$$(\alpha ,\beta )$$ ( α , β ) -Metrics Satisfying the T -Condition or the $$\sigma T$$ σ T -Condition

The Journal of Geometric Analysis https://doi.org/10.1007/s12220-020-00555-3 (˛, ˇ)-Metrics Satisfying the T -Condition or the T -Condition Salah G. Elgendi1 · László Kozma2 Received: 27 July 2020 / Accepted: 29 October 2020 © The Author(s) 2020 Abstract We describe the (α, β)-metrics whose the T -tensor vanishes (T -condition) and the and σ is a (α, β)-metrics that satisfy the σ T -condition σh Tihjk = 0, where σh = ∂∂σ xh 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. Keywords (α, β)-metrics · T -tensor · T -condition · σ T -condition · Landsberg space · Berwald space Mathematics Subject Classification 53B40 · 58B20 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 [1–3]. B László Kozma http://www.math.unideb.hu/kozma-laszlo/ Salah G. Elgendi ; http://www.bu.edu.eg/staff/salahali7 1 Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt 2 Department of Geometry, Institute of Mathematics, University of Debrecen, P. O. Box 400, 4002 Debrecen, Hungary 123 S.G. Elgendi, L. Kozma 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 F(v) := eσ ( p) F(v) if v ∈ T p M. (1.1) A Landsberg manifold remains of the same type under a conformal change (1.1) if and only if the T -tensor satisfies the condition r σr T jk = 0, σr := ∂σ . ∂ xr 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 r = 0 is satisfied for some conformal was studied in the case when the condition σr T jkh 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 := βα . We start by studying the Cartan tensor Ci jk of (α, β)metrics. We show that the Cartan tensor  Ci jk vanishes identically and hence the space is Riemannian if and only if φ(s) = k1 + k2 s 2 , where k1 and k2 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 cb2 −1 1 φ(s) = c3 s cb2 (cb2 − cs 2 ) 2cb2 . We introduce the notion of σ T -condition. We say that a Finsler space satisfies this . condition if it admits smooth function σ (x) such that σh Tihjk = 0, where σh = ∂∂σ xh We find necessary and sufficient conditions for an (α, β)-metric to satisfy the σ T condition. Moreover, we show that the (α, β)-metrics satisfy the σ T -condition if and only if the T -tensor vanishes (this is the trivial case) or φ(s) is given by  φ(s) = c3 exp 0 s  √ c1 b2 − t 2 + c2 t dt . √ t(c1 b2 − t 2 + c2 t) + 1 It is worthy to mention that the above special (α, β)-metrics have already been obtained by 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 σ 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 nonBerwaldian spaces is (closely) related to the question: 123 (α, β)-Metrics Satisfying the T -Condition or the σ T -Condition Is there any Finsler space admitting a smooth function σ such that σr Tirjk = 0, σr = ∂∂σ xr ? In this paper we confirm this claim in the almost regular case, since the class of (α, β)-metrics that satisfy the σ 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 (˛, ˇ)-Metrics Let M be an n-dimensional smooth manifold. The tangent space to M at p is denoted  T p M is the tangent bundle of M, τ : T M −→ M is the tangent by T p M; T M := p∈M bundle projection. 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 x i := u i ◦ τ, y i (v) := v(u i ) (v ∈ τ −1 (U)). 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, α = ai j d x i ⊗ d x j , β = bi d x i . (U ) (U ) The Riemannian metric α induces naturally a Finsler function Fα on T M given  by Fα (v) := ατ (v) (v, v). Similarly, the 1-form β can be interpreted as a smooth function β : T M −→ R, v −→ β(v) := βτ (v) (v). Locally, Fα = (U )  (ai j ◦ τ )y i y j , β = (bi ◦ τ )y i . (U ) In what follows, as usual, we shall simply write α and β instead of Fα and β, respectively. For any p ∈ M, we define β p α := β(v) . v∈T p M\{0 p } α(v) sup An (α, β)-metric for M is a function F on T M :=  (T p M\{0 p }) defined by p∈M F := αφ(s) := α(φ ◦ s), s := β , α 123 S.G. Elgendi, L. Kozma where φ : (−b0 , b0 ) −→ R is a smooth function (b0 > 0).  Now suppose that β p α < b0 for any p ∈ M. Then F = α φ ◦ βα is a (positive definite) Finsler function if and only if φ satisfies the following conditions: φ(t) > 0, φ(t) − tφ (t) + (x 2 − t 2 )φ (t) > 0, (2.1) where t and x are arbitrary real numbers with |t| < x < b0 . (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 ≤ b0 for all p ∈ M, then F = α φ ◦ βα is called almost regular (under condition (2.1)).  An almost regular (α, β)-metric F = α φ ◦ βα is positively almost regular if φ is defined only on (0, b0 ). 2 For an (α, β)-metric F = αφ(s), the components gi j = 21 ∂ y∂i ∂ y j F 2 of the fundamental tensor can be calculated by the formula gi j = ρai j + ρ0 bi b j + ρ1 (bi α j + b j αi ) + ρ2 αi α j , = where αi := ∂∂α yi (2.2) (ai j ◦τ ) j α y and ρ := φ 2 − sφφ , ρ0 := φ 2 + φφ , ρ1 := φφ − s(φ 2 + φφ ), ρ2 := s 2 (φ 2 + φφ ) − sφφ , see Chern-Shen [4, p. 179], where bi = a i j b j . Moreover, we have  det(gi j ) = φ n+1 (φ − sφ )n−2 (φ − sφ ) + (b2 − (...truncated)