On the Reversible Geodesics for a Finsler space with Randers change of Quartic metric

TL;DR

This paper investigates conditions under which a specific Finsler space with a Randers change of a quartic metric has reversible geodesics, exploring its geometric properties and the induced generalized distance.

Contribution

It provides new conditions for reversibility of geodesics in a Finsler space with a quartic Randers metric and studies related geometric properties.

Findings

Conditions for reversible geodesics are derived.

The Finsler metric induces a generalized weighted quasi-distance.

Geometric properties of the space with reversible geodesics are analyzed.

Abstract

In this paper, we consider a Finsler space with a Randers change of Quartic metric F = $\sqrt[4]{\alpha^4 + \beta^4} + \beta$. The conditions for this space to be with reversible geodesics are obtained. Further, we study some geometrical properties of F with reversible geodesics and prove that the Finsler metric F induces a generalized weighted quasi-distance $d_F$ on M.

Equations

$$L_F(\gamma )={\int }_0^{1}F(\gamma (t),\dot{\gamma }(t))dt,$$

$$G^i(x,y)={\Gamma }_{jk}^i(x,y)y^i{y}^j,$$

$$\displaystyle\tilde{\Gamma}\Bigl{(}\dfrac{\partial F}{\partial y^{i}}\Bigr{)}-\dfrac{\partial F}{\partial x^{i}}=0,$$

$$\displaystyle\tilde{\Gamma}\Bigl{(}\dfrac{\partial F}{\partial y^{i}}\Bigr{)}-\dfrac{\partial F}{\partial x^{i}}$$

$$\displaystyle=\tilde{\Gamma}\Bigl{(}F_{\alpha}\dfrac{\partial\alpha}{\partial y^{i}}+F_{\beta}\dfrac{\partial\beta}{\partial y^{i}}\Bigr{)}-F_{\alpha}\dfrac{\partial\alpha}{\partial x^{i}}-F_{\beta}\dfrac{\partial\beta}{\partial x^{i}}$$

$$\displaystyle=\tilde{\Gamma}(F_{\alpha})\dfrac{\partial\alpha}{\partial y^{i}}+F_{\alpha}\tilde{\Gamma}\Bigl{(}\dfrac{\partial\alpha}{\partial y^{i}}\Bigr{)}+\tilde{\Gamma}(F_{\beta})\Bigl{(}\dfrac{\partial\beta}{\partial y^{i}}\Bigr{)}+F_{\beta}\tilde{\Gamma}\Bigl{(}\dfrac{\partial\beta}{\partial y^{i}}\Bigr{)}-F_{\alpha}\dfrac{\partial\alpha}{\partial x^{i}}-F_{\beta}\dfrac{\partial\beta}{\partial x^{i}}$$

$$\displaystyle=\tilde{\Gamma}(F_{\alpha})\dfrac{\partial\alpha}{\partial y^{i}}+F_{\alpha}\Bigl{[}\tilde{\Gamma}\Bigl{(}\dfrac{\partial\alpha}{\partial y^{i}}\Bigr{)}-\dfrac{\partial\alpha}{\partial x^{i}}\Bigr{]}+\tilde{\Gamma}(F_{\beta})\dfrac{\partial\beta}{\partial y^{i}}+F_{\beta}\Bigl{[}\tilde{\Gamma}\Bigl{(}\dfrac{\partial\beta}{\partial y^{i}}\Bigr{)}-\dfrac{\partial\beta}{\partial x^{i}}\Bigr{]}.$$

$$\displaystyle\tilde{\Gamma}\Bigl{(}\dfrac{\partial F}{\partial y^{i}}\Bigr{)}-\dfrac{\partial F}{\partial x^{i}}$$

$$\frac{{\partial }^{2}F}{\partial x^m\partial y^k}{y}^m-\frac{\partial F}{\partial x^k}=0.$$

