The concept of (α, β) Finsler metrics was introduced by M. Matsumoto in 1972 as a generalization of the Randers metric [10]. The Randers metric is of the form F = α + β, where α is a Riemannian metric and β is a 1-form satisfying ||β|| = 1. It was first introduced by G. Randers regarding an asymmetric metric on four-dimensional space-time in general relativity [16]. The (α, β) Finsler metrics can be written as F = αφ(s), where cc is a smooth function satisfying

In the study of Finsler geometry, we often encounter long and complicated calculations. However, when we consider Finsler metrics with certain symmetries, things become much easier. In general relativity, the solution of vacuum Einstein field equations describing the gravitational field, which is spherically symmetric, we obtain the Schwarzschild metric in four dimensional space-time [21]. In this process, the condition of spherical symmetry plays a very important role. In 1996, S.F. Rutz [17] introduced a special class of Finsler metrics called spherically symmetric. A Finsler metric F on Bⁿ(δ) is called spherically symmetric if F(Ax, Ay) = F(x, y), for all n × n orthogonal matrix A, x = (xⁱ) ∈ Bⁿ(δ) and y = (yⁱ) ∈ T_xBⁿ(δ). Here Bⁿ(δ) denotes the Euclidean ball of radius δ around the origin and T_xBⁿ(δ) denotes the tangent space of Bⁿ(δ) at the point x. L. Zhou [22] proved that a Finsler metric F on Bⁿ(δ) is a spherically symmetric if and only if there exist a function φ : [0, δ) × ℝ → ℝ such that

where |.| denotes the Euclidean norm and 〈, 〉 denotes the Euclidean inner product on ℝⁿ.

In 2012, Yu and Zhu [20] introduced a new class of Finsler metrics, called general (α, β)-Finsler metrics given by F = αφ(b², s) where φ = φ(b², s) is a C^∞ positive function and b2:=‖β‖α2. This class of Finsler metrics not only generalizes (α, β)-metrics in a natural way, but also generalizes the spherically symmetric metrics. Moreover, this class of Finsler metric also include Finsler metrics constructed by R. Bryant [4, 5, 6]. Bryant’s metrics are rectilinear Finsler metrics on Sⁿ with flag curvature K = 1 and given by

where Q(X, Y) = x₀y₀ + e^ip₁x₁y₁ + e^ip₂x₂y₂ + ..... + e^ip_nx_ny_n is a complex quadratic form on ℝⁿ⁺¹ for n ≥ 2 with the parameters satisfying 0 ≤ p₁ ≤ p₂ ≤ … ≤ p_n < π and X = (x₀, …, x_n) ∈ Sⁿ, Y = (y₀, …, y_n) ∈ T_XSⁿ.

In 1978, Matsumoto and Shibata [12] introduced a special class of Finsler metric called semi-C-reducible. The concept of “semi-C-reducibility” is a generalization of the well-known C-reducibility. In 1992, Matsumoto and Shibata [11] proved that every (α, β)-metric is semi-C-reducible. In this paper, we prove the following results:

Theorem 1.1

Every general (α, β)-metric is semi C-reducible.

Corollary 1.2

A general (α, β)-metric can not be C2-like.

Recently, Finsler metrics satisfying Ricci flow equation has been an important topic of research. The Ricci flow equation introduced by R.S. Hamilton in 1981 [8, 9] which is an intrinsic flow that deforms the metric of a Riemannian manifold, in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric. Though it is a primary tool used in Grigori Perelman’s solution of the Poincare conjecture [13, 14, 15], it has various applications to dynamical systems, mathematical physics and even in cosmology. Sadegzadeh and Razavi studied C-reducible metrics satisfying Ricci flow equation [18], where as Tayebi, Payghan and Najafi studied semi C-reducible Finsler metrics satisfying Ricci flow equation [19]. In this paper we obtain the following:

Theorem 1.3

A general (α, β)-metric satisfying normal or unnormal Ricci flow equation is Einstein.

