Abstract
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 Shen and 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.
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1 Introduction
The T-tensor plays an interesting role in Finsler geometry and general relativity. It was introduced by Matsumoto [9]. 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 Szabó [12], a positive definite Finsler manifold with vanishing T-tensor is Riemannian. For further information, we refer to the papers [8, 9, 11]. 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 \(F\rightsquigarrow \overline{F}\) of F by a smooth function \(\sigma \) 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
Obviously, if this holds for every \(\sigma \in C^\infty (M)\), 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 \(\sigma _r T^r_{jkh}=0\) is satisfied for some conformal change by \(\sigma \) on M.
In this paper, we study the T-tensor of the \((\alpha ,\beta )\)-metrics. An \((\alpha ,\beta )\)-metric F is of the form \(F=\alpha \phi (s)\), \(s:=\frac{\beta }{\alpha }\). We start by studying the Cartan tensor \(C_{ijk}\) of \((\alpha ,\beta )\)-metrics. We show that the Cartan tensor \(C_{ijk}\) vanishes identically and hence the space is Riemannian if and only if \(\phi (s)= \sqrt{k_1+k_2s^2}\), where \(k_1\) and \(k_2\) are constants.
We calculate the T-tensor for the \((\alpha ,\beta )\)-metrics, and we find necessary and sufficient conditions for \((\alpha ,\beta )\)-metrics to satisfy the T-condition. By solving some ODEs, we show that an \((\alpha ,\beta )\)-metric satisfies the T-condition if and only if it is Riemannian or \(\phi (s)\) has the following form
We introduce the notion of \(\sigma T\)-condition. We say that a Finsler space satisfies this condition if it admits smooth function \(\sigma (x)\) such that \(\sigma _hT^h_{ijk}=0\), where \(\sigma _h=\frac{\partial \sigma }{\partial x^h}\). We find necessary and sufficient conditions for an \((\alpha ,\beta )\)-metric to satisfy the \(\sigma T\)-condition. Moreover, we show that the \((\alpha ,\beta )\)-metrics satisfy the \(\sigma T\)-condition if and only if the T-tensor vanishes (this is the trivial case) or \(\phi (s)\) is given by
It is worthy to mention that the above special \((\alpha ,\beta )\)-metrics have already been obtained by Shen [10]. Namely, the formulas of \(\phi (s)\) 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 \(\sigma T\)-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 \(\sigma \) such that \(\sigma _rT^r_{ijk}=0\), \(\sigma _r=\frac{\partial \sigma }{\partial x^r}\)?
In this paper we confirm this claim in the almost regular case, since the class of \((\alpha ,\beta )\)-metrics that satisfy the \(\sigma T\)-condition is the same as the class of non-Berwaldian Landsberg metrics obtained by Shen in his quoted paper [10].
2 The Cartan Tensor and T-Tensor of \((\alpha ,\beta )\)-Metrics
Let M be an n-dimensional smooth manifold. The tangent space to M at p is denoted by \(T_pM\); \(TM:={\mathop {\bigcup }\limits _{p\in M}}\,\,T_{p}M\) is the tangent bundle of M, \(\tau :TM\longrightarrow M\) is the tangent bundle projection. We fix a chart \((\mathcal {U}, (u^1,\ldots ,u^n))\) on M. It induces a local coordinate system \((x^1,\ldots ,x^n, y^1,\ldots ,y^n)\) on TM, where
By abuse of notation, we shall denote the coordinate functions \(u^i\) also by \(x^i\).
Let \(\alpha \) be a Riemannian metric, \(\beta \) a 1-form on M. Locally,
The Riemannian metric \(\alpha \) induces naturally a Finsler function \(F_\alpha \) on TM given by \(F_\alpha (v):=\sqrt{\alpha _{\tau (v)}(v,v)}\). Similarly, the 1-form \(\beta \) can be interpreted as a smooth function
Locally,
In what follows, as usual, we shall simply write \(\alpha \) and \(\beta \) instead of \(F_\alpha \) and \(\overline{\beta }\), respectively.
For any \(p\in M\), we define
An \((\alpha ,\beta )\)-metric for M is a function F on \({{\mathcal {T}}}M:={\mathop {\bigcup }\limits _{p\in M}}(T_pM\backslash \{0_p\})\) defined by
where \(\phi :(-b_0,b_0)\longrightarrow {\mathbb {R}}\) is a smooth function \((b_0>0)\).
