# On a Hypersurface of a Finsler Space with Randers Change of Matsumoto Metric

Authors: M. K. Gupta; Abhay Singh; P. N. Pandey
Journal: Geometry
 (2013)

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## Abstract

The present paper contains certain geometrical properties of a hypersurface of a Finsler space with Randers change of Matsumoto metric.


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## Body

## 1. Introduction

The concept of Finsler spaces with(α,β)-metric was introduced by Matsumoto [1]. It was later discussed by several authors such as Shibata and others [2–4]. It has plenty of applications in various fields such as physics, mechanics, seismology, biology, and ecology [5–8]. Matsumoto introduced a special type of (α,β)-metric of the form L=α2/(α-β), α=aij(x)yiyj, and β=bi(x)yi, which is slope-of-a-mountain metric and is known as Matsumoto metric [9]. This metric has enriched Finsler geometry and it has provided researchers an important tool to work with significantly in this field [7, 10].A change of Finsler metricL(x,y)→L-(x,y)=L(x,y)+bi(x)yi is called Randers change of metric. The notion of a Randers change was proposed by Matsumoto, named by Hashiguchi and Ichijyo [11] and studied in detail by Shibata [12]. A Randers change of Matsumoto metric is given by L(x,y)=α2/(α-β)+β. Recently, Nagaraja and Kumar [13] studied the properties of a Finsler space with the Randers change of Matsumoto metric.Matsumoto [14] presented the theory of Finslerian hypersurface. The present authors (Gupta and Pandey [15, 16]) obtained certain geometrical properties of hypersurfaces of some special Finsler spaces. Singh and Kumari [17] discussed a hypersurface of a Finsler space with Matsumoto metric.In this paper, we consider ann-dimensional Finsler space Fn=(Mn,L) with the Randers change of Matsumoto metric L=α2/(α-β)+β and find certain geometrical properties of a hypersurface of the Finsler space with above metric. The paper is organized as follows.Section2 consists of Preliminaries relevant to the subsequent sections. The induced Cartan connection for hypersurface of a Finsler space is defined in Section 3. Necessary and sufficient conditions under which the hypersurface of the above Finsler space is a hyperplane of first, second and third kind are obtained in Section 4.


## 2. Preliminaries

LetMn be an n-dimensional smooth manifold and let Fn=(Mn,L) be an n-dimensional Finsler space equipped with Randers change of Matsumoto metric function
(1)L=α2α-β+β.
The derivative of above Randers change of Matsumoto metric with respect to α and β is given by
(2)Lα=α2-2αβ(α-β)2,Lβ=2α2+β2-2αβ(α-β)2,Lαα=2β2(α-β)3,Lββ=2α2(α-β)3,Lαβ=-2αβ(α-β)3,
where
(3)Lα=∂L∂α,Lβ=∂L∂β,Lαα=∂Lα∂α,Lββ=∂Lβ∂β,Lαβ=∂Lα∂β.
The normalized element of support li=∂˙iL is given by
(4)li=α-1Lαyi+Lβbi,
where yi=aijyj. The angular metric tensor hij=L∂˙i∂˙jL is given by
(5)hij=paij+q0bibj+q1(biyj+bjyi)+q2yiyj,
where
(6)p=LLαα-1=α3-α2β+2β3-3αβ2(α-β)3,q0=LLββ=2α2(α2+αβ-β2)(α-β)4,q1=LLαβα-1=-2β(α2+αβ-β2)(α-β)4,q2=Lα-2(Lαα-Lαα-1)=(α2+αβ-β2)(3β-α)α(α-β)4.
The fundamental metric tensor gij=(1/2)∂˙i∂˙jL2 is given by
(7)gij=paij+p0bibj+p1(biyj+bjyi)+p2yiyj,
where
(8)p0=q0+Lβ2=6α4-6α3β+6α2β2-4αβ3+β4(α-β)4,p1=q1+L-1pLβ=α(2α2+3β2-8αβ)(α-β)4,p2=q2+p2L-2=β(8αβ-2α2-3β2)α(α-β)4.
Moreover, the reciprocal tensor gij of gij is given by
(9)gij=p-1aij-s0bibj-s1(biyj+bjyi)-s2yiyj,
where
(10)bi=aijbj,s0={pp0+(p0p2-p12)α2}ζp,s1={pp1-(p0p2-p12)β}ζp,s2={pp2+(p0p2-p12)b2}ζp,b2=aijbibj,ζ=p(p+p0b2+p1β)+(p0p2-p12)(α2b2-β2).
The Cartan tensor Cijk=(1/2)∂˙kgij is given by
(11)2pCijk=p1(hijmk+hjkmi+hkimj)+γ1mimjmk,
where
(12)γ1=p∂p0∂β-3p1q0,mi=bi-α-2βyi.Let{jki} be the components of Christoffel symbols of the associated Riemannian space Rn and let ∇k be the covariant differentiation with respect to xk relative to this Christoffel symbols. We will use the following tensors:
(13)2Eij=bij+bji,2Fij=bij-bji,
where bij=∇jbi.If we denote the Cartan connection inFn as CΓ=(Fjki,Gji,Cjki), then the difference tensor Djki=Fjki-{jki} of the Finsler space Fn is given by
(14)Djki=BiEjk+FkiBj+FjiBk+Bjib0k+Bkib0j-b0mgimBjk-CjmiAkm-CkmiAjm+CjkmAsmgis+λs(CjmiCskm+CkmiCsjm-CjkmCmsi),
where
(15)Bk=p0bk+p1yk,Bi=gijBj,Fik=gkjFji,Bij={p1(aij-α-2yiyj)+(∂p0/∂β)mimj}2,Akm=BkmE00+BmEk0+BkF0m+B0Fkm,Bik=gkjBji,λm=BmE00+2B0F0m.
The suffix “0” denotes the transvection by the supporting element yi except for the quantities p0, q0, and s0.