Let M be an n-dimensional C^∞-manifold. T_xM denotes the tangent space of M at x. The tangent bundle of M is the union of tangent spaces TM := ∪_x∈M T_xM. We denote the elements of TM by (x, y) where y ∈ T_xM and define TM₀ := TM {0}.

Definition 2.1

A Finsler metric on M is a function F : TM → [0, ∞) satisfying the following conditions:

• F is C^∞ on TM₀,

• F is a positively 1-homogeneous on the fibers of tangent bundle TM,

• The Hessian of F22 with element gij=12∂2F2∂yi∂yj is positive definite on TM₀.

The pair (M, F) is called a Finsler space, F is called the fundamental function and g_ij is called the fundamental tensor.

Let (M, F) be a Finsler space. For a vector y ∈ T_xM {0}, let

where u, v, w ∈ T_xM. Each C_y is a symmetric trilinear form on T_xM. We call the family C := {C_y : y ∈ T_xM {0}} the Cartan torsion. Denote the components of Cartan torsion C by C_ijk. Therefore, we have

The mean Cartan torsion I at x ∈ M is defined by

where, I_y(u) := g^ij(y)C_y(u, ∂_i, ∂_j), u ∈ T_xM. Denote the components of mean Cartan torsion I by I_i and therefore, we have I_i = g^jkC_ijk.

Roughly speaking the Cartan torsion measures how much a Finsler metric is far from being a Riemannian one. In particular, if C = 0, a Finsler metric reduces to a Riemannian metric. In general, the calculation with general form of Cartan torsion is very tedious. However, when we consider some special form of it sometimes we deduce some very interesting geometric properties of the Finsler space. To simplify the calculation and geometrical objects M. Matsumoto studied various special Finsler spaces [12]. For instance, C2-like Finsler spaces, C-reducible Finsler spaces, semi-C-reducible Finsler spaces etc.

Definition 2.2

A Finsler metric F is called Semi C-reducible if its Cartan torsion is given by

where h_ij is angular metric tensor given by hij=F∂2F∂yi∂yj, p = p(x, y) and q = q(x, y) are scalar functions on TM with p + q = 1 and I² = I_iIⁱ.

Remark 2.3

If p = 0, then F is called C2-like Finsler metric and if q = 0 then F is called C-reducible Finsler metric.

Remark 2.4

A two dimensional Finsler space is always C2-like as well as C-reducible where as, a three dimensional Finsler space is always semi C-reducible.

Definition 3.1.([20])

A Finsler metric F on a manifold M is called a general (α, β)-metric, if F can be expressed in the form F=αφ (b2,βα) for some C^∞ function φ := φ(b², s), where α is a Riemannian metric and β is a 1-form. In particular, if φ only depends on s, i.e., φ=φ (βα), then the Finsler metric F=αφ (βα)is called an (α, β)-metric.

You and Zhu [20] proved that the function φ in the general (α, β)-metric F=αφ (b2,βα) satisfies

or

where s and b are arbitrary numbers with |s| ≤ b < b₀. Here φ′ denotes the differentiation of φ with respect to s.

For a general (α, β)- metric F=αφ (b2,βα), the fundamental tensor g_ij is given in [20] as:

where ρ = φ (φ − sφ′), ρ₀ = φφ″ + φ′φ′, ρ₁ = (φ − sφ′) φ′ − sφφ″. Moreover,

and

where (g^ij) = (g_ij)⁻¹, (a^ij) = (a_ij)⁻¹, bⁱ = a^ijb_j, η=-φ″φ-sφ′+(b2-s2)φ″,η0=-(φ-sφ′)φ′-sφφ″φ(φ-sφ′+(b2-s2)φ″),η1=(sφ+(b2-s2)φ) ((φ-sφ′)φ′-sφφ″)φ2(φ-sφ′+(b2-s2)φ″).

Proposition 3.1

The Cartan torsion C_ijk of a general (α, β)-metric F=αφ (b2,βα)is given by

where (i → j → k → i) denotes cyclic permutation of indices i, j, k in the preceeding terms.