$$d_{\alpha }:M\times M⟶[0,\infty ),d_{\alpha }(x,y)=\underset{\gamma \in {\Gamma }_{xy}}{\inf }{\int }_a^b\alpha (\gamma (t),\dot{\gamma }(t))dt,$$

$$d_F:M\times M⟶[0,\infty ),d_F(x,y)=\underset{\gamma \in {\Gamma }_{xy}}{\inf }{\int }_a^{b}F(\gamma (t),\dot{\gamma }(t))dt.$$

$$d_F(x,y)$$

$$w_a(x)={\int }_{{\gamma }_{ax}}\beta -{\int }_{{\gamma }_{xa}}\beta =-2{\int }_{{\gamma }_{xa}}\beta ,$$

$$\displaystyle d_{F}(x,y)+w_{a}(x)=\int_{\gamma_{xy}}\Bigl{(}\sqrt[4]{(\alpha^{4}+\beta^{4})}\Bigr{)}-\int_{\gamma_{xa}}\beta-\int_{\gamma_{ya}}\beta,$$

$$\displaystyle d_{F}(y,x)+w_{a}(y)=\int_{\gamma_{yx}}\Bigl{(}\sqrt[4]{(\alpha^{4}+\beta^{4})}\Bigr{)}-\int_{\gamma_{ya}}\beta-\int_{\gamma_{xa}}\beta.$$

$$d(x,y)=\rho (x,y)+\frac{1}{2}[w(x)-w(y)],\forall x,y\in M,$$

$$\frac{1}{2}[w(x)-w(y)]\le \rho (x,y),\forall x,y\in M.$$

$$d_F(x,y)+d_F(y,z)+d_F(z,x)=d_F(x,z)+d_F(z,y)+d_F(y,x),\forall x,y,z\in M.$$

Full text

On the Reversible Geodesics for a Finsler space with Randers change of Quartic metric

Gauree Shanker, Ruchi Kaushik Sharma

Abstract. In this paper, we consider a Finsler space with a Randers change of Quartic metric F = $\sqrt[4]{\alpha^{4}+\beta^{4}}+\beta$. The conditions for this space to be with reversible geodesics are obtained. Further, we study some geometrical properties of F with reversible geodesics and prove that the Finsler metric F induces a generalized weighted quasi-distance $d_{F}$ on M.

M. S. C. 2010: 58B20, 53B21, 53B40, 53C60.

Keywords - Riemannian spaces; Reversible Geodesics; weighted quasi metric; Randers change; Quartic metric.

1 Introduction

An interesting topic in Finsler geometry is to study the reversible geodesics of a Finsler metric. Recall that, a Finsler space is said to have reversible geodesics if for every one of its oriented geodesic paths, the same path traversed in the opposite sense is also a geodesic. In the last decade many interesting and applicable results have been obtained on the theory of Finsler spaces with reversible geodesics. In [1], Crampin discussed Randers space with reversible geodesics. In ([4], [5]), Masca, Sabau and Shimada have discussed reversible geodesics with ($\alpha$, $\beta$)-metric and two dimensional Finsler space with ($\alpha$, $\beta$)-metric to be with reversible geodesic, respectively. In [6], Sabau and Shimada have given some important results on reversible geodesics. In [10], Shanker and Baby have discussed reversible geodesics for generalized ($\alpha,\beta$)-metric. Recently, Shanker and Rani [11] have studied weighted quasi metric associated with Finsler spaces with reversible geodesics.

In this paper, we find conditions for a Finsler space (M, F) with Randers change of Quartic metric F = $\sqrt[4]{\alpha^{4}+\beta^{4}}+\beta$ to be with reversible geodesics. The main results of this paper lies in theorem (3.1), (4.1), (4.2), (5.1) and (5.2).

