On generalized P-reducible Finsler manifolds

Seyyed Mohammad Zamanzadeh; Behzad Najafi; Megerdich Toomanian

doi:10.1515/math-2018-0065

Article Open Access

On generalized P-reducible Finsler manifolds

Seyyed Mohammad Zamanzadeh , Behzad Najafi and Megerdich Toomanian

Published/Copyright: July 4, 2018

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

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 a_i 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 = b_i(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 dimensionn ≥ 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||2−3(n+1)||J||2=λ2[||C||2−3(n+1)||I||2].(1)

In particular, if F has relatively isotropic Landsberg curvatureL = 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 y ∈ T_xM, the quadratic form g_y : T_xM ⊗ T_xM → ℝ on T_xM is positive definite,

gy(u,v):=12∂2∂s∂t[F2(y+su+tv)]|s,t=0,u,v∈TxM.

Let x ∈ M and F_x := F|_{T_xM}. To measure the non-Euclidean feature of F_x, define C_y : T_xM ⊗ T_xM ⊗ T_xM → ℝ by

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

The family C := {C_y}_y∈TM₀ 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 I_y. Let (M, F) be an n-dimensional Finsler manifold. For y ∈ T_xM₀, define I_y : T_xM → ℝ by

Iy(u)=∑i=1ngij(y)Cy(u,∂i,∂i),

where {∂_i} is a basis for T_xM at x ∈ M. The family I := {I_y}_y∈TM₀ is called the mean Cartan torsion.

For y ∈ T_xM₀, define the Matsumoto torsion M_y : T_xM ⊗ T_xM ⊗ T_xM → ℝ by M_y(u, v, w) := M_ijk(y)uⁱv^jw^k where

Mijk:=Cijk−1n+1{Iihjk+Ijhik+Ikhij},

and h_ij = g_ij – F_yⁱF_y^j is the angular metric. F is said to be C-reducible if M_y = 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 M_y = 0, ∀y ∈ TM₀.

The horizontal covariant derivative of the Cartan torsion C along geodesics gives rise to the Landsberg curvature L_y : T_xM ⊗ T_xM ⊗ T_xM → ℝ is defined by L_y(u, v, w) := L_ijk(y)uⁱv^jw^k, where

Lijk:=Cijk|sys.(3)

The family L := {L_y}_y∈TM₀ 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 J_y : T_xM → ℝ which is defined by J_y(u) := J_i(y)uⁱ, where

Ji:=Ii|sys.(4)

The family J := {J_y}_y∈TM₀ 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 a_i and λ are scalar functions on TM. By definition, if a_i = 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=xi∂∂xi 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 g^ij and using relations g^ijh_ij = n – 1 and g^ij(a_ih_jk) = g^ij(a_jh_ik) = a_k imply that

Jk=λIk+(n+1)ak.(10)

Then

ai=1n+1[Ji−λIi].(11)

Putting (11) into (9) yields

Lijk−1n+1(Jihjk+Jjhki+Jkhij)=λ{Cijk−1n+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 x ∈ M is given by

∥M∥x:=supy,u,v,w∈TxM∖{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)uiujuk≠0

for some y, u ∈ T_xM₀ 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)=e−∫0tλ(t)dtM(0).

By assumption, 𝓜(0) = M_y(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 g^mig^pjg^qk yields

Lijk=λCijk+aihjk+ajhki+akhij,(15)

where aⁱ = g^ima_m. The following hold

Cijkhij=Ik,hijhjk=hik,hijhij=n−1.(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 ∥a∥² := a_sa^s. Thus

6λIkak=||L||2−λ2||C||2−3(n+1)||a||2.(18)

Contracting (14) with g^jk implies that

Jk=λIk+(n+1)ak.(19)

Multiplying (19) with g^ik 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||2−λ2||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)

XhPhijk−Cijk=0,(24)

XhShijk=0.(25)

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

XiCijk=0.(26)

It is well known that

Phijk=Cijk|h−Chjk|i+ChjrCik|0r−CijrChk|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 X_i(x) denote the covariant components of Xⁱ. Define a 1-form β as follows β := X_iyⁱ. Put mi:=Xi−βyiF2. Then, it is proved that the following hold

hijXj=mi≠0,(30)

hijXiXj=m2≠0,(31)

where m² = g_ijmⁱm^j and mⁱ = g^ijm_j [12]. Contracting (5) by XⁱX^j and using (7) and (11), we obtain

XiXjhijJk=0.

Using (31), we get J_k = λI_k. Hence, we get Theorem 1.3.

References

[1] Heydari A., Peyghan E., Tayebi A., Generalized P-reducible Finsler metrics, Acta Math. Hungarica, 2016, 149(2), 286-296.10.1007/s10474-016-0615-0Search in Google Scholar

[13] Clote P., Kranakis E., Boolean functions, invariance groups, and parallel complexity, SIAM J. Comput., 1991, 20, 553-59010.1137/0220036Search in Google Scholar

[2] Tayebi A., Sadeghi H., Generalized P-reducible (α, β)-metrics with vanishing S-curvature, Ann. Polon. Math., 2015, 114(1), 67-79.10.4064/ap114-1-5Search in Google Scholar

[3] Tayebi A., Nankali A., On generalized Einstein Randers metrics, Int. J. Geom. Meth. Modern. Phys, (2015), 12(9), 1550105.10.1142/S0219887815501054Search in Google Scholar

[4] Shen Z., On Finsler geometry of submanifolds, Math. Annal, 1998, 311, 549–576.10.1007/s002080050200Search in Google Scholar

[5] Yano K., Sur le parallélisme et la concourance dans l’espace de Riemann, Proc. Imp. Acad. Tokyo, (1943), 19, 189-197.10.3792/pia/1195573583Search in Google Scholar

[6] Brickell F., Yano K., Concurrent vector fields and Minkowski structures, Kodai Math. Sem. Rep., 1974, 26, 22-28.10.2996/kmj/1138846943Search in Google Scholar

[7] Matsumoto M. and Hōjō S., A conclusive theorem for C-reducible Finsler spaces, Tensor. N. S., 1978, 32, 225-230.Search in Google Scholar

[8] Peyghan E., Tayebi A., Generalized Berwald metrics, Turkish. J. Math, 2012, 36, 475-484.10.3906/mat-1003-221Search in Google Scholar

[9] Tayebi A., On the class of generalized Landsbeg manifolds, Periodica Math. Hungarica, 2016, 72, 29-36.10.1007/s10998-015-0108-xSearch in Google Scholar

[10] Izumi H., On ^*P-Finsler spaces I., Memoirs of the Defense Academy, Japan, 1977, 16, 133-138.Search in Google Scholar

[11] Chen BY., Deshmukh S., Ricci solitons and concurrent vector fields, Balkan Journal of Geometry and Its Applications, 2015, 20(1), 14-25.Search in Google Scholar

[12] Rastogi S. C., Dwivedi A. K., On the existence of concurrent vector fields in a Finsler space, Tensor (N.S.), 2004, 65 no.1, 48-54.Search in Google Scholar

Received: 2017-08-22

Accepted: 2017-10-12

Published Online: 2018-07-04

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0065

Keywords for this article

Generalized P-reducible metric; Randers metric; Landsberg metric

Creative Commons

BY-NC-ND 4.0