$(\alpha,\beta)$-metrics satisfying the $T$-Condition or the $\sigma T$-Condition

We describe the $(\alpha,\beta)$-metrics whose the $T$-tensor vanishes ($T$-condition) and the $(\alpha,\beta)$-metrics that satisfy the $\sigma T$-condition $\sigma_hT^h_{ijk}=0$, where $\sigma_h=\frac{\partial \sigma}{\partial x^h}$ and $\sigma$ is a smooth function on $M$. These classes have already been obtained by Z. Shen and G. S. Asanov in a completely different approach. The Finsler metrics of the first class are Berwaldian, the metrics of the second class are almost regular non-Berwaldian Landsberg metrics.


Introduction
The T -tensor plays an interesting role in Finsler geometry and general relativity. It was introduced by M. Matsumoto [9]. M. Hashiguchi [6] showed that a Landsberg space remains a Landsberg space under all conformal changes of the Finsler function if and only if its T -tensor vanishes. By a famous observation of Z. I. Szabó [12], a positive definite Finsler manifold with vanishing T -tensor is Riemannian. For further information, we refer to the papers [11,8,9]. Moreover, for the physical point of view, we refer, for example, to [1,2,3] Let (M, F ) be a Finsler manifold. We recall that a conformal change F F of F by a smooth function σ on M is given by A Landsberg manifold remains of the same type under a conformal change (1.1) if and only if the T -tensor satisfies the condition σ r T r jkℓ = 0, σ r := ∂σ ∂x r . Obviously, if this holds for every σ ∈ C ∞ (M ), then T = 0 and (M, F ) is Riemannian by Szabó's observation. So it will be more beneficial to consider the case when a Landsberg space remains Landsberg under some conformal transformation. In [5], it was studied in the case when the condition σ r T r jkh = 0 is satisfied for some conformal change by σ on M .
In this paper, we study the T -tensor of the (α, β)-metrics. An (α, β)-metric F is of the form F = αφ(s), s := β α . We start by studying the Cartan tensor C ijk of (α, β)-metrics. We show that the Cartan tensor C ijk vanishes identically and hence the space is Riemannian if and only if φ(s) = √ k 1 + k 2 s 2 , where k 1 and k 2 are constants. We calculate the T -tensor for the (α, β)-metrics, and we find necessary and sufficient conditions for (α, β)-metrics to satisfy the T -condition. By solving some ODEs, we show that an (α, β)-metric satisfies the T -condition if and only if it is Riemannian or φ(s) has the following form We introduce the notion of σT -condition. We say that a Finsler space satisfies this condition if it admits smooth function σ(x) such that σ h T h ijk = 0, where σ h = ∂σ ∂x h . We find necessary and sufficient conditions for an (α, β)-metric to satisfy the σT -condition. Moreover, we show that the (α, β)-metrics satisfy the σT -condition if and only if the T -tensor vanishes (this is the trivial case) or φ(s) is given by It is worthy to mention that the above special (α, β)-metrics have already been obtained by Z. Shen [10]. Namely, the formulas of φ(s) that characterized the T -condition produce positively almost regular Berwald metrics. One can predict that the metric is not regular in this case because the T -tensor vanishes (by Szabó's observation). In his paper, Shen showed this almost regular property. The non-trivial formula that characterized the σT -condition (with some restrictions) provides the class of (almost regular) Landsberg metrics which are not Berwaldian.
In [5], it was claimed that the long existing problem of regular Landsberg non-Berwaldian spaces is (closely) related to the question: Is there any Finsler space admitting a smooth function σ such that σ r T r ijk = 0, σ r = ∂σ ∂x r ?
In this paper we confirm this claim in the almost regular case, since the class of (α, β)-metrics that satisfy the σT -condition is the same as the class of non-Berwaldian Landsberg metrics obtained by Z. Shen in his quoted paper [10].
2. The Cartan tensor and T -tensor of (α, β)-metrics Let M be an n-dimensional smooth manifold. The tangent space to M at p is denoted by T p M ; We fix a chart (U, (u 1 , ..., u n )) on M . It induces a local coordinate system (x 1 , ..., x n , y 1 , ..., y n ) on T M , where By abuse of notation, we shall denote the coordinate functions u i also by x i .
Let α be a Riemannian metric, β a 1-form on M . Locally, The Riemannian metric α induces naturally a Finsler function F α on T M given by ). Similarly, the 1-form β can be interpreted as a smooth function Locally, In what follows, as usual, we shall simply write α and β instead of F α and β, respectively. For any p ∈ M , we define is a (positive definite) Finsler function if and only if φ satisfies the following conditions: where t and x are arbitrary real numbers with |t| < x < b 0 . (For a proof, see Shen [10], Lemma 2.1) In this case we say that F is a regular (α, β)-metric. If β p α ≤ b 0 for all p ∈ M , then F = α(φ• β α ) is called almost regular (under condition (2.1)). An almost regular (α, β)-metric F = α(φ • β α ) is positively almost regular if φ is defined only on (0, b 0 ).
Proposition 2.1. For an (α, β)-metric F = αφ(s), the inverse (g ij ) of the matrix (g ij ) is given by and Remark 2.2. It should be noted that the choice φ(s) = c 1 s + c 2 √ b 2 − s 2 , c 1 and c 2 are constants is excluded. Indeed, the function ρ + φφ ′′ m 2 appearing in the denominators of µ 0 , µ 1 and µ 2 can be written as follows which contradicts to condition (2.1). To avoid not only this contradiction, but also the dividing by zero (in µ 0 , µ 1 and µ 2 ), we must exclude the choice of φ for which ρ + φφ ′′ m 2 = 0. Since φ cannot be zero, we have where c 1 and c 2 are constants.
It should be noted that, in the literature, the metric F = αφ(s), φ(s) = k 1 s + k 2 √ 1 + k 3 s 2 , k 1 > 0 is a Finsler metric of Randers-type. But with certain choice of the constant k 3 , we can get the case where the metric tensor is singular ( det(g ij ) = 0). For example, Example 1. Let M = R n , α = |y| and β = εy 1 , ε is a constant. Then, we have 3), is singular in the sense that its metric tensor has vanishing determinant.
Proof. Differentiating (2.2) with respect to y k and taking into account that ∂s ∂y k = m k α , we have Remark 2.4. The covariant vector m i satisfies the properties where ζ(x, y) and η(x, y) are smooth functions on T M . Then ζ and η must vanish. And the contraction by g ij gives (2.5) (n + 1)ζ + ηm 2 = 0. Now, taking the fact that n ≥ 3, subtracting (2.4) and (2.5) we get ζ = 0 and η = 0.

