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Kyungpook Mathematical Journal 2024; 64(4): 579-590

Published online December 31, 2024 https://doi.org/10.5666/KMJ.2024.64.4.579

Copyright © Kyungpook Mathematical Journal.

Projective Changes Between the Polynomial (α, β)-metric and the Generalized Kropina Metric

Ramdayal Singh Kushwaha∗ and Renu

Department of Mathematics and Statistics, Banasthali Vidyapith, Jaipur 304022, India
e-mail : bhuramdayal@gmail.com and renu3119@gmail.com

Received: December 18, 2021; Revised: August 7, 2022; Accepted: August 23, 2022

In this paper, we look at the projective relativity of (α, β)-metrics. Let α and α¯ be Riemannian metrics, and β and β¯ be non-zero one-forms on a manifold M of dimension at least 2. We give necessary and sufficient conditions for an (α, β)-metric F and an (α¯, β¯)-metric ¯ F to be projectively related. In particular this gives conditions for the polynomial (α, β)-metric and a generalized Kropina (α¯, β¯)-metric to be projectively related. We then investigate these conditions in more detail when F is assigned special curvature properties.

Keywords: Finsler metric, (&alpha,, &beta,)-metric, Projective change, Douglas metric and S-curvature

Generalizing Rander's metric, Matsumoto [10] introduced the concept of an (α, β)-metric. The concept has been used in Biology and Physics as ecological metrics [1, 17] and has been extensively explored by many researchers [7, 9, 19] who have developed various (α, β)-metrics such as the polynomial (α, β)-metric, the Square metric, the generalized Kropina metric, the exponential metric, and the Matsumoto metric.

Projective relativity is an important topic in Finsler geometry. A Finsler metric is said to be projectively related or projectively equivalent to another Finsler metric if they have the same geodesics as point sets. In 1961, Rapscák [16] presented a crucial result associated to projective equivalence and established necessary and sufficient conditions for two metrics to be projectively related. Based on the concept of projective equivalence, H. S. Park and I. Y. Lee [13] derived a relation, on a a Finsler space, between an (α, β)-metric and the associated Riemannian metric. This replacing a metric with one that is projectively related is called projective change. Projective change between metrics of a Finsler space has been extensively investigated by many geometers [2, 5, 6, 8, 12, 20-22].

Here, in the present paper, we find equations to characterize the projective equivalence of a polynomial (α, β)-metric F=α+εβ+κβ2α and a generalized Kropina metric F¯=α¯m+1β¯m on a manifold M with dimension n>2, where α and α¯ are two Riemannian metrics, β and β¯ are two non-zero one-forms. Additionally, we characterized the projective equivalence of two metrics on F when F has some special curvature properties.

Suppose that M is an n-dimensional C manifold and TxM is the tangent space of M at a point x. The tangent bundle TM:=xMTxM on M is the union of the tangent spaces. The elements in TM are denoted by (x,y), where yTxM is known as the supporting element.

Definition 2.1. A Finsler metric on M is a continuous function F:TM[0,) with the following properties:

  • F(x,y) is C on TM\{0}.

  • Fx(y):=F(x,y) is a Minkowski norm on TxM, for any xM.

The pair (M,F)=Fn is called a Finsler space of dimension n, where F is the fundamental function and gij is the fundamental metric tensor. In [3], the fundamental metric tensor gij for the Finsler space Fn is the positive definite at every point of TM\{0}, where

gij(x,y):=12[F2]yiyj(x,y).

For a given Finsler metric F=F(x,y), the geodesics of Fn are given by system of differential equations [3]:

d2xidt2+2Gi(x,dxdt)=0,

where Gi=Gi(x,y) are known as the spray coefficients of F and are given by

Gi:=14gil(x){[F2]xmylym-[F2]xl}.

In Riemannian geometry, two Riemannian metrics α and α¯ are projectively related if and only if their spray coefficients are given by [5]:

Gαi=Gα¯i+λxkykyi,

where λ=λ(x) is a scalar function.

Two Finsler metrics F and F¯ are projectively related if and only if their spray coefficients have the relation [16]:

Gi=G¯i+P(x,y)yi,

where P(x,y) is a scalar function on TM\{0} that is homogeneous of degree one in y.