2 Preliminaries

Let $F^{n}$ = ($M,F$) be a connected n-dimensional Finsler manifold and let TM = $\bigcup\limits_{x\in M}T_{x}M$ denotes the tangent bundle of M with local coordinates u = (x, y) = $(x^{i},y^{i})\in TM$, where i = 1, …, n, y = $y^{i}\dfrac{\partial}{\partial x^{i}}$.

If $\gamma$ : [0, 1] $\longrightarrow$ M is a piecewise $C^{\infty}$ curve on M, then its Finslerian length is defined as

[TABLE]

and the Finslerian distance function $d_{F}$ : M $\times$ M $\longrightarrow$ [0, $\infty$) is defined by $d_{F}(p,q)=inf_{\gamma}L$, where infimum is taken over all piecewise $C^{\infty}$ curves $\gamma$ on M joining the points p, q $\in$ M. In general, this is not symmetric.

A curve $\gamma$ : [0, 1] $\longrightarrow$ M is called a geodesic of (M, F) if it minimizes the Finslerian length for all piecewise $C^{\infty}$ curves that keep their endpoints fixed. We denote the reverse Finsler metric of F as $\tilde{F}$ : TM $\longrightarrow$ (0, $\infty$), given by $\tilde{F}(x,y)$ = $F(x,-y)$. One can easily see that $\tilde{F}$ is also a Finsler metric.

Lemma 2.1

A Finsler metric is with a reversible geodesic if and only if for any geodesic $\gamma$(t) of F, the reverse curve $\tilde{\gamma}(t)$ = $\gamma$(1-t) is also a geodesic of F.

Lemma 2.2

Let $(M,F)$ be a connected, complete Finsler manifold with associated distance function $d_{F}$ : M $\times$ M $\longrightarrow$ [0, $\infty$). Then, $d_{F}$ is a symmetric distance function on M $\times$ M if and only if F is a reversible Finsler metric, i.e., $F(x,y)=F(x,-y)$.

Lemma 2.3

A smooth curve $\gamma$ : [0, 1] $\longrightarrow$ M is a constant Finslerian speed geodesic of $(M,F)$ if and only if it satisfies $\ddot{\gamma}$ + 2 $G^{i}$($\gamma(t)$, $\dot{\gamma}(t)$) = 0, i = 1, …, n, where the functions $G^{i}$ : TM $\longrightarrow\mathbb{R}$ are given by

[TABLE]

with \Gamma_{jk}^{i}(x,y)=\dfrac{g^{is}}{2}\Bigl{(}\dfrac{\partial g_{sj}}{\partial x^{k}}+\dfrac{\partial g_{sk}}{\partial x^{j}}-\dfrac{\partial g_{jk}}{\partial x^{s}}\Bigr{)}.

Remark 1. It is well known [7] that the vector field $\Gamma$ = $y^{i}\dfrac{\partial}{\partial x^{i}}-2G^{i}\dfrac{\partial}{\partial y^{i}}$, is a vector field on TM, whose integral lines are the canonical lifts $\tilde{\gamma}(t)$ = ($\gamma$(t), $\dot{\gamma}(t)$) of the geodesics of $\gamma$. This vector field $\Gamma$ is called the canonical geodesics spray of the Finsler space $(M,F)$ and $G^{i}$ are called the coefficients of the geodesics spray $\Gamma$.

Definition 2.1

If F and $\tilde{F}$ are two different fundamental Finsler functions on the same manifold M, then they are said to be projectively equivalent if their geodesics coincide as set points.

Lemma 2.4

A Finsler structure ($M,F$) is with a reversible geodesic if and only if F and its reverse function $\tilde{F}$ are projectively equivalent.

3 Reversible Geodesics for a Finsler space with Randers change of Quartic metric.

Consider a Finsler space ($M,F$) with a special ($\alpha,\beta$)-metric F = $\sqrt[4]{\alpha^{4}+\beta^{4}}+\beta$. Here, F can be treated as the Randers change of Quartic-metric $\tilde{F}=\sqrt[4]{\alpha^{4}+\beta^{4}}$. One can easily see that $\tilde{F}(x,-y)=\tilde{F}(x,y)$.