and the following combination is satisfied
Proof. Assume that Contracting the above equation by b i b j and using Remark 2.4, we obtain where we use the notations A β := A i b i and B β := B i b i . Using the facts that y i A i = 0, y i B i = 0, the contraction by a ij gives Again, contracting the equations (2.6) and 2.7 by b k gives rise to Multiplying (2.8) by 3 and subtracting it from (2.9), then using the fact that n > 2, we get that A β = 0, B β = 0. By substitution into (2.6) and (2.7) and repeating the last process we obtain that A k = 0 and B k = 0.
By the help of Lemma 2.5, one can easily prove the following theorem.
Theorem 2.7. For the (α, β)-metrics with n ≥ 3, the following assertions are equivalent: For a Finsler manifold (M, F ), the T -tensor is defined by [7] (2.10) The T -tensor is totally symmetric in all of its indices.
Theorem 2.8. The T -tensor of an (α, β)-metric takes the form: , Proof. By using Lemma 2.3 and making use of the fact that∂ i s = mi α , we havė where n ij := α i m j + α j m i . By making use of the fact that K 1 and K 2 satisfy we have Now, taking the fact that F = αφ into account, the T -tensor of the space (M, F ) is given by For an (α, β)-metric, one can calculate Φ, Ψ and Ω to obtain the formula for its T -tensor. Or one can, easily, use Maple program for these calculations, for example we have the following corollary.
The T -tensor of Randers metric, (F = α(1 + s), φ(s) = 1 + s), is given by It is to be noted that the T -tensor of Kropina metric is also obtained by Shibata [11] and [13]. The T -tensor of Randers metric has been studied by Matsumoto [8].