Definition 2.2. A Finsler metric is called a projectively flat metric if it is projectively related to a locally Minkowskian metric.

Definition 2.3.([3]) An (α, β)-metric expressed as

F=αϕ(s), s=βα,

is a regular (α, β)-metric if the function φ=φ(s) is a C-function on an open interval (-b0,b0) satisfying

ϕ(s)sϕ'(s)+(b2s2)ϕ''(s)>0, sb<b0.

where β satisfies βxα.1cmb0, for any xM.

In this case, the metric tensor induced by F is positive definite. Let rij:=12(bij+bji),sij:=12(bij-bji), where bij denote the coefficients of the covariant derivatives of β with respect to α. The one-form β is closed if and only if sij=0. Furthermore, we denote rji:=aikrkj,sji:=aikskj,r00:=rijyiyj,ri0:=rijyj,si:=bjsij,s0:=siyi,r:=rijbibj,si0=sijyj. In [3], the spray coefficients Gi of F are related to Gαi of α by:

Gi=Gαi+αQs0i+{-2Qαs0+r00}{Ψbi+Θα-1yi},

where

Q=φ'φ-sφ',Θ=φφ'-s(φφ''+φ'φ')2φ((φ-sφ')+(b2-s2)φ''),Ψ=φ''2((φ-sφ')+(b2-s2)φ'').

Definition 2.4. The tensor D:=Djkliidxjdxkdxl is known as Douglas tensor, where

Djkli:=3yjykyl(Gi-1n+1Gmymyi).

A Finsler metric is known as Douglas metric if its Douglas tensor vanishes.

Projective invariance is a property of the Douglas tensor [14]. Since the spray coefficients of a Riemannian metric are in quadratic form, Douglas tensor vanishes for the Riemannian metrics. So, the Douglas tensor is a non-Riemannian quantity. The fundamental fact is that all Berwald metrics must be Douglas metrics.

The Douglas tensor of an (α, β)-metric is determined by

Djkli=3yjykyl(Ti-1n+1Tmymyi),

where

Ti=αQs0i+Ψ{-2Qαs0+r00}bi,

and

Tymm=Q's0+Ψ'α-1(b2-s2)[r00-2Qαs0] +2Ψ[r0-Q'(b2-s2)s0-Qss0)].

For (α, β)-metrics F and F¯, let they have the same Douglas tensor, that is, Djkli=D¯jkli. Then from 2.6 and 2.7, we have

3yjykyl{Ti-T¯i-1n+1(Tymm-T¯ymm)yi}=0.

Thus, there exists a class of scalar functions Hjki:=Hjki(x), such that

Ti-T¯i-1n+1(Tymm-T¯ymm)yi=H00i,

where H00i:=Hjki(x)yjyk,Ti and Tymm are given by the equations (2.8) and 2.9 respectively.

In this section, we characterize the projective equivalence of the polynomial (α, β)-metric F=α+εβ+κβ2α and the generalized Kropina metric F¯=α¯m+1β¯m on a manifold with dimension n>2.

For the polynomial (α, β)-metric F=α+εβ+κβ2α, let b0=b0(k)>0 be the largest number such that 1+2κb2+3κs2>0,sb<b0. Then F is a Finsler metric if and only if one-form β satisfies the condition βα<b0, for any xM and the quantities given in 2.5 are:

Q=ε+2κs1-κs2,Θ=ε-3εκs2-4κ2s32(1+2κb2-3κs2)(1+εs+κs2),Ψ=κ(1+2κb2-3κs2).

Now, for the generalized Kropina metric F¯=α¯m+1β¯m, where m0,-1. It is easy to see that F¯ is not a regular (α, β)-metric but the relation given in 2.3 is still true for s>0. For this metric, the quantities in 2.5 are given by:

Q¯=-ms¯(m+1),Θ¯=-ms¯mb¯2-(m-1)s¯2,Ψ¯=m2{mb¯2-(m-1)s¯2}.

Lemma 3.1.([11]) A generalized Kropina metric F=αm+1βm,(n>2) with non-zero b2, is a Douglas metric if and only if

sij=1b2(bisj-bjsi).

