# On an R-Randersmth-Root Space

Authors: P. N. Pandey; Shivalika Saxena
Journal: Geometry
 (2013)

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## Abstract

We consider ann-dimensional Finsler space Fn(n>2) with the metric L(x,y)=F(x,y)+α(x,y), where F is an mth-root metric and α is a Riemannian metric. We call such space as an R-Randers mth-root space. We obtain the expressions for the fundamental metric tensor, Cartan tensor, geodesic spray coefficients, and the coefficients of nonlinear connection in an R-Randers mth-root space. Some other properties of such space have also been discussed.


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## Body

## 1. Introduction

The theory of anmth-root metric was introduced by Shimada [1] in 1979. By introducing the regularity of the metric, various fundamental quantities of a Finsler metric could be found. In particular, the Cartan connection of a Finsler space with mth-root metric was introduced from the theoretical standpoint. Matsumoto and Okubo [2] studied Berwald connection of a Finsler space with mth-root metric and gave main scalars in two dimensional cases and defined higher-order Christoffel symbols. The mth-root metric is used in many problems of theoretical physics [3]. Pandey et al. [4] studied three-dimensional Finsler space with mth-root metric. To discuss general relativity with the electromagnetic field, Randers [5] introduced a metric of the form L(x,y)=α(x,y)+β(x,y), where α is a square root metric and β is a differential one form. In his honor, this metric is called Randers metric, and it has been extensively studied by several geometers and physicists [6–8].Munteanu and Purcaru [9] first defined complex Finsler spaces by reducing the scalar λ∈R in the homogeneity condition of fundamental function, that is, F(z,λη,z¯,λη¯)=|λ|F(z,η,z¯,η¯),∀λ∈R, and named such space as an R-complex Finsler space. Aldea and Purcaru [10] introduced the concept of R-complex Finsler spaces with (α,β)-metrics. In 2011, Purcaru [11] also discussed the notion of R-complex Finsler space with Kropina metric. Lungu and Nimineţ [12] studied a special Finsler space with the metric L of the form L(x,y)=F(x,y)+α(x,y),∀y∈TM, where F is a quartic root metric and α is a square root metric. They regarded this space as an R-Randers quartic space and obtained many results related to it. The aim of the present paper is to study a more general space with the metric L(x,y)=F(x,y)+α(x,y), where F is an mth-root metric and α is a Riemannian metric. We call the space endowed with this metric as an R-Randers mth-root space.The paper is organized as follows. Section2 deals with some preliminary concepts required for the discussion of the following sections. It includes the notion of an R-Randers mth-root space. In Section 3, we derive certain identities satisfied in an R-Randers mth-root space. We obtain the fundamental metric tensor gij, its inverse gij, and the Cartan tensor Cijk for an R-Randers mth-root space. In Section 4, we obtain the spray coefficients of an R-Randers mth-root space. It includes the equations of the geodesics. Section 5 discusses the nonlinear connection in an R-Randers mth-root space.


## 2. Preliminaries

LetFn=(M,L(x,y))(n>2) be an n-dimensional Finsler space. The mth-root metric on M is defined as Lm=ai1i2⋯im(x)yi1yi2⋯yim, where ai1i2⋯im(x) are components of an mth-order covariant symmetric tensor.In case ofm=2, the metric L is Riemannian, and in the cases m=3 and m=4 these metrics are called cubic and quartic, respectively.The covariant symmetric metric tensorgij of Fn(M,L(x,y)) is defined by
(1)gij=:12∂˙i∂˙jL2,∂˙i≡∂∂x˙i.
This tensor is positively homogeneous of degree zero in x˙i. From the metric tensor gij, we construct the Cartan tensor Cijk by
(2)Cijk=:12∂˙kgij.
The tensor Cijk is symmetric in its lower indices and is positively homogeneous of degree -1 in x˙i. Due to its homogeneous and symmetric properties, it satisfies the following:
(3)Cijkx˙i=Ckijx˙i=Cjkix˙i=0.
The angular metric tensor of a Finsler space is written as
(4)hij=gij-lilj,
where li=∂˙iL.The geodesic of a Finsler space is given by(5)d2xidt2+2Gi=0,
where Gi is the geodesic spray coefficients given by
(6)2Gi=12gij{yk∂˙j∂kL2-∂jL2},∂j≡∂∂xj.
The nonlinear connection of a Finsler space is defined as
(7)Nji=∂˙jGi.
In the present paper, we study the space whose fundamental function is given by
(8)L(x,y)=F(x,y)+α(x,y),
where
(9)Fm=ai1i2⋯im(x)yi1yi2⋯yim
is  an  mth-root  metric and
(10)α2=bi1i2(x)yi1yi2
is a Riemannian metric. We call this space an R-Randers mth-root space.


