑥,𝜔)
 admitting h-metrical d-connection is a Berwald space  

In [12], it is deduced that a locally Minkowski space is a Berwald space in which curvature tensor 

vanishes. Hence, from the Propositions 3.1 and 3.2, we have following theorem: 

Theorem 3.3  A Cartan space 𝐶  with the (𝛼, 𝛽) -metric 𝐾(𝑥, 𝜔) = 𝛼(𝑥, 𝜔) + 𝜖𝛽(𝑥, 𝜔) +

2𝑘
𝛽2(𝑥,𝜔)

𝛼(𝑥,𝜔)
−

𝑘2

3

𝛽4(𝑥,𝜔)

𝛼3(𝑥,𝜔)
 admitting ℎ-metrical 𝑑-connection is locally Minkowski space if and only if the 

associated Riemannian space is locally flat.  

 



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4  Conformal change of Cartan space with an (𝜶, 𝜷)-metric 

In this section our aim is to conformally transform a Cartan space (𝑀, 𝐾(𝑥, 𝜔)) to another Cartan 

space (𝑀, �̃�(𝑥, 𝜔))  and then to determine the nature of curvature tensor �̃�ℎ𝑗𝑘
𝑖   in the conformally 

transformed space (𝑀, �̃�(𝑥, 𝜔))  under the influence of ℎ -metrical 𝑑 -connection on the original 

Cartan space (𝑀, 𝐾(𝑥, 𝜔)). That is, we are going to determine the shape of conformally transformed 

space (𝑀, �̃�(𝑥, 𝜔)) under the stipulation of ℎ-metrical 𝑑-connection on (𝑀, 𝐾(𝑥, 𝜔)). 

For that, consider an 𝑛-dimensional Cartan space 𝐶 = (𝑀, 𝐾(𝑥, 𝜔)) equipped with a real smooth 𝑛-

manifold 𝑀  and the (𝛼, 𝛽) -metric 𝐾(𝑥, 𝜔) = 𝛼(𝑥, 𝜔) + 𝜖𝛽(𝑥, 𝜔) + 2𝑘
𝛽2(𝑥,𝜔)

𝛼(𝑥,𝜔)
−

𝑘2

3

𝛽4(𝑥,𝜔)

𝛼3(𝑥,𝜔)
 , where 

𝛼 = (𝑎𝑖𝑗(𝑥, 𝜔)𝜔𝑖𝜔𝑗)
1

2  and 𝛽 = 𝜔𝑖𝑏
𝑖(𝑥) . By a conformal change 𝜎: 𝐾 → �̃�  such that �̃�(�̃�, 𝛽) =

𝑒𝜎𝐾(𝛼, 𝛽), we have the another Cartan space �̃�𝑛 = (𝑀, �̃�(�̃�, 𝛽)), where �̃� = 𝑒𝜎𝛼 and 𝛽 = 𝑒𝜎𝛽. 

Putting 𝛼 = (𝑎𝑖𝑗(𝑥, 𝜔)𝜔𝑖𝜔𝑗)
1

2 and 𝛽 = 𝜔𝑖𝑏
𝑖(𝑥) in the above relations, we get  

 �̃� = 𝑒𝜎𝛼 

 �̃� = 𝑒𝜎(𝑎𝑖𝑗(𝑥, 𝜔)𝜔𝑖𝜔𝑗)
1

2 

 �̃� = (𝑒2𝜎𝑎𝑖𝑗(𝑥, 𝜔)𝜔𝑖𝜔𝑗)
1

2 

 �̃� = (�̃�𝑖𝑗𝜔𝑖𝜔𝑗)
1

2 

 �̃�𝑖𝑗 = 𝑒2𝜎𝑎𝑖𝑗(𝑥, 𝜔) 

and  

 𝛽 = 𝑒𝜎𝛽 

 𝛽 = 𝑒𝜎𝜔𝑖𝑏
𝑖(𝑥) 

 𝛽 = 𝜔𝑖𝑒
𝜎𝑏𝑖(𝑥) 

 𝛽 = 𝜔𝑖𝑏
𝑖 

 �̃�𝑖 = 𝑒𝜎𝑏𝑖(𝑥) 

Now we calculate the Christoffel symbols �̃�𝑟𝑘
𝑝

 in conformally transformed space (𝑀, �̃�(𝑥, 𝜔)) 

as follows: 

We know from Riemannian geometry Christoffel symbols of second kind 𝛾𝑟𝑘
𝑝

  from fundamental 

metric tensor 𝑎𝑝𝑞(𝑥, 𝜔) can be defined as  

 𝛾𝑞𝑘
𝑝 =

1

2
𝑎𝑙𝑝 (

𝜕𝑎𝑘𝑙

𝜕𝑥𝑞 +
𝜕𝑎𝑙𝑞

𝜕𝑥𝑘 −
𝜕𝑎𝑞𝑘

𝜕𝑥𝑙 ) 