First, we have the following lemma:

Lemma 3.2. Let F=α+εβ+κβ2α be a polynomial (α, β)-metric and F¯=α¯m+1β¯m be a generalized Kropina metric on a manifold M with dimension n>2, where α and α¯ are two Riemannian metrics, β and β¯ are two non-zero collinear one-forms. Then F and F¯ have the same Douglas tensor if and only if they are Douglas metrics.

Proof. The proof of sufficiency is easy. Now, we prove the necessary condition. If F and F¯ have the same Douglas tensor, then 2.10 holds. Substituting Q,Θ,Ψ from 3.1 and Q¯,Θ¯,Ψ¯ from 3.2 in 2.10, we get

H00i=Aiα7+Biα6+Ciα5+Diα4+Eiα3+Fiα2+JiNα6+Oα4+Pα2+R +A¯iα¯6+B¯iα¯4+C¯iα¯2+D¯i2β¯(N¯α¯4+O¯α¯2+P¯),

where

Ai=ε(1+2κb2){s0i(1+2κb2)-2κs0bi},Bi=κ(1+2κb2){r00bi-2λyi(r0+s0)+2βs0i(2κb2+1)-4κβbis0},Ci=-6εκβ{βs0i((1+2κb2))-κs0(2b2λyi+βbi)},Di=-2κ2β[-λyi{2b2κβ(7s0+r0)-3b2r00+4β(s0+r0)} +6β2s0i(1+2κb2)+βbi{r00(2+b2κ)-6βκs0}],Ei=3εκ2β3(-4λyis0+3βs0i),Fi=3κ2β3[2λyi{(b2κ+1)r00-κβ(r0+5s0)}+βκ(6βs0i+bir00)],Ji=-6κ3λyiβ5r00,A¯i=2m3b¯2(b¯2s¯0i-b¯is¯0),B¯i=m2β¯[2b¯2λyi{(3m-1)s¯0+(m+1)r¯0}-b¯i{(m+1)b¯2r¯00 -2β¯s¯0(m-1)}-4(m-1)b¯2β¯s¯0i],C¯i=mβ¯2[2λyi{(m2-1)b¯2r¯00-β¯(3m2-2m-1)s¯0+β¯(1-m2)r¯0} +β¯(m2-1)b¯ir¯00+2β¯2(m2-2m+1)s¯0i],D¯i=2m(1-m2)β¯4λr¯00yi,λ=1n+1,andN=(1+2κb2)2,O=-κβ2(1+2κb2)(7+2κb2),P=3κ2β4(5+4κb2),R=-9κ3β6,N¯=(m+1)m2b¯4,O¯=-2mb¯2(m2-1)β¯2,P¯=(m-1)2β¯4.

Now, the equation (3.3) becomes:

2β¯{H00i(Nα6+Oα4+Pα2+R) -(Aiα7+Biα6+Ciα5+Diα4+Eiα3+Fiα2+Ji)}(N¯α¯4+O¯α¯2+P¯) =(A¯iα¯6+B¯iα¯4+C¯iα¯2+D¯i)(Nα6+Oα4+Pα2+R).

It is clear that RHS of the above equation can be divided by β¯. Since β=μβ¯, therefore A¯iα¯6Nα6 can be divided by β¯. Because β¯ is prime with respect to α and α¯, therefore A¯i=2m3b¯2(b¯2s¯0i-b¯is¯0) can be divided by β¯. Hence, there is a scalar function ψi(x) such that

b¯2s¯0i-b¯is¯0=β¯ψi(x).

Contracting 3.4 by y¯i:=a¯ijy¯j, we get ψi(x)=-s¯i. Then, we have

s¯ij=1b¯2(b¯is¯j-b¯js¯i),

provided b¯20. Thus, by Lemma 3.1, F¯=α¯m+1β¯m is a Douglas metric. Since F and F¯ have the same Douglas tensor, they are Douglas metrics. Hence, the lemma is proved completely.

Now, we prove the following theorem:

Theorem 3.3. Let F=α+εβ+κβ2α and F¯=α¯m+1β¯m be two (α, β)-metrics on a manifold M with dimension n>2, where α and α¯ are two Riemannian metrics, β and β¯ are two non-zero collinear one-forms. Then, F is projectively related to F¯ if and only if the following relations hold

Gαi=G¯α¯i-2κτα2bi+ηα¯2s¯i+η{(m-1)α¯2s¯0+(m+1)r¯002}b¯i+θyi,bi|j=2τ{(1+2κb2)aij-3κbibj},s¯ij=1b2¯(b¯is¯j-b¯js¯i),

where bi:=aijbj,b¯i:=a¯ijb¯j,b¯2:=β¯α¯,τ:=τ(x) is a scalar function, and θ:=θiyi is a one-form on M.