## 3. Induced Cartan Connection

A hypersurfaceMn-1 of the underlying manifold Mn may be represented parametrically by xi=xi(uα), where uα are the Gaussian coordinates on Mn-1 (Latin indices run from 1 to n, while Greek indices take values from 1 to n-1). We assume that the matrix of projection factors Bαi=∂xi/∂uα is of rank n-1. If the supporting element yi at a point u=(uα) of Mn-1 is assumed to be tangent to Mn-1, we may then write yi=Bαi(u)vα so that v=(vα) is thought of as the supporting element of Mn-1 at the point uα. Since the function L_(u,v)=L(x(u),y(u,v)) gives rise to a Finsler metric on Mn-1, we get an (n-1)-dimensional Finsler space Fn-1=(Mn-1,L_(u,v)). The metric tensor gαβ and the Cartan tensor Cαβγ are given by
(16)gαβ=gijBαiBβj,Cαβγ=CijkBαiBβjBγk.
At each point uα of Fn-1, a unit normal vector Ni(u,v) is defined by
(17)gijBαiNj=0,gijNiNj=1.
For the angular metric tensor hij, we have
(18)hαβ=hijBαiBβj,hijBαiNj=0,hijNiNj=1.
The inverse projection factors Biα(u,v) of Bαi are defined as
(19)Biα=gαβgijBβj,
where gαβ is the inverse of the metric tensor gαβ of Fn-1.From (17) and (19), it follows that
(20)BαiBiβ=δαβ,BαiNi=0,NiBiα=0,NiNi=1,
and further
(21)BαiBjα+NiNj=δji.
For the induced Cartan connection ICΓ=(Fβγα,Gβα,Cβγα) on Fn-1, the second fundamental h-tensor Hαβ and the normal curvature vector Hα are given by
(22)Hαβ=Ni(Bαβi+FjkiBαjBβk)+MαHβ,Hα=Ni(B0αi+GjiBαj),
where Mα=CijkBαiNjNk, Bαβi=∂2xi/∂uα∂uβ, and B0αi=Bβαivβ. It is clear that Hαβ is not symmetric and
(23)Hαβ-Hβα=MαHβ-MβHα.
Equation (22) yields
(24)H0α=Hβαvβ=Hα,Hα0=Hαβvβ=Hα+MαH0.
The second fundamental v-tensor Mαβ is defined as:
(25)Mαβ=CijkBαiBβjNk.
The relative h- and v-covariant derivatives of Bαi and Ni are given by
(26)Bα|βi=HαβNi,Bαi|β=MαβNi,N|βi=-HαβBjαgij,Ni|β=-MαβBjαgij.
Let Xi(x,y) be a vector field of Fn. The relative h- and v-covariant derivatives of Xi are given by
(27)Xi|β=Xi|jBβj+Xi|jNjHβ,Xi|β=Xi|jBβj.Matsumoto [14] defined different kinds of hyperplanes and obtained their characteristic conditions, which are given in the following lemmas.Lemma 1.
A hypersurfaceFn-1 is a hyperplane of the first kind if and only if Hα=0 or equivalently H0=0.Lemma 2.
A hypersurfaceFn-1 is a hyperplane of the second kind if and only if Hαβ=0.Lemma 3.
A hypersurfaceFn-1 is a hyperplane of the third kind if and only if Hαβ=0=Mαβ.


