We consider an n-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.

1. Introduction

The theory of an mth-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. Section 2 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

Let Fn=(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 of m=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 tensor gij 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 tensor gij, 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.

Take ρ-(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.

Differentiating μ-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-Randers mth-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-Randers mth-root space is given by (25).

Theorem 3.

In an R-Randers mth-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).

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 an R-Randers mth-root space, the angular metric tensor hij is given by (28).

Differentiating ρ-(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-Randers mth-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 of ai 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-Randers mth-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-Randers mth-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-Randers mth-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-Randers mth-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-Randers mth-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-Randers mth-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, differentiating ρ-(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).

Differentiating A0=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-Randers mth-root space are given by (57).

Acknowledgments

The authors are thankful to the reviewers for their valuable comments and suggestions. S. Saxena gratefully acknowledges the financial support provided by the UGC, Government of India.

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