Similarly, we can also define the Christoffel symbols �̃�𝑟𝑘
𝑝

  in conformally transformed space 

(𝑀, �̃�(𝑥, 𝜔)) as  

 �̃�𝑞𝑘
𝑝 =

1

2
�̃�𝑙𝑝 (

𝜕�̃�𝑘𝑙

𝜕𝑥𝑞 +
𝜕�̃�𝑙𝑞

𝜕𝑥𝑘 −
𝜕�̃�𝑞𝑘

𝜕𝑥𝑙 ) 



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 =
1

2
𝑒2𝜎𝑎𝑙𝑝(𝑥, 𝜔) (

𝜕𝑒2𝜎𝑎𝑘𝑙(𝑥,𝜔)

𝜕𝑥𝑞 +
𝜕𝑒2𝜎𝑎𝑙𝑞(𝑥,𝜔)

𝜕𝑥𝑘 −
𝜕𝑒2𝜎𝑎𝑞𝑘(𝑥,𝜔)

𝜕𝑥𝑙
) 

 =
1

2
𝑒2𝜎𝑎𝑙𝑝 [(𝑒2𝜎 𝜕𝑎𝑘𝑙

𝜕𝑥𝑞 + 𝑎𝑘𝑙
𝜕𝑒2𝜎

𝜕𝑥𝑞 ) + (𝑒2𝜎 𝜕𝑎𝑙𝑞

𝜕𝑥𝑘 + 𝑎𝑙𝑞
𝜕𝑒2𝜎

𝜕𝑥𝑘 ) − (𝑒2𝜎 𝜕𝑎𝑗𝑞

𝜕𝑥𝑙 + 𝑎𝑞𝑘
𝜕𝑒2𝜎

𝜕𝑥𝑙 )] 

 =
1

2
𝑒2𝜎𝑎𝑙𝑝 [(𝑒2𝜎 𝜕𝑎𝑘𝑙

𝜕𝑥𝑞 + 2𝑒2𝜎𝑎𝑘𝑙
𝜕𝜎

𝜕𝑥𝑞) + (𝑒2𝜎 𝜕𝑎𝑙𝑞

𝜕𝑥𝑘 + 2𝑒2𝜎𝑎𝑙𝑞
𝜕𝜎

𝜕𝑥𝑘) − 

     (𝑒2𝜎 𝜕𝑎𝑞𝑘

𝜕𝑥𝑙 + 2𝑒2𝜎𝑎𝑞𝑘
𝜕𝜎

𝜕𝑥𝑙)] 

 =
1

2
𝑒4𝜎𝑎𝑙𝑝 [(

𝜕𝑎𝑘𝑙

𝜕𝑥𝑞 +
𝜕𝑎𝑙𝑞

𝜕𝑥𝑘 −
𝜕𝑎𝑞𝑘

𝜕𝑥𝑙 ) + (2𝑎𝑘𝑙
𝜕𝜎

𝜕𝑥𝑞 + 2𝑎𝑙𝑞
𝜕𝜎

𝜕𝑥𝑘 − 2𝑎𝑞𝑘
𝜕𝜎

𝜕𝑥𝑙)] 

 = 𝑒4𝜎 [
1

2
𝑎𝑙𝑝 (

𝜕𝑎𝑘𝑙

𝜕𝑥𝑞 +
𝜕𝑎𝑙𝑞

𝜕𝑥𝑘 −
𝜕𝑎𝑞𝑘

𝜕𝑥𝑙 ) + (𝑎𝑙𝑝𝑎𝑘𝑙𝜎𝑞 + 𝑎𝑙𝑝𝑎𝑙𝑞𝜎𝑘 − 𝑎𝑙𝑝𝑎𝑞𝑘𝜎𝑙)] 

 = 𝑒4𝜎[𝛾𝑞𝑘
𝑝 + (𝛿𝑘

𝑝𝜎𝑞 + 𝛿𝑞
𝑝𝜎𝑘 − 𝑎𝑞𝑘𝜎𝑖)] 

Hence, the components of Christoffel symbols �̃�𝑞𝑘
𝑝

, constructed from �̃�𝑝𝑞, in conformally transformed 

space are given by  

 �̃�𝑞𝑘
𝑝 = 𝛾𝑞𝑘

𝑝 + 𝐵𝑞𝑘
𝑝 , (19) 

where 𝐵𝑞𝑘
𝑝 = 𝜎𝑘𝛿𝑞

𝑝 + 𝜎𝑞𝛿𝑘
𝑝 − 𝑎𝑘𝑞𝜎𝑝, 𝜎𝑝 = 𝜎𝑞𝑎𝑝𝑞 . 