## 3. Fundamental Metric Tensor and Cartan Tensor

In this section, we find the fundamental metric tensorgij, its inverse gij, angular metric tensor hij, and the Cartan tensor Cijk for an R-Randers mth-root space.Differentiating (9) partially with respect to x˙i, we get
(11)Fm-1(∂˙iF)=ai,
where ai(x,y)=aii2⋯im(x)yi2yi3⋯yim.Differentiating (11) partially with respect to x˙j, we find
(12)(m-1)Fm-2(∂˙jF)(∂˙iF)+Fm-1(∂˙j∂˙iF)=(m-1)aij,
where aij(x,y)=aiji3⋯im(x)yi3yi4⋯yim.Takeρ-(m-1)=(m-1)F-(m-1) and ρ-(2m-1)=(m-1)F-(2m-1), where the subscripts denote the degree of homogeneity of the corresponding entities with respect to x˙i. In view of this, (11) and (12) give
(13)∂˙iF=ρ-(m-1)(m-1)ai,(14)∂˙j∂˙iF=ρ-(m-1)aij-ρ-(2m-1)aiaj,
where
(15)∂˙jρ-(m-1)=-(m-1)ρ-(2m-1)aj.
Now, differentiating (10) partially with respect to x˙i, we get
(16)∂˙iα=μ-1bi,
where bi(x,y)=bii2(x)yi2 and μ-1=α-1.Further differentiating (16) partially with respect to x˙j, we find
(17)∂˙j∂˙iα=μ-1bij+μ-3bibj,
where μ-3=-α-3.Differentiatingμ-1=α-1 partially with respect to x˙j, we get
(18)∂˙jμ-1=μ-3bj.
Thus, we have the following.Proposition 1.
In an R-Randersmth-root space, the following identities hold:
(19)∂˙iF=ρ-(m-1)(m-1)ai,∂˙j∂˙iF=ρ-(m-1)aij-ρ-(2m-1)aiaj,∂˙iα=μ-1bi,∂˙j∂˙iα=μ-1bij+μ-3bibj,∂˙jρ-(m-1)=-(m-1)ρ-(2m-1)aj,∂˙jμ-1=μ-3bj.Differentiating (8) partially with respect to x˙i and using (13) and (16), we have
(20)∂˙iL=ρ-(m-1)(m-1)ai+μ-1bi.
Further, differentiating (20) partially with respect to x˙j and using (14) and (17), we get
(21)∂˙j∂˙iL=ρ-(m-1)aij-ρ-(2m-1)aiaj+μ-1bij+μ-3bibj.
From (1), we have
(22)gij=(∂˙iL)(∂˙jL)+L(∂˙i∂˙jL).
Using (20) and (21), (22) yields
(23)gij=(ρ-(m-1)(m-1)ai+μ-1bi)(ρ-(m-1)(m-1)aj+μ-1bj)+L(ρ-(m-1)aij-ρ-(2m-1)aiaj+μ-1bij+μ-3bibj).
Let us take dij=ρ-(m-1)aij+μ-1bij and ci=q0ai+q-1bi, where q0 and q-1 satisfy
(24)q02=(ρ-(m-1)2(m-1)2-Lρ-(2m-1)),q-12=(μ-12+Lμ-3),q0q-1=ρ-(m-1)(m-1)μ-1.
The matrix (dij) is nonsingular, in general, for aij's are functions of x and y whereas bij's are functions of x only.Equation (23) takes the form
(25)gij=Ldij+cicj.
Thus, we have the following.Theorem 2.
The fundamental metric tensor of an R-Randersmth-root space is given by (25).Theorem 3.
In an R-Randersmth-root space, the inverse gij of the fundamental metric tensor gij is given by
(26)gij=1L(dij-1L+c2cicj),
where ci=dijcj and c2=cici.Proof.