As we know that [6] if $(M,F)$ is a non-Riemannian n(n $\geq$ 2)-dimensional Finsler space with ($\alpha,\beta$)-metric, which is not absolute homogeneous, then F is with reversible geodesics if and only if $F(\alpha,\beta)=F_{0}(\alpha,\beta)+\epsilon\beta$, where $F_{0}$ is absolute homogeneous ($\alpha,\beta$)-metric, $\epsilon$ is a non-zero constant and $\beta$ is a closed 1-form on the Manifold M.

In our case, $F_{0}$ = $\tilde{F}$, which is absolute homogeneous. If $\beta$ is a closed 1-form, then F is with reversible geodesics. Further, a necessary and sufficient condition for F to have reversible geodesics is that [6]

[TABLE]

where $\tilde{\Gamma}$ is the reverse of $\Gamma$, the geodesic spray of F; moreover $\tilde{\Gamma}$ is geodesic spray of $\tilde{F}$. We have F = $\tilde{F}$ + $\beta$, where, $\tilde{F}=\sqrt[4]{\alpha^{4}+\beta^{4}}$.

Therefore,

[TABLE]

For the Riemannian metric $\alpha$, the Euler-Lagrange equation gives \tilde{\Gamma}\Bigl{(}\dfrac{\partial\alpha}{\partial y^{i}}\Bigr{)}-\dfrac{\partial\alpha}{\partial x^{i}} = 0. Also, one knows [6] that if (M, F($\alpha,\beta$)) is a Finsler space with ($\alpha$, $\beta$)-metric, then $f(x,y)\dfrac{\partial\alpha}{\partial y^{i}}+g(x,y)b_{i}$ = 0, $\forall$ i = 1, 2, …, n, implies that f = g = 0, for any smooth functions f and g on TM. It is known that, if $\beta$ is closed and F is projectively equivalent to the Riemannian metric $\alpha$, then

$\tilde{\Gamma}(F_{\alpha})\dfrac{\partial\alpha}{\partial y^{i}}+\tilde{\Gamma}(F_{\beta})b_{i}$ = 0

and hence by using lemma 2.7 of [6], we find that $\tilde{\Gamma}(F_{\alpha})=0,\tilde{\Gamma}(F_{\beta})=0.$

Again, since F = $\sqrt[4]{\alpha^{4}+\beta^{4}}+\beta$, therefore $F_{\beta}=1+\dfrac{\beta^{3}}{(\alpha^{4}+\beta^{4})^{\frac{3}{4}}}$.

Now, using the above results, the equation (3.2) reduces to the form

[TABLE]

Now, $1+\dfrac{\beta^{3}}{(\alpha^{4}+\beta^{4})^{\frac{3}{4}}}$ can not be zero. Therefore, from equation (3.1) and (3.3) we conclude that F is with reversible geodesics if and only if \Bigl{(}\dfrac{\partial b_{i}}{\partial x^{j}}-\dfrac{\partial b_{j}}{\partial x^{i}}\Bigr{)}y^{j} = 0.

i.e., $\tilde{F}$ is with reversible geodesic if and only if $\beta$ is closed 1-form. Hence, we have the following theorem:

Theorem 3.1

A Finsler space $(M,F)$ with Randers change of Quartic metric F = $\tilde{F}+\beta,where,\tilde{F}=\sqrt[4]{(\alpha^{4}+\beta^{4})}$, is with reversible geodesics if and only if the differential 1-form $\beta$ is closed on M.

4 Projective Flatness of Randers change of Quartic metric

A Finsler space $(M,F)$ is called (locally) projectively flat if all its geodesics are straight lines [8]. An equivalent condition is that the spray coefficients $G^{i}$ of F can be expressed as $G^{i}=P(x,y)y^{i}$, where $P(x,y)$ = $\dfrac{1}{2F}\dfrac{\partial F}{\partial x^{k}}y^{k}$.