Now suppose that \(\Vert \beta _p\Vert _\alpha < b_0\) for any \(p\in M\). Then \(F=\alpha \left( \phi \circ \frac{\beta }{\alpha }\right) \) is a (positive definite) Finsler function if and only if \(\phi \) 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 \((\alpha ,\beta )\)-metric. If \(\Vert \beta _p\Vert _\alpha \le b_0\) for all \(p\in M\), then \(F=\alpha \left( \phi \circ \frac{\beta }{\alpha }\right) \) is called almost regular (under condition (2.1)). An almost regular \((\alpha ,\beta )\)-metric \(F=\alpha \left( \phi \circ \frac{\beta }{\alpha }\right) \) is positively almost regular if \(\phi \) is defined only on \((0,b_0)\).
For an \((\alpha ,\beta )\)-metric \(F=\alpha \phi (s)\), the components \(g_{ij}=\frac{1}{2}\frac{\partial ^2}{\partial y^i\partial y^j} F^2\) of the fundamental tensor can be calculated by the formula
where \(\alpha _i:=\frac{\partial \alpha }{\partial y^i}=\frac{(a_{ij}\circ \tau )}{\alpha }y^j\) and
see Chern-Shen [4, p. 179], where \(b^i=a^{ij}b_j\).
Moreover, we have
where \( b^2:=b^ib_i\).
The formula for the inverse metric \(g^{ij}\) can be found in [4] as follows.
Proposition 2.1
For an \((\alpha ,\beta )\)-metric \(F=\alpha \phi (s)\), the inverse \((g^{ij})\) of the matrix \((g_{ij})\) is given by
where \(\mu _o:=-\frac{\phi \phi ''}{\rho (\rho +\phi \phi ''m^2)}\), \(\mu _1:=-\frac{\rho _1}{\rho (\rho +\phi \phi ''m^2)}\), \(\mu _2:=\frac{\rho _1(s\rho +(\rho _1+s\phi \phi '')m^2)}{\rho ^2(\rho +\phi \phi ''m^2)}\) and \(m^2:=b^2-s^2\).
Remark 2.2
It should be noted that the choice \(\phi (s)=c_1s+c_2\sqrt{b^2-s^2}, \, c_1 \,\, \text {and} \, c_2 \,\text {are constants} \) is excluded. Indeed, the function \(\rho +\phi \phi ''m^2\) appearing in the denominators of \(\mu _0\), \(\mu _1\) and \(\mu _2\) can be written as follows
So \(\rho +\phi \phi ''m^2=0\) yields
which contradicts to condition (2.1). To avoid not only this contradiction, but also the dividing by zero (in \(\mu _0\), \(\mu _1\) and \(\mu _2\)), we must exclude the choice of \(\phi \) for which \(\rho +\phi \phi ''m^2=0\). Since \(\phi \) cannot be zero, we have
The solution of this ODE is the function
where \( c_1\) and \( c_2\) are constants.
It should be noted that, in the literature, the metric \(F=\alpha \phi (s)\), \(\phi (s)= k_1s+k_2\sqrt{1+k_3s^2}\), \(k_1>0\) 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 \(M={\mathbb {R}}^n\), \(\alpha =|y|\) and \(\beta =\varepsilon y^1\), \(\varepsilon \) is a constant. Then, we have
Then the metric \(F=\alpha \phi (s)\), \(\phi (s)=c_1 s+c_2 \sqrt{\varepsilon ^2 -s^2}\), by (2.3), is singular in the sense that its metric tensor has vanishing determinant.
Lemma 2.3
The components \(C_{ijk}=\frac{1}{2}\frac{\partial g_{ij}}{\partial y^k}\) of the Cartan tensor of an \((\alpha ,\beta )\)-metric are given by
where \(h_{ij}=a_{ij}-\alpha _i\alpha _j\) and \(m_i:=b_i-s\alpha _i\).
Proof
Differentiating (2.2) with respect to \(y^k\) and taking into account that \(\frac{\partial s}{\partial y^k}=\frac{m_k}{\alpha }\), we have
Since
the result follows. \(\square \)
Remark 2.4
The covariant vector \(m_i\) satisfies the properties
where \(m^2=b^2-s^2\).
Lemma 2.5
Let (M, F) be an \((\alpha ,\beta )\)-metric with \(n\ge 3\) such that
where \(\zeta (x,y)\) and \( \eta (x,y) \) are smooth functions on \({{\mathcal {T}}}M\). Then, \(\zeta \) and \( \eta \) must vanish.