Let(dij) be the inverse of nonsingular matrix (dij). Suppose that (gij) is given by (26).
Now,(27)gijgjk=(Ldij+cicj)1L(djk-1L+c2cjck)=dijdjk-dijL+c2cjck+1Lcicjdjk-1L(L+c2)cicjcjck=δki-cickL+c2+1Lcick-1L(L+c2)c2cick=δki.
Therefore,(gij) given by (26) is the inverse of the matrix (gij). This also shows that (gij) is nondegenerate.Using (20) and (23) in (4), we get the angular metric tensor of an R-Randers mth-root space:
(28)hij=L(ρ-(m-1)aij-ρ-(2m-1)aiaj+μ-1bij+μ-3bibj).
Thus, we have the following.Theorem 4.
In anR-Randers mth-root space, the angular metric tensor hij is given by (28).Differentiatingρ-(2m-1)=(m-1)F-(2m-1) partially with respect to x˙k, we get
(29)∂˙kρ-(2m-1)=-(2m-1)ρ-(3m-1)ak,
where ρ-(3m-1)=(m-1)F-(3m-1).Also, we have(30)∂˙kμ-3=3μ-5bk,
where μ-5=α-5.Thus, we have the following.Proposition 5.
In an R-Randersmth-root space, the following holds good:
(31)∂˙kρ-(2m-1)=-(2m-1)ρ-(3m-1)ak,∂˙kμ-3=3μ-5bk.Differentiating (23) partially with respect to x˙k, we get
(32)∂˙kgij=2ρ-(m-1)(∂˙kρ-(m-1))aiaj+(ρ-(m-1))2(aj∂˙kai+ai∂˙kaj)+[(∂˙kρ-(m-1))μ-1+ρ-(m-1)(∂˙kμ-1)](biaj+bjai)+ρ-(m-1)μ-1[aj∂˙kbi+bi∂˙kaj+ai∂˙kbj+bj∂˙kai]+2μ-1(∂˙kμ-1)bibj+μ-12[bj∂˙kbi+bi∂˙kbj]+(∂˙kL)[ρ-(m-1)aij-ρ-(2m-1)aiaj+(∂˙kL)+μ-1bij+μ-3bibj]+L[(∂˙kρ-(m-1))aij+(∂˙kaij)ρ-(m-1)+L-(∂˙kρ-(2m-1))aiaj-ρ-(2m-1)+L×((∂˙kai)aj+(∂˙kaj)ai)+(∂˙kμ-1)bij+L+(∂˙kbij)μ-1+(∂˙kμ-3)bibj+μ-3(∂˙kbi)bj+(∂˙kbj)bi].Partial differentiation ofai and bi, with respect to x˙k, yields
(33)∂˙kai=(m-1)aik,∂˙kbi=bik.
Further, differentiation of aij and bij with respect to x˙k gives
(34)∂˙kaij=(m-2)aijk,∂˙kbij=0,
where aijk(x,y)=aijki4⋯im(x)yi4⋯yim.Thus, we have the following.Proposition 6.
In an R-Randersmth-root space the following holds good:
(35)∂˙kai=(m-1)aik,∂˙kaij=(m-2)aijk,∂˙kbi=bik,∂˙kbij=0.If we use (2), (15), (18), (20), (29), (30), (33), and (34) in (32), on simplification it follows that
(36)2Cijk=ρ-(2m-2)akaij+ξ-(2m-2)(ajaik+aiajk)+μ-2(bibjk+bjbik+bkbij)+μ-4bibjbk+ρ-(3m-2)aiajak+ρ-(m-2)aijk+ρ-(m-1)μ-1×{1m-1akbij+bkaij+ajbik+(m-1)(biajk+bjaik)+aibjk1m-1}-(m-1)ρ-(2m-1)μ-1(biaj+bjai)ak+ρ-(m-1)μ-3(biaj+bjai)bk,
where
(37)ρ-(2m-2)=ρ-(m-1)2m-1-L(m-1)ρ-(2m-1),ξ-(2m-2)=(m-1)(ρ-(m-1)2-Lρ-(2m-1)),μ-2=(μ-12+Lμ-3),μ-4=(3Lμ-5+2μ-1μ-3),ρ-(m-2)=L(m-2)ρ-(m-1),ρ-(3m-2)=(2m-1)Lρ-(3m-1)-2(m-1)ρ-(m-1)ρ-(2m-1).
Thus, we have the following.Theorem 7.
In an R-Randersmth-root space, the Cartan tensor Cijk is given by (36).