Proposition 3.2.
The T -tensor T h ijk := g hr T rijk is given by Proof. The proof is a straightforward calculations by using Proposition 2.1. Proof. By using Proposition 3.2, we have where σ 0 := σ i y i and σ β : The above equation can be written in the following form Putting By using the above quantities, σ h T h ijk can be written as follows Contracting the above two equations by b i and then by σ i := a ij σ j , respectively, we have where τ β := τ i b i , m σ := m i σ i and τ σ := τ i σ i . Now, we claim that both sides of the four equalities in (3.4) must vanish. We prove this claim via contradiction, so we assume that, for example, the sides of the third equality are non zero, hence by dividing the first equality on the third one, we can get where σ 2 := σ i σ i . Now, simplifying the above equation we get the following Making use of the facts that ∂σ0 ∂y i = σ i and the functions σ 2 , σ β , b 2 are functions on M , that is, they are functions of (x i ) only, then differentiating the above equation with respect to y i , we have By using the properties α = a ij y i y j and ∂(ααi) ∂y j = a ij , differentiating the above equation with respect to y j and then by y k , we get σ i a jk + σ j a ki + σ k a ij = 0.
Conversely, if the conditions (a), (b) and (c) are satisfied, then the result is obviously obtained.
is equivalent to σ j = e a(x) b j . Indeed, where a(x) is an arbitrary, locally defined function on M .

Some ODEs
In this section, we focus our study on the T -condition and σT -condition. By solving some ODEs, we find explicit formulas for (α, β)-metrics that satisfy the T -condition and σT -condition.
We define a function Q(s) as follows The function Q(s) simplifies and helps to solve the ODEs that will be treated in this section. Moreover, φ is given by Theorem 4.1. An (α, β)-metric with n ≥ 3 satisfies the T -condition if and only if it is Riemannian or φ is given by Proof. By Theorem 3.1, any (α, β)-metric satisfies the T -condition if and only if Φ = 0. So, taking the fact that φ − sφ = 0 into account, the ODE s + αK 1 m 2 = 0 can be rewritten as follows This is a first order linear differential equation and has the solution Hence, .
Proof. First we should write Φ and Ψ in terms of Q(s) and its derivations with respect to s, as follows Now, making use of the condition (2.1), Remark 2.2 and the fact that φ − sφ ′ = 0, the condition Φ + m 2 Ψ = 0 gives the following two possible ODEs The ODE (4.5) can be given in the form which gives the trivial case, that is, the T -tensor vanishes. The ODE (4.4) has the solution By using (4.1), φ(s) is given by (4.3).

Examples and Concluding remarks
We start by giving two classes of examples satisfying the σT -condition.
Example 2. Let M = R n and α, β be given by where |y| is the Euclidean norm and f (x 1 ) is arbitrary function on M . Then, the class satisfies the σT -condition, where p and q are arbitrary constants. Indeed, in this class, one can see that the function φ(s) is given by Using the formula of φ(s), it is much simpler to use the Maple program to show that Φ + m 2 Ψ = 0, m 2 Ω + 3Ψ = 0.
Since β = y 2 , then we have b 2 = 1, b 1 = b 3 = 0. As in the previous example, we can have σ(x) = θ(x 2 ) for some functions θ(x 2 ) on M . Or instead, using the Finsler package and Maple program, we obtain that T 2 ijk = 0, for all i, j, k = 1, 2, 3. And since, σ = θ(x 2 ), then σ 2 = ∂θ ∂x 2 and hence Finally, we have the following remarks: • Consider the conformal transformation of a Finsler function F , that is, F = κ(x)F, where κ(x) is positive smooth function on M . Then, by simple and straightforward calculations, one can obtain that the T -tensor is transformed by the formula T h ijk = κ(x)T h ijk . In Example 2, one can see that the conformal transformation of F by any positive smooth function κ(x 1 ) still satisfying the σT -condition, that is, F = κ(x 1 )F satisfies the σT -condition. Also, in Example 3, the Finsler function F = κ(x 2 )F satisfies the σT -condition.
It should be noted that this irregularity is studied by Z. Shen [10]. Here we confirm that this metric is not regular Finsler metric because it has vanishing T -tensor. This because of Z. Szabó's result, that is, positive definite Finsler metric with vanishing T -tensor is Riemannian.
• If b(x) = b 0 , then (4.3) can be rewritten an follows We notice that the above formulae for φ is the same as the one obtained in [10] ( (1.3) in Theorem 1.2). Under some restrictions on β, this represents a class of Landsberg non-Berwaldian Finsler spaces. Also, with a special choice of the constants, b 0 = 1 and c 1 = 0, we obtain which is obtained by Asanov [2].
• Summarizing above, the classes (4.2) and (4.3) are almost regular (α, β)-metrics. Moreover, the class (4.3) of (α, β)-metrics that satisfies the σT -condition, when b(x) = b 0 for some constant b 0 , is the same as the class which is obtained by Z. Shen in [10, Theorem 1.2]. This confirms our previous claim in [5] that the long existing problem of regular Landsberg non-Berwaldian spaces is (closely) related to the question: Is there any Finsler space admitting functions σ r (x) such that σ r T r ijk = 0?