Proof
Assume that
Contracting the above equation by \(b^ib^j\) and using Remark 2.4, we obtain
And the contraction by \(g^{ij}\) gives
Now, taking the fact that \(n\ge 3\), subtracting (2.4) and (2.5) we get \(\zeta =0\) and \(\eta =0\). \(\square \)
Lemma 2.6
Let (M, F) be an \((\alpha ,\beta )\)-metric with \(n\ge 3\). If there exist covectors \(A_i\) and \(B_j\) on TM such that \(y^iA_i=0\), \(y^iB_i=0\) and the following combination is satisfied
then \(A_i\) and \( B_i\) must vanish at each point of TM, that is, \(A_i\) and \( B_i\) are zero covectors.
Proof
Assume that
Contracting the above equation by \(b^ib^j\) and using Remark 2.4, we obtain
where we use the notations \(A_\beta :=A_ib^i\) and \(B_\beta :=B_ib^i\). Using the facts that \(y^iA_i=0\), \(y^iB_i=0\), the contraction by \(a^{ij}\) gives
Again, contracting the Eqs. (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 \(A_\beta =0\), \(B_\beta =0\). By substitution into (2.6) and (2.7) and repeating the last process we obtain that \(A_k=0\) and \(B_k=0\). \(\square \)
By the help of Lemma 2.5, one can easily prove the following theorem.
Theorem 2.7
For the \((\alpha ,\beta )\)-metrics with \(n\ge 3\), the following assertions are equivalent:
-
(a)
\(\rho _1=0\).
-
(b)
\(\rho _2=0\).
-
(c)
\((\alpha ,\beta )\)-metric is Riemannian.
-
(d)
\(\phi =\sqrt{k_1s^2+k_2}\).
For a Finsler manifold (M, F), the T-tensor is defined by [7]
where \(\ell _j:=\dot{\partial }_j F\), \( C_{rijk}:=\dot{\partial }_rC_{ijk}\) and \(\dot{\partial }_j\) is the differentiation with respect to \(y^j\). The T-tensor is totally symmetric in all of its indices.
Theorem 2.8
The T-tensor of an \((\alpha ,\beta )\)-metric takes the form:
where
Proof
By using Lemma 2.3 and making use of the fact that \(\dot{\partial }_is=\frac{m_i}{\alpha }\), we have
where \(n_{ij}:=\alpha _im_j+\alpha _jm_i\). By making use of the fact that \(K_1\) and \(K_2\) satisfy
we have
Since \( \ell _i:=\dot{\partial }_iF= \phi \alpha _i+\phi 'm_i\), we get
Now, taking the fact that \(F=\alpha \phi \) into account, the T-tensor of the space (M, F) is given by
\(\square \)
For an \((\alpha ,\beta )\)-metric, one can calculate \(\Phi \), \(\Psi \) and \(\Omega \) 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.
Corollary 2.9
The T-tensor of Kropina metric, \((F=\frac{\alpha }{s} ,\phi (s)=1/s)\), is given by
The T-tensor of Randers metric, \(\left( F={\alpha }(1+s), \phi (s)=1+s\right) \), 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].
3 The T-Condition and \(\sigma T\)-Conditions
The Finsler spaces with vanishing T-tensor are called Finsler spaces satisfying the T-condition, for example, see [3]. In a similar manner, we will call the Finsler spaces admitting a function \(\sigma (x)\) such that \(\sigma _hT^h_{ijk}=0\), \(\sigma _h:=\frac{\partial \sigma }{\partial x^h} \) Finsler spaces satisfying the \(\sigma T\)-condition. In this section, we characterize the \((\alpha ,\beta )\)-metrics which satisfy the T-condition and the \(\sigma T\)-condition.
Theorem 3.1
The \((\alpha ,\beta )\)-metrics with \(n\ge 3\) satisfy the T-condition if and only if \(\Phi =0\).
Proof
Let \(T_{hijk}=0\), then we have
Contracting the above equation by \(b^h\), we get
Since \(n\ge 3\), Lemma 2.5 implies
Again, contraction (3.1) by \(a^{hi}\), we obtain
Then, taking the fact that \(n\ge 3\) into account, we get
Now, solving the Eqs. (3.2) and (3.3) for \(\Phi \), \(\Psi \) and \(\Omega \), we have \(\Phi =0\), \(\Psi =0\) and \(\Omega =0\).
Conversely, let \(\Phi =0\), then we have either \(\rho _1=0\) or \(s+\alpha k_1m^2=0\). If \(\rho _1=0\) (the space is Riemannian), then \(\rho _0'=0\) and hence \(\Psi =0\) and \(\Omega =0\). And if \(s+\alpha K_1m^2=0\), one can conclude that \(\Psi =0\) and \(\Omega =0\) (see the proof of Theorem 4.1). \(\square \)
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. \(\square \)
Theorem 3.3
The \((\alpha ,\beta )\)-metrics with \(n\ge 3\) satisfies the \(\sigma T\)-condition if and only if
-
(a)
\(\Phi +m^2\Psi =0.\)
-
(b)
\(m^2\Omega +3 \Psi =0.\)
-
(c)
\(\sigma _j-\frac{\sigma _0}{ s\alpha }b_j=0\).