## 4. Spray and Equation of Geodesics

In this section, we discuss the spray of an R-Randersmth-root space and obtain its local coefficients. We also obtain the equation of geodesics in such space.If we differentiate (9) partially with respect to xj, we get
(38)∂jF=ρ-(m-1)m(m-1)Aj,
where
(39)Aj=(∂jai1⋯im)yi1⋯yim.
Further, differentiating (13) partially with respect to xk and utilizing ∂kρ-(m-1)=(-(m-1)/m)ρ-(2m-1)Ak, we have
(40)∂k∂˙iF=-1mρ-(2m-1)Akai+ρ-(m-1)(m-1)∂kai.
Differentiating (10) partially with respect to xj, we get
(41)∂jα=12μ-1Bj,
where
(42)Bj=(∂jbi1i2)yi1yi2.
Also, we have
(43)∂kμ-1=12μ-3Bk.
Next, differentiating (16) partially with respect to xk and using (43), we obtain
(44)∂k∂˙iα=12μ-3biBk+μ-1∂kbi.
Thus, we have the following.Proposition 8.
In an R-Randersmth-root space, the following holds good:
(45)∂jF=ρ-(m-1)m(m-1)Aj,∂jα=12μ-1Bj,∂k∂˙iF=-1mρ-(2m-1)Akai+ρ-(m-1)(m-1)∂kai,∂k∂˙iα=12μ-3biBk+μ-1∂kbi.If we differentiate (8) partially with respect to xk and use (38) and (41), it follows that
(46)∂kL=ρ-(m-1)m(m-1)Ak+12μ-1Bk.
Next, differentiating (20) partially with respect to xk and using (40) and (44), we get
(47)∂k∂˙iL=-1mρ-(2m-1)Akai+ρ-(m-1)(m-1)∂kai+12μ-3biBk+μ-1∂kbi.
In view of (46) and (47), (6) gives
(48)Gi=14gij[2m(ρ-(m-1)2(m-1)2-Lρ-(2m-1))ajAkyk+(μ-12+Lμ-3)bjBkyk+ρ-(m-1)(m-1)μ-1(2bjAk+aJBk)yk+2Lρ-(m-1)(m-1)(∂kaj)yk+2Lμ-1(∂kbj)yk-2Lρ-(m-1)(m-1)Aj-Lμ-1Bj{(ρ-(m-1)2(m-1)2-Lρ-(2m-1))}],
that is
(49)Gi=14gij[2m(m-1)ρ-(2m-2)ajA014gij+μ-2bjB0+1m-1ρ-(m-1)μ-1(2bjA0+ajB0)14gij+2(m-1)(m-2)ρ-(m-2)((∂kaj)yk-Aj)14gij+η-2(2(∂kbj)yk-Bj){2m(m-1)}],
where
(50)A0=Akyk,B0=Bkyk,η-2=Lμ-1,
and gij is given by (26). Thus, we have the following.Theorem 9.
In an R-Randersmth-root space, the spray coefficients are given by (49).In view of (5) and Theorem 9, we have the following.Corollary 10.
In an R-Randersmth-root space, the equation of geodesics is given by
(51)d2xidt2+2Gi=0,
where the spray coefficients Gi are given by (49).