An equivalent characterization of projective flatness is the Hamel’s relation [2]

[TABLE]

Recall that ([10], [11]) if F = $F_{0}$ + $\epsilon$ $\beta$ is a Finsler metric, where $F_{0}$ is an absolute homogeneous ($\alpha$, $\beta$)-metric, then any two of the following properties imply the third one:

(1) F is projectively flat;

(2) $F_{0}$ is projectively flat;

(3) $\beta$ is closed.

In our case, F = $\tilde{F}$ + $\beta$, where $\tilde{F}=\sqrt[4]{(\alpha^{4}+\beta^{4})}$, which is absolute homogeneous. Hence, we have the following:

Theorem 4.1

Let $(M,F)$ be a Finsler space with Randers change of Quartic metric F = $\sqrt[4]{(\alpha^{4}+\beta^{4})}+\beta$. Then, F is projectively flat if and only if $\tilde{F}$ is projectively flat.

Proof: Let $(M,F)$ be projectively flat, then by Hamel’s relation for projective flatness, we have

[TABLE]

The proof of the theorem directly follows from it.

Theorem 4.2

Let $(M,F)$ be a Finsler space with Randers change of Quartic metric. If F is projectively flat, then it is with reversible geodesics.

Proof. Applying Hammel’s equation, one can easily see that F is projectively flat if and only if $\tilde{F}$ is projectively flat, which implies that both F and $\tilde{F}$ are projectively equivalent to the standard Euclidean metric and therefore F must be projective to $\tilde{F}$. Thus, F must be with a reversible geodesic.

5 Weighted quasi metric associated with Randers change of Quartic metric

It is well known that the Riemannian spaces can be represented as metric spaces. Indeed, for a Riemannian space (M, $\alpha$), one can define the induced metric space (M, $d_{\alpha}$) with the metric

[TABLE]

where $\Gamma_{xy}$ = {$\gamma$ : [a, b] $\longrightarrow$ M $\mid$ $\gamma$ is piecewise, $\gamma$(a) = x, $\gamma$(b) = y} is the set of curves joining x and y, $\dot{\gamma}(t)$ is the tangent vector to $\gamma$ at $\gamma$(t). Then $d_{\alpha}$ is a metric on M satisfying the following conditions:

1. Positiveness : $d_{\alpha}(x,y)$ $>$ 0, if x $\neq$ y, $d_{\alpha}$(x, x) = 0, x, y $\in$ X.

2. Symmetry : $d_{\alpha}$(x, y) = $d_{\alpha}$(y, x), $\forall$ x, y $\in$ M.

3. Triangle inequality: $d_{\alpha}$(x, y) $\leq$ $d_{\alpha}$(x, z) + $d_{\alpha}$(z, y), $\forall$ x, y, z $\in$ M.

Similar to the Riemannian space, one can induce the metric $d_{F}$ to a Finsler space (M, F), given by

[TABLE]

But unlike the Riemannian case, here $d_{F}$ lacks the symmetric condition. In fact, $d_{F}$ is a special case of quasi metric defined below:

Definition 5.1

*A quasi metric d on a set X is a function d : X $\times$ X $\longrightarrow$ [0, $\infty$) that satisfies the following axioms:

1. Positiveness : d(x, y) $>$ 0, if x $\neq$ y, d(x, x) = 0, x, y $\in$ X.

2. Triangle inequality : d(x, y) $\leq$ d(x, z) + d(z, y), $\forall$ x, y, z $\in$ X.

3. Separation axiom : d(x, y) = d(y, x) = 0 $\Rightarrow$ x = y, $\forall$ x, y $\in$ X.

One special class of quasi metric spaces are the so called weighted quasi metric spaces (M, d, w), where d is a quasi-metric on M and for each d, there exist a function w : M $\longrightarrow$ [0, $\infty$), called the weight of d that satisfies

4. Weightability : d(x, y) + w(x) = d(y, x) + w(y), $\forall$ x, y $\in$ M.

In this case, the weight function w is $\mathbb{R}$-valued, and is called generalized weight.