The covariant derivative of �̃�𝑝 with respect to �̃�𝑟𝑘
𝑝

, yields  

 �̃�:𝑘
𝑝 = 𝑒𝜎(𝑏:𝑘

𝑝 + 2𝜎𝑘𝑏𝑝 + 𝑏𝑟𝜎𝑟𝛿𝑘
𝑝 − 𝜎𝑝𝑏𝑟𝑎𝑟𝑘). (20) 

Transvecting the equation (20) by �̃�𝑘, and putting  

 𝑀𝑝 =
1

𝐵2
(𝑏𝑘𝑏:𝑘

𝑝 −
𝑏:𝑟

𝑟 𝑏𝑝

𝑛+4
), (21) 

we have 𝜎𝑝 = �̃�𝑝 − 𝑀𝑝, from which we get 𝜎𝑝 = �̃�𝑝 − 𝑀𝑝. Substituting the values of 𝜎𝑝 and 𝜎𝑝 

in equation  (19) and using 𝐷ℎ𝑞
𝑝 = 𝛾ℎ𝑞

𝑝 + 𝛿ℎ
𝑝𝑀𝑞 + 𝛿ℎ

𝑝𝑀𝑞 + 𝛿𝑞
𝑝𝑀ℎ − 𝑀𝑝𝑎ℎ𝑞 , we find  

 �̃�ℎ𝑞
𝑝 = 𝐷ℎ𝑞

𝑝 . (22) 

Here 𝐷ℎ𝑞
𝑝

 is a symmetric and conformally invariant linear connection on 𝑀. 

The whole discussion can be summarized in the following proposition.  

Proposition 4.1  Let 𝐶 = (𝑀, 𝐾(𝑥, 𝜔)) be a Cartan space the (𝛼, 𝛽)-metric 𝐾(𝑥, 𝜔) = 𝛼(𝑥, 𝜔) +

𝜖𝛽(𝑥, 𝜔) + 2𝑘
𝛽2(𝑥,𝜔)

𝛼(𝑥,𝜔)
−

𝑘2

3

𝛽4(𝑥,𝜔)

𝛼3(𝑥,𝜔)
 . hhen, there eiists a conformally invariant symmetric linear 

connection 𝐷𝑞𝑘
𝑝

 on 𝑀.  

Next, if we denote the curvature tensor of 𝐷𝑞𝑘
𝑝

 by 𝐷ℎ𝑞𝑘
𝑝

, then from the equation (22), we get  

 �̃�ℎ𝑞𝑘
𝑝 = 𝐷ℎ𝑞𝑘

𝑝 . (23) 

Since 𝑏:𝑘
𝑝 = 0, from equation (21), we get 𝑀𝑖 = 0. Hence, we deduce that 𝐷𝑞𝑘

𝑝 = 𝛾𝑞𝑘
𝑝

 and 𝐷ℎ𝑞𝑘
𝑝 =



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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𝑅ℎ𝑞𝑘
𝑝

. 

Thus we have the following proposition.  

Proposition 4.2  Let 𝐶 = (𝑀, 𝐾)  be a Cartan space the (𝛼, 𝛽) -metric 𝐾(𝑥, 𝜔) = 𝛼(𝑥, 𝜔) +

𝜖𝛽(𝑥, 𝜔) + 2𝑘
𝛽2(𝑥,𝜔)

𝛼(𝑥,𝜔)
−

𝑘2

3

𝛽4(𝑥,𝜔)

𝛼3(𝑥,𝜔)
  admitting ℎ -metrical 𝑑 -connection. hhen, there eiists a 

conformally invariant symmetric linear connection 𝐷𝑞ℎ
𝑝

  such that 𝐷𝑞𝑘
𝑝 = 𝛾𝑞𝑘

𝑝
  and it’s curvature 

tensor 𝐷ℎ𝑞𝑘
𝑝 = 𝑅ℎ𝑞𝑘

𝑝
.  

Next, if the associated Riemannian space (𝑀, 𝛼)  is locally flat, that is, 𝑅ℎ𝑞𝑘
𝑝 = 0 , then from 

Proposition 4.2 and equation (23), we deduce that �̃�ℎ𝑞𝑘
𝑝

= 0, that is, the space 𝐶 is conformally flat. 

Thus we have the following theorem.  

Theorem 4.3 Let 𝐶 = (𝑀, 𝐾) be a Cartan space the (𝛼, 𝛽)-metric 𝐾(𝑥, 𝜔) = 𝛼(𝑥, 𝜔) + 𝜖𝛽(𝑥, 𝜔) +

2𝑘
𝛽2(𝑥,𝜔)

𝛼(𝑥,𝜔)
−

𝑘2

3

𝛽4(𝑥,𝜔)

𝛼3(𝑥,𝜔)
 admitting ℎ-metrical 𝑑-connection. hhen the space 𝐶 is conformally flat if and 

only if the associated Riemannian space is locally flat.  

Conclusion: The conditions derived clarify how the (α,β)-metric 𝐾(𝑥, 𝜔) = 𝛼(𝑥, 𝜔) + 𝜖𝛽(𝑥, 𝜔) +

2𝑘
𝛽2(𝑥,𝜔)

𝛼(𝑥,𝜔)
−

𝑘2

3

𝛽4(𝑥,𝜔)

𝛼3(𝑥,𝜔)
 influences the geometric structure of the Cartan space, particularly regarding its 

transformation into a locally Minkowski space. Moreover, it identifies the criteria under which the 

space becomes conformally flat, suggesting that the curvature can be transformed to zero by a 

conformal transformation under h-metrical d-connection.  These resuluts may have implications for 

theoretical physics, particularly in contexts like general relativity, where understanding the structure 

of spacetime is crucial. 

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