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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1


Volume 13 (2015)

On generalized P-reducible Finsler manifolds

Seyyed Mohammad Zamanzadeh / Behzad Najafi / Megerdich Toomanian
Published Online: 2018-07-04 | DOI: https://doi.org/10.1515/math-2018-0065


The class of generalized P-reducible manifolds (briefly GP-reducible manifolds) was first introduced by Tayebi and his collaborates [1]. This class of Finsler manifolds contains the classes of P-reducible manifolds, C-reducible manifolds and Landsberg manifolds. We prove that every compact GP-reducible manifold with positive or negative character is a Randers manifold. The norm of Cartan torsion plays an important role for studying immersion theory in Finsler geometry. We find the relation between the norm of Cartan torsion, mean Cartan torsion, Landsberg and mean Landsberg curvatures of the class of GP-reducible manifolds. Finally, we prove that every GP-reducible manifold admitting a concurrent vector field reduces to a weakly Landsberg manifold.

Keywords: Generalized P-reducible metric; Randers metric; Landsberg metric

MSC 2010: 53B40; 53C60

1 Introduction

In [2], Tayebi and Sadeghi studied a class of Finsler metrics which contains the class of P-reducible metrics. They called Finsler metrics in this class by GP-reducible metrics. A Finsler metric F on a manifold M is called a GP-reducible metric if its Landsberg curvature is given by


where ai and λ are scalar functions on TM. In this case, (M, F) is called a GP-reducible manifold and λ is called the character of F. An longstanding open problem in Finsler geometry is to find a Landsberg manifold which is not Berwald manifold. Matsumoto and Shimada proposed a generalization of this problem in the context of P-reducible maniolfds. The class of (α, β)-metrics is a rich class of Finsler metrics with a diverse areas of application. In [2], it is proved that there is no P-reducible (α, β)-metric with vanishing S-curvature which is not C-reducible, thus a partial answer to Matsumoto and Shimada’s problem.

The class of Randers metrics are natural Finsler metrics which were first introduced by G. Randers and derived from the research on the four-space of general relativity. His metric is in the form F = α + β, where α=aij(x)yiyj is gravitational field and β = bi(x)yi is the electromagnetic field. Randers metrics have been widely applied in many areas of natural science, including seismic ray theory, biology and physics, etc [3].

We study compact GP-reducible manifold and prove the following rigidity theorem.

Theorem 1.1

Every compact GP-reducible manifold of dimension n ≥ 3 with negative or positive character is a Randers manifold.

In [4], Shen proved that a Finsler manifold with unbounded Cartan torsion can not be isometrically imbedded into any Minkowski space. Thus, the norm of Cartan torsion plays an important role for studying immersion theory in Finsler geometry. We find the following result on the norm of impotrtant non-Riemannian quantities for a GP-reducible manifold.

Theorem 1.2

Let (M, F) be a GP-reducible Finsler manifold. Then the following holds


In particular, if F has relatively isotropic Landsberg curvature L = cFC, then (1) reduces to


provided that λ ≠ ±cF.

It is known that for a P-reducible manifold (1) holds with λ = 0. Thus, we can consider (1) as a generealization of this well-known fact.

Studying geometric vector fields plays a prominent role in Differential Geometry. In [5], K. Yano introduced the notion of concurrent vector fields on an affine manifold (M, ∇). Then, F. Brickell and K. Yano studied this kind of geometric vector fields in Riemannian geometry considering the Levi-Civita connection as ∇ [6]. Here, we prove that addmiting a concurrect vector field on a GP-reducible manifold reduces it to a relatively isotropic mean Landsberg metric manifold. More precisely, we have the following.

Theorem 1.3

Let (M, F) be a GL-reducible Finsler manifold admitting a concurrent vector field. Then F is a weakly Landsberg.

Throughout this paper, we use the Cartan connection and the h- and v- covariant derivatives of a Finsler tensor field are denoted by “ | ” and “, ” respectively.

2 Preliminaries

A Finsler metric on an n-dimensional C manifold M is a function F : TM → [0, ∞) which has the following properties: (i) F is C on TM0 = TM ∖ {0}, (ii) F is positively 1-homogeneous on the fibers of tangent bundle TM, (iii) for each yTxM, the quadratic form gy : TxMTxM → ℝ on TxM is positive definite,


Let xM and Fx := F|TxM. To measure the non-Euclidean feature of Fx, define Cy : TxMTxMTxM → ℝ by


The family C := {Cy}yTM0 is called the Cartan torsion. It is well known that C = 0 if and only if F is Riemannian.

Taking a trace of Cartan torsion yields the mean Cartan torsion Iy. Let (M, F) be an n-dimensional Finsler manifold. For yTxM0, define Iy : TxM → ℝ by


where {i} is a basis for TxM at xM. The family I := {Iy}yTM0 is called the mean Cartan torsion.

For yTxM0, define the Matsumoto torsion My : TxMTxMTxM → ℝ by My(u, v, w) := Mijk(y)uivjwk where


and hij = gijFyiFyj is the angular metric. F is said to be C-reducible if My = 0.