Proof. First, we proof the necessity part. If F is projectively related to F¯, then they have the same Douglas tensor. By Lemma 3.2, F and F¯ are both Douglas metrics.

By [15], the polynomial (α, β)-metric F=α+εβ+κβ2α(n>2), is a Douglas metric if and only if

bi|j=2τ{(1+2κb2)aij-3κbibj},

for some scalar function τ=τ(x), where bi|j denote the coefficients of the covariant derivatives of β with respect to α. In this case, β is closed. If β is closed, then sij=0bi|j=bj|i. Thus, s0i=0 and s0=0.

By using 3.7, we have r00=2τ{(1+2κb2)α2-3κβ2}. Substituting all these in 2.4, we get

Gi=Gαi+(εα3-3εκαβ2-4κ2β3)α2+εαβ+κβ2τyi+2κτα2bi.

From Lemma 3.1, F¯=α¯m+1β¯m(n>2) is a Douglas metric if and only if

s¯ij=1b¯2(b¯is¯j-b¯js¯i).

On the other hand, putting Q¯,Θ¯,Ψ¯ and s¯ij from 3.2 and 3.9 into 2.4, we get

Gi¯=G¯α¯i+ηα¯2s¯i+η{(m-1)α¯2s¯0+(m+1)r¯002}b¯i -η{2ms¯0+(m+1)β¯α¯2r¯00}yi,

where

η=mα¯2(m+1)(mb¯2α¯2-(m-1)β¯2).

Since F and F¯ are projectively related, there is a scalar function P=P(x,y) on TM0 such that

Gi=G¯i+Pyi.

Using 3.8, 3.10 and 3.11, we have

[P-(εα3-3εκαβ2-4κ2β3)α2+εαβ+κβ2τ-η{2ms¯0+(m+1)β¯α¯2r¯00}]yi=Gαi-G¯α¯i+2κτα2bi-ηα¯2s¯i-η{(m-1)α¯2s¯0+(m+1)r¯002}b¯i.

The RHS of the above equation is a quadratic form in y. So, there exist a one-form θ=θiyi on M such that

P-(εα3-3εκαβ2-4κ2β3)α2+εαβ+κβ2τ-η{2ms¯0+(m+1)β¯α¯2r¯00}=θ.

Then, we have

Gαi=G¯α¯i-2κτα2bi+η{(m-1)α¯2s¯0+(m+1)r¯002}b¯i+ηα¯2s¯i+θyi.

From 3.7, 3.9 and 3.14, the proof of necessity part is completed.

Conversely, from 3.6, 3.8, and 3.10, we have

Gi=G¯i+[θ+(εα3-3εκαβ2-4κ2β3)α2+εαβ+κβ2τ +η{2ms¯0+(m+1)β¯α¯2r¯00}]yi,

this implies that F is projectively related to F¯.

From the above theorem, we get the following corollary:

Corollary 3.4. Let F=α+εβ+κβ2αand F¯=α¯m+1β¯m be two (α, β)-metrics on a manifold M with dimension n>2, where α and α¯ are two Riemannian metrics, β and β¯ are two non-zero one-forms. Then, F is projectively related to F¯ if and only if they are Douglas metrics and the spray coefficients of α and α¯ have the following relation

Gαi=G¯α¯i-2κτα2bi+ηα¯2s¯i+η{(m-1)α¯2s¯0+(m+1)r¯002}b¯i+θyi,

where bi:=aijbj,τ:=τ(x) is a scalar function, and θ:=θiyi is a one-form on M with dimension n>2.

It is well known that the Berwald curvature tensor of a Finsler metric F is defined by B:=Bjklidxjidxkdxl, where Bjkli=3Giyjykyl=[Gi]yjykyl and Gi are the spray coefficients of F.

A Finsler metric F is of isotropic Berwald curvature if

Bjkli=c(Fyjykδli+Fyjylδki+Fykylδji+Fyjykylyi),

where c=c(x) is a scalar function on M [3].

