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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

Issues

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

Abstract

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

Lijk=λCijk+aihjk+ajhki+akhij,

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

||L||23(n+1)||J||2=λ2[||C||23(n+1)||I||2].(1)

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

||C||=3n+1||I||,(2)

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,

gy(u,v):=122st[F2(y+su+tv)]|s,t=0,u,vTxM.

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

Cy(u,v,w):=12ddt[gy+tw(u,v)]|t=0,u,v,wTxM.

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

Iy(u)=i=1ngij(y)Cy(u,i,i),

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

Mijk:=Cijk1n+1{Iihjk+Ijhik+Ikhij},

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

Lijk:=Cijk|sys.(3)

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

Ji:=Ii|sys.(4)

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

Lijk=λCijk+aihjk+ajhki+akhij,(5)

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

X|ji=δji,(6)

X,ji=0.(7)

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

Mijk|sys=λ(x,y)Mijk.(8)

Proof

Let F be a GP-reducible metric

Lijk=λCijk+aihjk+ajhki+akhij.(9)

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

Jk=λIk+(n+1)ak.(10)

Then

ai=1n+1[JiλIi].(11)

Putting (11) into (9) yields

Lijk1n+1(Jihjk+Jjhki+Jkhij)=λ{Cijk1n+1(Iihjk+Ijhki+Ikhij)}.(12)

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.

Proof

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

Mx:=supy,u,v,wTxM{0}F(y)|My(u,v,w)|gy(u,u)gy(v,v)gy(w,w).

Suppose that the Matsumoto torsion satisfies

My(u,u,u)=Mijk(x,y)uiujuk0

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

M(t):=Mσ˙(t)(U(t),U(t),U(t))=Mijk(σ(t),σ˙(t))Ui(t)Uj(t)Uk(t).

It followos from (8)

M(t)+˘(t)M(t)=0.(13)

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

M(t)=e0tλ(t)dtM(0).

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.

Proof

Let F be a GP-reducible metric

Lijk=λCijk+aihjk+ajhki+akhij.(14)

Multiplying (14) with gmigpjgqk yields

Lijk=λCijk+aihjk+ajhki+akhij,(15)

where ai = gimam. The following hold

Cijkhij=Ik,hijhjk=hik,hijhij=n1.(16)

Multiplying (14) with (15) implies that

LijkLijk=(λCijk+aihjk+ajhki+akhij)(λCijk+aihjk+ajhki+akhij)

equivalently

||L||2=λ2||C||2+6λIkak+3(n+1)||a||2,(17)

where ∥a2 := asas. Thus

6λIkak=||L||2λ2||C||23(n+1)||a||2.(18)

Contracting (14) with gjk implies that

Jk=λIk+(n+1)ak.(19)

Multiplying (19) with gik yields

Jk=λIk+(n+1)ak.(20)

By (19) and (20), we get

JkJk=(λIk+(n+1)ak)(λIk+(n+1)ak)

equivalently

||J||2=λ2||I||2+2(n+1)λIkak+(n+1)2||a||2.(21)

Thus

2(n+1)λIkak=||J||2λ2||I||2(n+1)2||a||2.(22)

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:

XhRhijk=0,(23)

XhPhijkCijk=0,(24)

XhShijk=0.(25)

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

XiCijk=0.(26)

It is well known that

Phijk=Cijk|hChjk|i+ChjrCik|0rCijrChk|0r.(27)

From (7), one can see that

XhChjk|i=Cijk.(28)

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

XhCijk|h=0.(29)

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

hijXj=mi0,(30)

hijXiXj=m20,(31)

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

XiXjhijJk=0.

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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