Lemma 2.1

([7]) A Finsler metric F on a manifold M of dimension n ≥ 3 is a Randers metric if and only if My = 0, ∀yTM0.

The horizontal covariant derivative of the Cartan torsion C along geodesics gives rise to the Landsberg curvature Ly : TxMTxMTxM → ℝ is defined by Ly(u, v, w) := Lijk(y)uivjwk, where


The family L := {Ly}yTM0 is called the Landsberg curvature. A Finsler metric is called a Landsberg metric if L = 0 (see [8, 9]).

The horizontal covariant derivative of the mean Cartan torsion I along geodesics gives rise to the mean Landsberg curvature Jy : TxM → ℝ which is defined by Jy(u) := Ji(y)ui, where


The family J := {Jy}yTM0 is called the mean Landsberg curvature. A Finsler metric is called a weakly Landsberg metric if J = 0.

A Finsler metric F is called GP-reducible if its Landsberg curvature is given by


where ai and λ are scalar functions on TM. By definition, if ai = 0 then F reduces to a general relatively isotropic Landsberg metric and if λ = 0 then F is P-reducible [10]. Thus, the study of this class of Finsler spaces will enhance our understanding of the geometric meaning of C-reducible and P-reducible metrics.

Remark 2.2

Let F be a GP-reducible metric with character λ. Then for every positive constant c, the Finsler metric cF is also GP-reducible metric with character c2λ. Hence, the class of GP-reducible metrics is closed under homothetic transformations.

Finsler geometry is a natural generalization of Riemannain one. Study of geometric vector fields has been extended from Riemannian geometry to Finsler geometry. Concurrent vector fields in Finslerian setting have been studied extensively (for example see [11, 12]). Let us consider a vector field Xi(x) in (M, F). This field is called concurrent if it satisfies the following



One can also use covariant derivative with respect to Berwald connection to define concurrent vector fields. It is well knwon that the two defintions are the same.

Let us mention a standard example of concurrent vector field. Let V be a vector space and (xi) be a standard chart of V. Suppose that F is a Minkowski norm on V. Then, the radial vector field X=xixi is a concurrent vector field of (V, F).

3 Proof of Theorem 1.1

In this section, we will prove a generalized version of Theorem 1.1. Indeed we study complete GP-reducible manifold of positive (or negative) character with the assumption that F has bounded Matsumoto torsion. For this aim, we remark the following.

Lemma 3.1

([2]) Let (M, F) be a GP-reducible Finsler manifold. Then the Matsumoto torsion of F satisfies the following



Let F be a GP-reducible metric


Contracting (9) with gij and using relations gijhij = n – 1 and gij(aihjk) = gij(ajhik) = ak imply that




Putting (11) into (9) yields


It is easy to see that (12) is equivalent to (8). □

Now, we are going to prove the main result of this section.

Theorem 3.2

Let (M, F) be an n-dimensional complete GP-reducible manifold (n ≥ 3) with negative or positive character. Suppose that F has bounded Matsumoto torsion. Then F is a Randers metric.


By definition, the norm of the Matsumoto torsion at xM is given by


Suppose that the Matsumoto torsion satisfies


for some y, uTxM0 with F(x, y) = 1. Let σ(t) be the unit speed geodesic with σ(0) = x and σ̇(0) = y. Let U(t) denote the linear parallel vector field along σ with U(0) = u. Let


It followos from (8)


Remark 2.2 permits us to assume that λ(t) ≤ –1 or λ(t) ≥ 1. The general solution of (13) is given by


By assumption, 𝓜(0) = My(u, u, u) ≠ 0. Letting t → –∞ or t → ∞ implies that 𝓜(t) is unbounded, contradicting with boundednes of Matsumoto’s torsion. Thus the Matsumoto torsion of F vanishes. By Lemma 2.1, F reduces to a Randers metric. □

4 Proof of Theorem 1.2

In this section, we prove Theorem 1.2.


Let F be a GP-reducible metric


Multiplying (14) with gmigpjgqk yields


where ai = gimam. The following hold


Multiplying (14) with (15) implies that




where ∥a2 := asas. Thus


Contracting (14) with gjk implies that


Multiplying (19) with gik yields


By (19) and (20), we get






By (18) and (22), we get (1). □

5 Proof of Theorem 1.3

Suppose that Xi(x) is a concurrent vector field on a Finsler manifold, then from Ricci identities, we get the following integrability conditions:




Since Phijk are skew symmetric in h and i, we have from (24)


It is well known that


From (7), one can see that


Plugging (27) and (28) into (24), we get


Let Xi(x) denote the covariant components of Xi. Define a 1-form β as follows β := Xiyi. Put mi:=XiβyiF2. Then, it is proved that the following hold



where m2 = gijmimj and mi = gijmj [12]. Contracting (5) by XiXj and using (7) and (11), we obtain


Using (31), we get Jk = λIk. Hence, we get Theorem 1.3.


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About the article

Received: 2017-08-22

Accepted: 2017-10-12

Published Online: 2018-07-04

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 718–723, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0065.

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© 2018 Zamanzadeh et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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