## 4. HypersurfaceFn-1(c) of the Finsler Space with Randers Change of Matsumoto Metric

Let us consider the Randers change of Matsumoto metricL=α2/(α-β)+β with the gradient bi(x)=∂ib for a scalar function b(x) and a hypersurface Fn-1(c) given by the equation b(x)=c (constant). From parametric equation xi=xi(uα) of Fn-1(c), we get ∂αb(x(u))=0=biBαi, so that bi(x) are regarded as covariant components of a normal vector field of Fn-1(c). Therefore, along Fn-1(c), we have
(28)biBαi=0,biyi=0.
Therefore the induced metric L_(u,v) of Fn-1(c) is given by
(29)L_(u,v)=aαβvαvβ,aαβ=aijBαiBβj,
which is a Riemannian metric.At a point ofFn-1(c), from (6), (8), and (10), we have
(30)p=1,q0=2,q1=0,q2=-1α2,p0=6,p1=2α,p2=0,ζ=1+2b2,s0=2(1+2b2),s1=2α(1+2b2),s2=-4b2α2(1+2b2).
Therefore (9) gives
(31)gij=aij-2(1+2b2)bibj-2α(1+2b2)×(biyj+bjyi)+4b2α2(1+2b2)yiyj,
using (28) we get
(32)gijbibj=b21+2b2,
which gives
(33)bi(x(u))=b21+2b2Ni,
where b is the length of the vector bi. Using (31) and (33) we get
(34)bi=aijbj=b2(1+2b2)Ni+2b2α-1yi.Theorem 4.
LetFn be a Finsler space with Randers change of Matsumoto metric L=α2/(α-β)+β with a gradient bi(x)=∂ib(x) and let Fn-1(c) be a hypersurface of Fn, which is given by b(x)=c (constant). Then the induced metric on Fn-1(c) is Riemannian metric given by (29), and the scalar function b(x) is given by (33) and (34).AlongFn-1(c), the angular metric tensor and the metric tensor of Fn are given by
(35)hij=aij+2bibj-yiyjα2,(36)gij=aij+6bibj+2α(biyj+bjyi).
If hαβ(a) denote the angular metric tensor of the Riemannian metric aij(x), then, using (28), (35), and (18), we get
(37)hαβ=hαβ(a).
From (8), we get
(38)∂p0∂β=18α4-6α3β(α-β)5.
Thus, along Fn-1(c), ∂p0/∂β=18/α, and therefore (12) gives γ1=6/α,mi=bi. Then the Cartan tensor becomes
(39)Cijk=1α(hijbk+hjkbi+hkibj)+3αbibjbk,
and therefore, using (18), (25), and (28), we get
(40)Mαβ=1αb21+2b2hαβ,Mα=0,
and hence from (23) it follows that Hαβ is symmetric. Thus we have the following.Theorem 5.
The second fundamental v-tensor of Finsler hypersurfaceFn-1(c) of Finsler space with Randers change of Matsumoto metric, is given by (40), and the second fundamental h-tensor is symmetric.Takingh-covariant derivative of (28) with respect to the induced connection, we get
(41)bi|βBαi+biBα|βi=0.
Applying (27) for the vector bi, we get
(42)bi|β=bi|jBβj+bi|jNjHβ.
Using this and Bα|βi=HαβNi, (41) becomes
(43)bi|jBαiBβj+bi|jBαiNjHβ+biHαβNi=0.
Since bi|j=-bhCijh, using (33) and (40), we get
(44)bi|jBαiNj=-b21+2b2Mα=0.
Thus (43) gives
(45)b21+2b2Hαβ+bi|jBαiBβj=0.
Since Hαβ is symmetric, it is clear that bi|jis symmetric. Further contracting (45) with vβ and then with vα, we get
(46)b21+2b2Hα+bi|jBαiyj=0,b21+2b2H0+bi|jyiyj=0.