The mean Berwald curvature tensor is defined by E:=Eijdxidxj, where

Eij:=122yiyj(Gmym)=12Bmijm.

A Finsler metric is said to be of isotropic mean Berwald curvature if

Eij=n+12c(x)Fyiyj,

where c=c(x) is a scalar function on M [18]. Clearly, the Finsler metric of isotropic Berwald curvature must be of isotropic mean Berwald curvature.

In [3], let V be an n-dimensional real vector space and F be a Minkowski norm on V. For a basis ei of V, let

σF:=Vol(Bn)Vol(yi)RnF(yiei)<1,

where Vol means the volume of a subset in the standard Euclidean space Rn and Bn is the open ball of radius one. This quantity is generally dependent on the choice of the basis ei. But it is easily seen that

τ:=lndet(gij(y))σF,yV\{0},

is independent of the choice of the basis. τ=τ(y) is called the distortion of (V,F).

Now, let (M,F) be a Finsler space and τ(x,y) be the distortion of the Minkowski norm Fx on TxM. For yTxM\{0}, let τ(t) be the geodesic with τ(0)=x and τ˙(0)=y. Then the quantity

S(x,y):=ddt[τ(σ(t),σ˙(t))]t=0,

is called the S-curvature of the Finsler space (M,F). A Finsler space (M,F) is said to have almost isotropic S-curvature if there exists a smooth function c(x) on M and a closed one-form η such that:

S(x,y)=(n+1)(c(x)F(y)+η(y)),xM,yTxM.

In the above equation, if η=0, then (M,F) is said to have isotropic S-curvature i.e. S=(n+1)c(x)F, for some scalar function c(x) on M.

Lemma 4.1. ([4]) For (α, β)-metric F=α+εβ+κβ2α, where ε,κ are non-zero constants, the following are equivalent:

  • F is of isotropic S-curvature, that is, S=(n+1)c(x)F

  • F is of isotropic mean Berwald curvature, E=n+12c(x)F-1h

  • F has vanished S-curvature, that is, S=0

  • F is a weak Berwald metric, that is, E=0

  • β is a Killing 1-form of constant length with respect to α, that is, r00=0 and s0=0, where c=c(x) is a scalar function.

Theorem 4.2. Let F=α+εβ+κβ2α be projective equivalent to F¯=α¯m+1β¯m and F¯ has isotropic Berwald curvature. Then F has isotropic Berwald curvature if and only if F has isotropic S-curvature.

Proof. Suppose F has isotropic Berwald curvature, then F has isotropic mean Berwald curvature. Hence by Lemma 4.1, F is of isotropic S-curvature.

Conversely, suppose F and F¯ are projectively equivalent, that is, 2.1 holds and F has isotropic S-curvature, that is,

S=(n+1)c(x)F.

By Lemma 4.1, F is of isotropic mean Berwald curvature, that is,

Eij=n+12c(x)Fyiyj.

Given that F¯ has isotropic Berwald curvature, then

B¯jkli=c¯(F¯yjykδli+F¯yjylδki+F¯ykylδji+F¯yjykylyi),

where c¯=c¯(x) is a scalar function on M. Hence, by the definition of the mean Berwald tensor, it follows from 2.1 that cFyiyj=c¯F¯yiyj+Pyiyj, which gives that cFyiyjyk=c¯F¯yiyjyk+Pyiyjyk. Now, we have

Bjkli=3Giyjykyl=B¯jkli+(Pyjykδli+Pyjylδki+Pykylδji+Pyjykylyi) =c¯(F¯yjykδli+F¯yjylδki+F¯ykylδji+F¯yjykylyi) +(Pyjykδli+Pyjylδki+Pykylδji+Pyjykylyi) =c(Fyjykδli+Fyjylδki+Fykylδji+Fyjykylyi),

this implies that F has isotropic Berwald curvature. We complete the proof.

By the above methods, we could obtain the following theorem.

Theorem 4.3. Let F=α+εβ+κβ2α be projective equivalent to F¯=α¯m+1β¯m and F has isotropic Berwald curvature. Then F¯ has isotropic Berwald curvature if and only if F¯ has isotropic S-curvature.

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