Proof
By using Proposition 3.2, we have
where \(\sigma _0:=\sigma _iy^i\) and \(\sigma _\beta :=\sigma _ib^i\). Using the fact that \(m_i=b_i-s\alpha _i\), we get
The above equation can be written in the following form
Putting \(A_i:=A m_i+\Phi \tau _i, \quad B_i:=\frac{B}{3}m_i+\Psi \tau _i,\) where \(\tau _i:=\frac{1}{\rho }\left( \sigma _i-\frac{\sigma _0}{ s\alpha }b_i\right) \) and
By using the above quantities, \(\sigma _h T^h_{ijk}\) can be written as follows
Now, putting \(\sigma _h T^h_{ijk}=0\) and since \(y^iA_i=0\), \(y^iB_i=0\), one can use Lemma 2.6 to conclude that
Contracting the above two equations by \(b^i\) and then by \(\sigma ^i:=a^{ij}\sigma _j\), respectively, we have
where \(\tau _\beta :=\tau _ib^i\), \(m_\sigma :=m_i\sigma ^i\) and \(\tau _\sigma :=\tau _i\sigma ^i\). 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
From which, we have
where \(\sigma ^2:=\sigma _i\sigma ^i\). Now, simplifying the above equation we get the following
Making use of the facts that \(\frac{\partial \sigma _0}{\partial y^i}=\sigma _{i}\) and the functions \(\sigma ^2\), \(\sigma _\beta \), \(b^2\) 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 \(\alpha =\sqrt{a_{ij}y^iy^j}\) and \(\frac{\partial (\alpha \alpha _i)}{\partial y^j}=a_{ij}\), differentiating the above equation with respect to \(y^j\) and then by \(y^k\), we get
Contracting the above equation by \(a^{ij}\), we get
which gives \(\sigma _k=0\) and this means that \(\sigma \) is constant and this is a contradiction. Consequently, all sides of the equalities in (3.4) are zero. That is,
Since \(m^2\ne 0\) and \(\Phi \), \(\Psi \) can not be zero, then \(A=0\), \(B=0\), \(\tau _\beta =0\), \(\tau _\sigma =0\) and hence \(\tau _i=0\). In other words, we have
Therefore \(\sigma _\beta =\frac{\sigma _0b^2}{s\alpha }\) and taking the fact that \(\sigma \ne 0\) into account, we get
Now, the choice \( \frac{1}{s\alpha \rho }+\mu _0\frac{b^2}{s\alpha }+\frac{\mu _1}{\alpha }=0\) gives the ODE \(\rho -s\rho _1-s^2\phi \phi ''=0\), which has the solution \(\phi =k s\). This solution is just again the same background Riemannian metric \(\alpha \) up to some constants. So, we should have \(\Phi +m^2\Psi =0\) and \(\Omega m^2+3\Psi =0.\)
Conversely, if the conditions (a), (b) and (c) are satisfied, then the result is obviously obtained. \(\square \)
Remark 3.4
The condition
is equivalent to \(\sigma _j=e^{a(x)}b_j\). Indeed,
where a(x) is an arbitrary, locally defined function on M.
4 Some ODEs
In this section, we focus our study on the T-condition and \(\sigma T\)-condition. By solving some ODEs, we find explicit formulas for \((\alpha ,\beta )\)-metrics that satisfy the T-condition and \(\sigma T\)-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, \(\phi \) is given by
Theorem 4.1
An \((\alpha ,\beta )\)-metric with \(n\ge 3\) satisfies the T-condition if and only if it is Riemannian or \(\phi \) is given by
Proof
By Theorem 3.1, any \((\alpha ,\beta )\)-metric satisfies the T-condition if and only if \(\Phi =0\). So, taking the fact that \(\phi -s\phi \ne 0\) into account, the ODE \(s+\alpha K_1m^2=0\) can be rewritten as follows
This is a first order linear differential equation and has the solution
Hence,
By using (4.1), \(\phi (s)\) is given by (4.2). Plugging \(\phi (s)\) in \(\Psi \) and \(\Omega \), we have \(\Psi =0\) and \(\Omega =0\). \(\square \)
Theorem 4.2
An \((\alpha ,\beta )\)-metric with \(n\ge 3\) satisfies the \(\sigma T\)-condition if and only if it satisfies the T-condition or \(\phi \) is given by
Proof
First we should write \(\Phi \) and \(\Psi \) 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 \(\phi -s\phi '\ne 0\), the condition \(\Phi +m^2\Psi =0\) gives the following two possible ODEs
or
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), \(\phi (s)\) is given by (4.3). \(\square \)
5 Examples and Concluding Remarks
We start by giving two classes of examples satisfying the \(\sigma T\)-condition.