Theorem 5.1

Let (M, F) be an n-dimensional simply connected smooth Finsler manifold with F as Randers change of Quartic metric. Then, F induces generalized weighted quasi metric $d_{F}$ on M.

Proof. We consider that (M, F) is a Finsler space with F = $\beta+\sqrt[4]{(\alpha^{4}+\beta^{4})}$, which can be written as F = $\tilde{F}$ + $\beta$, where $\tilde{F}$ = $\sqrt[4]{(\alpha^{4}+\beta^{4})}$ is an absolute homogeneous Finsler metric on M and $\beta$ an exact 1-form.

Let $\gamma_{xy}$ $\in$ $\Gamma_{xy}$ be an F-geodesic, which is at the same time $\tilde{F}$-geodesic, then from equation (5.2), we get

[TABLE]

Consider a fixed point $a\in M$ and define the function $w_{a}:M\longrightarrow\mathbb{R}$ by $w_{a}(x):=d_{F}(a,x)-d_{F}(x,a)$.

From the equation (5.3) it follows that

[TABLE]

where we have used the Stokes theorem for the 1-form $\beta$ on the closed domain D with boundary $\partial$D := $\gamma_{ax}\bigcup\gamma_{xa}$.

It can be easily seen that $w_{a}$ is an anti-derivative of $\beta$. This is well defined if and only if the integral in the R.H.S. of equation(5.4) is path independent, i.e., $\beta$ must be exact.

Then $d_{F}$ is a weighted quasi-metric with generalized weight $w_{a}$. Next we have

[TABLE]

where we have again used the Stokes theorem for the one form $\beta$ on the closed domain with boundary $\gamma_{ax}\bigcup\gamma_{xy}\bigcup\gamma_{ya}$.

Similarly,

[TABLE]

From the equations (5.5) and (5.6) we conclude that $d_{F}$ is weighted quasi-metric with generalized weight $w_{a}$.

This completes the proof.

Next, recall the following:

Lemma 5.1

([3], [9]) Let (M, d) be any quasi-metric space. Then d is weightable if and only if there exists w : M $\longrightarrow$ [0, $\infty$) such that

[TABLE]

where $\rho$ is the symmetrized distance function of d. Moreover, we have

[TABLE]

The proof is trivial from the definition of weighted quasi-metric.

Remark 2. If (M, F) is a Finsler space with a special ($\alpha$, $\beta$)-metric F = $\sqrt[4]{(\alpha^{4}+\beta^{4})}+\beta$, then the induced quasi-metric $d_{F}$ and the symmetrized metric $\rho$ induce the same topology on M. This follows immediately from ([7], [8]).

Remark 3. From Lemma 5.1, It can be seen that the assumption of w to be smooth is not essential.

Next, we discuss an interesting geometric property concerning the geodesic triangles.

Theorem 5.2

Let (M, F) be a Finsler space with the Randers change of Quartic-metric F = $\sqrt[4]{(\alpha^{4}+\beta^{4})}+\beta$. Then the parameteric length of any geodesic triangle on M does not depend on the orientation, that is,

[TABLE]

Proof. Since the Randers change of Quartic metric F = $\sqrt[4]{(\alpha^{4}+\beta^{4})}+\beta$ can be treated as the Randers change of absolute homogeneous Finsler metric $\tilde{F}=\sqrt[4]{(\alpha^{4}+\beta^{4})}$, i.e., F = $\tilde{F}$ + $\beta$ with d$\beta$ = 0, from theorem 5.1, it follows that the quasi-metric is weightable and therefore equation (5.7) holds good. By using the formula (5.7), a simple calculation gives the required result.