In view of Lemma 1, the hypersurface Fn-1(c) is a hyperplane of the first kind if and only if bi|jyiyj=0. Here bi|j being the covariant derivative with respect to the Cartan connection of Fn may depend on yi.Sincebi is a gradient vector, from (13), we have Eij=bij,Fij=0. Thus (14) reduces to
(47)Djki=Bibjk+Bjib0k+Bkib0j-b0mgimBjk-CjmiAkm-CkmiAjm+CjkmAsmgis+λs(CjmiCskm+CkmiCsjm-CjkmCmsi).
In view of (30) and (31), the relations in (15) become to
(48)Bi=6bi+2αyi,Bi=21+2b2bi+2α(1+2b2)yi,Bij=1αaij-1α3yiyj+9αbibj,Bji=gkiBkj,Akm=Bkmb00+Bmbk0,λm=Bmb00.
By virtue of (28) we have Bi0=0, B0i=0which leads A0m=Bmb00. Therefore we have
(49)Dj0i=Bibj0+Bjib00-BmCjmib00,D00i=Bib00=[2bi1+2b2+2yiα(1+2b2)]b00.
Using the relation (28), we get
(50)biDj0i=2b21+2b2bj0+1+9b2α(1+2b2)b00bj-21+2b2bmbiCjmib00,(51)biD00i=2b21+2b2b00.bi|j is the covariant derivative of bi with respect to xj relative to the Cartan connection of Fn, and bij=∇jbi is the covariant derivative of bi with respect to xj relative to the Riemannian connection:
(52)bi|j-bij=(∂jbi-Fijrbr)-(∂jbi-{ijr}br)=-(Fijr-{ijr})br=-Dijrbr,thatis,bi|j=bij-Dijrbr.
Using (51), we get
(53)bi|jyiyj=b00-D00rbr=11+2b2b00.
Consequently, (46) may be written as
(54)b21+2b2H0+11+2b2b00=0.
Thus the condition H0=0 is equivalent to b00=0, where bij does not depend upon yi. Since yi is to satisfy (28), the condition is written as bijyiyj=(biyi)(cjyj) for some cj(x), so that we have
(55)2bij=bicj+bjci.
Using (28), it follows that
(56)b00=0,bijBαiBβj=0,bijBαiyj=0.
Again (48) and (55) gives
(57)bi0bi=c0b22,λm=0,AjiBβj=0,BijBαiBβj=1αhαβ.
Using the (25), (31), (34), (40), and (47), we get
(58)brDijrBαiBβj=-c0b22α(1+2b2)2hαβ.
Therefore in view of (52), (45) reduces to
(59)b21+2b2Hαβ+c0b22α(1+2b2)2hαβ=0.Theorem 6.
The necessary and sufficient condition for the hypersurfaceFn-1(c) of Finsler space with Randers change of Matsumoto metric to be hyperplane of the first kind is (55) and in this case the second fundamental h-tensor of hypersurface Fn-1(c) is proportional to its angular metric tensor.In view of Lemma2, the hypersurface Fn-1(c) is a hyperplane of the second kind if and only if Hαβ=0. Thus from (59) we get c0=ci(x)yi=0. Therefore there exists a function e(x) such that ci(x)=e(x)bi(x). Thus (55) gives
(60)bij=ebibj.Theorem 7.
The necessary and sufficient condition for the hypersurfaceFn-1(c) of Finsler space with Randers change of Matsumoto metric, to be hyperplane of the second kind is (60).In view of (40) and Lemma 3, we have the following.Theorem 8.
The hypersurfaceFn-1(c) of Finsler space with Randers change of Matsumoto metric is not a hyperplane of the third kind.



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