Example 2
Let \(M={\mathbb {R}}^n\) and \(\alpha \), \(\beta \) be given by
where |y| is the Euclidean norm and \(f(x^1)\) is arbitrary function on M. Then, the class
satisfies the \(\sigma T\)-condition, where p and q are arbitrary constants. Indeed, in this class, one can see that the function \(\phi (s)\) is given by
Using the formula of \(\phi (s)\), it is much simpler to use the Maple program to show that
Moreover, since \(\beta = f(x^1)y^1\), then we have \(b_1=f(x^1)\), \(b_2=\cdots =b_n=0\) and taking Remark 3.4 into account, \(\sigma _1=\frac{\partial \sigma }{\partial x^1}=\omega (x^1)f(x^1)\), for some function \(\omega (x^1)\) on M, therefore one can see that \(\sigma (x)=\theta (x^1)\) where \(\theta (x^1)\) is an arbitrary function on M. Another way, one can use the Finsler package and Maple program to calculate the T-tensor, but in this case we have to choose the dimension, say \(n=3\), then one can find that
And since, \(\sigma =\theta (x^1)\), then \(\sigma _1=\frac{\partial \theta }{\partial x^1}\) and hence
Example 3
Let \(M= \mathbb {R}^3\), and \(\alpha =\sqrt{(y^2)^2+e^{2x^2}((y^1)^2+(y^3)^2)}\), \(\beta =y^2\). Then, the class
satisfies the \(\sigma T\)-condition. As in the previous example, we repeat the same process. So, one can see that the function \(\phi (s)\) is given by
Using Maple program, or by hand, we can show that
Since \(\beta = y^2\), then we have \(b_2=1\), \(b_1=b_3=0\). As in the previous example, we can have \(\sigma (x)=\theta (x^2)\) for some functions \(\theta (x^2)\) on M. Or instead, using the Finsler package and Maple program, we obtain that
And since, \(\sigma =\theta (x^2)\), then \(\sigma _2=\frac{\partial \theta }{\partial x^2}\) and hence
Finally, we have the following remarks:
\(\bullet \) Consider the conformal transformation of a Finsler function F, that is, \(\overline{F}=\kappa (x) F,\) where \(\kappa (x)\) is positive smooth function on M. Then, by simple and straightforward calculations, one can obtain that the T-tensor is transformed by the formula
In Example 2, one can see that the conformal transformation of F by any positive smooth function \(\kappa (x^1)\) still satisfying the \(\sigma T\)-condition, that is, \(\overline{F}=\kappa (x^1)F\) satisfies the \(\sigma T\)-condition. Also, in Example 3, the Finsler function \(\overline{F}=\kappa (x^2)F\) satisfies the \(\sigma T\)-condition.
\(\bullet \) By the following special choice \(c_2:=-c\) and \(c_1:=cb^2-1\) (\(b^2\) is constant), the class (4.2) becomes
which is the same as the one obtained by [10] ( (7.4) in Theorem 7.2). Moreover, this metric is positively almost regular Berwaldian.
It should be noted that this irregularity is studied by 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.
\(\bullet \) If \(b(x)=b_0\), then (4.3) can be rewritten an follows
We notice that the above formulae for \(\phi \) is the same as the one obtained in [10] ( (1.3) in Theorem 1.2). Under some restrictions on \(\beta \), 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].
\(\bullet \) Summarizing above, the classes (4.2) and (4.3) are almost regular \((\alpha ,\beta )\)-metrics. Moreover, the class (4.3) of \((\alpha ,\beta )\)-metrics that satisfies the \(\sigma T\)-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 \(\sigma _r(x)\) such that \(\sigma _rT^r_{ijk}=0\)?
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Elgendi, S.G., Kozma, L. \((\alpha ,\beta )\)-Metrics Satisfying the T-Condition or the \(\sigma T\)-Condition. J Geom Anal 31, 7866–7884 (2021). https://doi.org/10.1007/s12220-020-00555-3
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DOI: https://doi.org/10.1007/s12220-020-00555-3
Keywords
- (\(\alpha , \beta \))-metrics
- T-tensor
- T-condition
- \(\sigma T\)-condition
- Landsberg space
- Berwald space