The present paper contains certain geometrical properties of a hypersurface of a Finsler space with Randers change of Matsumoto metric.
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 metric L(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 an n-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.
Section 2 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
Let Mn 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 in Fn 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 hypersurface Mn-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 hypersurface Fn-1 is a hyperplane of the first kind if and only if Hα=0 or equivalently H0=0.
Lemma 2.
A hypersurface Fn-1 is a hyperplane of the second kind if and only if Hαβ=0.
Lemma 3.
A hypersurface Fn-1 is a hyperplane of the third kind if and only if Hαβ=0=Mαβ.
4. Hypersurface Fn-1(c) of the Finsler Space with Randers Change of Matsumoto Metric
Let us consider the Randers change of Matsumoto metric L=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 of Fn-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.
Let Fn 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).
Along Fn-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 hypersurface Fn-1(c) of Finsler space with Randers change of Matsumoto metric, is given by (40), and the second fundamental h-tensor is symmetric.
Taking h-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|j is 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.
Since bi 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 hypersurface Fn-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 Lemma 2, 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 hypersurface Fn-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 hypersurface Fn-1(c) of Finsler space with Randers change of Matsumoto metric is not a hyperplane of the third kind.
Acknowledgment
The authors are thankful to the referee for his/her valuable suggestions.
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