## 5. Nonlinear Connection

In this section, we obtain the coefficients of nonlinear connection of the space under consideration.Differentiating (49) partially with respect to x˙k, we have
(52)∂˙kGi=14(∂˙kgij)[2m(m-1)ρ-(2m-2)ajA0+μ-2bjB014(∂˙kgij)+1m-1ρ-(m-1)μ-1(2bjA0+ajB0)14(∂˙kgij)+2(m-1)(m-2)ρ-(m-2)((∂kaj)yj-Aj)14(∂˙kgij)+η-2(2(∂kbj)yk-Bj){2m(m-1)}]+14gij[2m(m-1){(∂˙kρ-(2m-2))ajA0+14gij2m(m-1)+ρ-(2m-2)(A0∂˙kaj+aj∂˙kA0)}+14gij+(∂˙kμ-2)bjB0+μ-2(bj∂˙kB0+B0∂˙kbj)+14gij+1m-1{(∂˙kρ-(m-1))μ-1+14gij+1m-1+ρ-(m-1)(∂˙kμ-1)}(2bjA0+ajB0)+14gij+1m-1ρ-(m-1)μ-1+14gij×{2(∂˙kbj)A0+2bj(∂˙kA0)+14gij+(∂˙kaj)B0+aj(∂˙kB0)}+14gij+2(m-1)(m-2)+14gij×{(∂˙kρ-(m-2))((∂laj)yl-Aj)+ρ-(m-2)+14gij×((∂˙k∂laj)yl+(∂laj)(∂˙kyl)-∂˙kAj)}+14gij+(∂˙kη-2)(2(∂kbj)yk-Bj)+η-2+14gij×(2(∂˙k∂kbj)yk+2(∂kbj)(∂˙kyk)-∂˙kBj){2m(m-1)}].
Differentiation of gijgjm=δmi partially with respect to x˙k yields
(53)∂˙kgil=-2Ckmigml,
where Ckmi=Cjkmgij.Next, differentiatingρ-(2m-2)=(ρ-(m-1)2/(m-1))-L(m-1)ρ-(2m-1) partially with respect to x˙k and using (15), (20), and (29), we get
(54)∂˙kρ-(2m-2)=(m-1)(-3ρ-(3m-2)ak-ρ-(2m-1)μ-1bk+(2m-1)Lρ-(3m-1)ak),
where ρ-(3m-2)=(m-1)F-(3m-2).DifferentiatingA0=Alyl and B0=Blyl partially with respect to x˙k, we, respectively, have
(55)∂˙kA0=m(∂lak)yl+Ak,∂˙kB0=2(∂lbk)yl+Bk.
Also, we have
(56)∂˙k(∂laj)=(m-1)∂lajk,∂˙k(∂lbj)=∂lbjk,∂˙kAj=m∂jak,∂˙kBj=2∂jbk.
Using (7), (15), (18), (29), (30), (33), and (53)–(56) in (52), we have
(57)Nki=-2CkmiGm+14gij[2m(-3ρ-(3m-2)-(2m-1)ρ-(3m-1))akajA0+14gij-ρ-(2m-1)μ-1(bkajA0+2akbjA0+akajB0)+14gij+2ρ-(2m-2)m(m-1)(m(∂lak)ylaj+14gij+2ρ-(2m-2)m(m-1)+Akaj+(m-1)AjkA0+14gij+2ρ-(2m-2)m(m-1)+(∂laj)ylak-Ajak)+14gij+(3μ-4-μ-1μ-3)bkbjB0+14gij+1(m-1)ρ-(m-1)μ-3(akbjB0+14gij+1(m-1)ρ-(m-1)μ-3+(2bjA0+ajB0)bk)+14gij+μ-2(bjkB0+2(∂lbk)ylbj+14gij+μ-2+Bkbj+bk(2(∂lbj)yl-Bj))+14gij+ρ-(m-1)μ-1(m-1){2bjkA0+2mbj(∂lak)yl+14gij+ρ-(m-1)μ-1(m-1)+2bjAk+(m-1)ajkB0+14gij+ρ-(m-1)μ-1(m-1)+2aj(∂lbk)yl+Bkaj+14gij+ρ-(m-1)μ-1(m-1)+2(∂lbj)ylak-Bjak+14gij+ρ-(m-1)μ-1(m-1)+2(∂laj)ylbk-2Ajbk}+14gij+2ρ-(m-2)(m-1)(m-2){(m-1)(∂lajk)yl+14gij+2ρ-(m-2)(m-1)(m-2)+(∂kaj)-m(∂jak)}+14gij+η-2{2(∂lbjk)yl+2(∂kbj)-(∂jbk)}{2m(-3ρ-(3m-2)-(2m-1)ρ-(3m-1))}].Thus, we have the following.Theorem 11.
The local coefficients of the nonlinear connection of an R-Randersmth-root space are given by (57).



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