Proposition 3.2

The mean Cartan torsion I_i of a general (α, β)-metric F=αφ (b2,βα)is given by

Proposition 3.3

The angular metric tensor h_ij of a general (α, β)-metric F=αφ (b2,βα)

is given by

For a semi C-reducible Finsler metric F, the Cartan torsion is given by equation (2.1). Equation (3.7) can be rewritten as

where P=1ρ[ρ1{(n+1)+3η(b2-s2)}+(b2-s2)T{1+η(b2-s2)}]. Using equation (3.2) and (3.7) we obtain

From equation (3.7) and equation (3.8) we get

Further from equations (4.1) and (4.2) we have

where

Using equations (4.3) and (4.4), we have

The general (α, β)-metric will be semi C-reducible if equations (3.3) and (4.6) are identical.

Now comparing the coefficients of these two equations we have,

Equations (4.8), (4.9) and (4.11) can be obtained from equations (4.7) and (4.10). So, a general (α, β)-metric will be semi C-reducible if it satisfies equations (4.7) and (4.10). Dividing equation (4.7) by equation (4.10), we have

Now solving for p we have,

As p + q = 1, using (4.13) we have

Since we can find the value of p and q uniquely, the theorem follows.

Proof of corollary 1.2

For C2-like Finsler metric we have, p = 0. In view of equation (4.13) and (4.5), since Q is non-zero, we have ρ₁ = 0. Then from equation (3.7) we get T = 0 and hence we have I_i = 0. This is a contradiction to the assumption that F is non-Riemannian. So the result follows.

The geometric evolution equation

is known as the un-normalized Ricci flow equation in Riemannian geometry. In principle, the same equation can be used in Finsler setting, because both g_ij and Ric_ij have been generalised to the broader framework, albeit gaining a y dependence in the process.

A deformation of Finsler metric means a 1-parameter family of metrics g_ij(x, y, t), with parameter t ∈ [−ε, ε] and ε is sufficiently small positive number. For such a metric ω = F_yⁱdxⁱ, the volume element as well as connections attached to it depend on t. Instead of the above tensor evolution equation, we use the following form of it. Contracting equation (5.1) with yⁱ and y^j respectively and using Euler’s theorem, we get

where Ric is the Ricci scalar function. That is,

This scalar equation directly addresses the evolution of the Finsler metric F and makes geometrical sense on both the manifold of nonzero tangent vectors TM₀ and the manifold of rays. It is therefore suitable as an un-normalized Ricci flow in Finsler geometry.

If M is compact, then so is SM and we can normalize the above equation by requiring that the flow keeps the volume of SM constant. Recalling the Hilbert form ω := F_yⁱdxⁱ, the volume of SM is given by

During the evolution, the Finsler metric F, the Hilbert form ω, the volume form dV_SM and consequently the volume Vol_SM, all depend on t. On the other hand, the domain of integration SM; being the quotient space of TM₀ under the equivalence relation z ~ y, if and only if, z = λy for some λ > 0; is totally independent of any Finsler metric and hence does not depend on t. We have from [1]

A normalized Ricci flow for Finsler metrics is introduced by Bao [2] and given by

where the given manifold M is compact. In differential geometry and mathematical physics, an Einstein manifold is a Riemannian manifold whose Ricci tensor is proportional to the metric tensor and this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations. In Finsler geometry the Einstein metric is defined as follows:

Definition 5.1

A Finsler metric is said to be an Einstein metric if the Ricci scalar function Ric depends only on x, i.e, Ric_ij = Ric(x)g_ij.

In general semi C-reducible Finsler metrics are not Einstein though they becomes Einsten when they satisfies the Ricci flow equation. More precisely, we have the following:

Lemma 5.2.([19])

• Every semi C-reducible Finsler metric satisfying Un-normalize Ricci flow equation is Einstein.

• Every semi C-reducible Finsler metric satisfying Normalize Ricci flow equation is Einstein.