On Tzitéeica type nonlinear equations for multiple soliton solutions in nonlinear optics

1.   Introduction

Finsler geometry extends the classical Riemannian geometry by considering more general metric structures. A very important class of Finsler metrics is known as $(\alpha, \beta)$-metrics, which were introduced by M. Matsumoto in $1972$. An $(\alpha, \beta)$-metric can be expressed as $F = \alpha \phi(s)$, where $\alpha$ is a Riemannian metric and $s = \frac{\beta}{\alpha}$, $\beta$ is a $1$-form. Randers metric, Kropina metric, exponential metric, Matsumoto metric, and cubic metric are important classes of $(\alpha, \beta)$-metric ^[9].

To study the curvature characteristics is a central problem in Finsler geometry. The Ricci curvature and S-curvature are very important non-Riemannian quantities in the Finslerian manifold ^[2]. The Ricci curvature in Finsler geometry is a natural extension of the Ricci curvature in Riemannian geometry and is defined as the trace of the Riemann curvature ^[5]. The S-curvature is a mathematical quantity and measures the rate of change of volume form of a Finsler space along the geodesics. Recent studies in differential geometry, such as those on Ricci solitons and conformal structures, have highlighted the importance of Ricci-type curvatures in understanding the geometric flow and structure of manifolds ^[6,7,8]. In Finsler geometry, the study of curvature involves understanding the deviation from flatness. The projective Ricci curvature is one aspect of this analysis. The concept of projective Ricci curvature in Finsler geometry is introduced by X. Cheng ^[1] in $2017$. Projective geometry deals with the properties that are invariant under projective transformations. The projective Ricci curvature measures the deviation of the Finsler metric from being projectively flat. Projective Ricci curvature has applications in various areas of mathematics and physics. It plays a crucial role in understanding the geometry of Finsler manifolds and connects to the problems in the calculus of variations, differential equations, and geometric optics.

In $2020$, H. Zhu ^[15] gave an expression of projective Ricci curvature for an $(\alpha, \beta)$-metric. Later on, many geometers ^[4,12,13] have studied the geometric properties of projective Ricci curvature. In this article, we obtain the geometric properties and flatness condition of projective Ricci curvature for the cubic Finsler metric, which is defined as $F = \alpha \phi(s)$ with

i.e., $F = \frac{(\alpha+\beta)^3}{\alpha^2}$. Cubic metric is Finsler metric for $b^2 < \frac{1}{4}$ ^[14].

The following notations will be used to state our main result:

where "$;$" denotes the covariant derivative with respect to the Levi-Civita connection of the Riemannian metric $\alpha$.

A $1$-form $\beta$ is said to be a Killing form if $r^{}_{ij} = 0$. The $1$-form $\beta$ is said to be a constant Killing form if it is a Killing form and constant length concerning $\alpha$, equivalently $r^{}_{ij} = 0$ and $s^{}_{i} = 0$.

In this paper we will use the following lemma:

Lemma 1.1. If $\alpha^2 = 0 (mod \beta)$, that is, $a_{ij}y^{i}y^{j}$ contains $b^{}_i(x)y^i$ as a factor, then the dimension is equal to two and $b^2$ vanishes. In this case, we have $\delta = d^{}_i(x)y^i$ satisfying $\alpha^2 = \beta\delta$ and $d^{}_ib^i = 2$.

We first prove the following result:

Theorem 1.1. For the cubic Finsler metric $F = \frac{(\alpha+\beta)^3}{\alpha^2}$ on an $n$-dimensional $(n > 2)$ Finsler manifold $M$, the S-curvature vanishes if and only if $\beta$ is a constant Killing form.

Next, we obtain the flatness condition for the projective Ricci curvature as

Theorem 1.2. If the $n$-dimensional $(n > 2)$ Finsler space with cubic metric $F = \frac{(\alpha+\beta)^3}{\alpha^2}$ is projective Ricci-flat $(PRic = 0),$ then $\beta$ is parallel with respect to the Riemannian metric $\alpha$.

In view of the above result, we obtain

Corollary 1.1. If the $n$-dimensional $(n > 2)$ Finsler space with cubic metric $F = \frac{(\alpha+\beta)^3}{\alpha^2}$ is projective Ricci flat then, it vanishes the S-curvature. Therefore, the Riemannian metric of $\alpha$ is Ricci flat $(Ric^{\alpha}$$=$$0)$.

We also prove the following result:

Theorem 1.3. The $n$-dimensional $(n > 2)$ Finsler space with cubic metric $F = \frac{(\alpha+\beta)^3}{\alpha^2}$ is weak PRic-curvature if and only if it is a PRic-flat metric.

2.   Preliminaries

Let $F$ be an $n$-dimensional Finsler manifold, and let $G^{j}$ be the geodesic coefficients of $F$, which are defined as

The geodesic coefficients of an $(\alpha, \beta)$-metric are given as ^[3]

where $G^{i}_{\alpha}$ denotes the geodesic coefficients of the Riemannian metric $\alpha$ and

For any $x\epsilon M$ and $y\epsilon T^{}_{x}M\backslash\{0\}$, the Riemann curvature $R^{}_{y}$ is defined as

where

The trace of Riemann curvature is called Ricci curvature $Ric = R^{m}_{m}$, which is a mathematical object that regulates the rate at which a metric ball's volume in a manifold grows. A Finsler metric $F$ is called an Einstein metric if Ricci curvature satisfies the equation $Ric(x, y) = (n-1)\gamma F^{2}$, where $\gamma = \gamma(x)$ is a scalar function.

In $1997$, Z. Shen ^[11] discussed S-curvature, which measures the average rate of change of $(T_{x}M; F_{x})$ in the direction $y \epsilon T^{}_{x}M$ and is defined as

where $\sigma ^{}_{F}$ is defined as

and Vol denotes the Euclidean volume, and $B^{n}(1)$ denotes the unit ball in $R^{n}$.

The expression of S-curvature for an $(\alpha, \beta)$-metric is given as ^[10]

where

The projective Ricci curvature is first defined by X. Cheng ^[1] as

where "$\vert$" denotes the horizontal covariant derivative with respect to the Berwald connections of $F$. A Finsler space $F$ is called weak projective Ricci curvature if

where $\gamma = \gamma(x)$ is a scalar function and $\theta = \theta_{i}(x)y^{i}$ is a $1$-form. If $\gamma =$ constant, then $F$ is called constant projective Ricci curvature. If $\theta = 0$, then $F$ is called isotropic projective Ricci curvature PRic = $(n-1)\gamma F^{2}$.

In $2020$, H. Zhu ^[15] gave an expression of the projective Ricci curvature for the $(\alpha, \beta)$-metrics as

where

3.   S-curvature of cubic metrics

For Eq (1.1), we obtain the following values:

By using Eqs (2.1) and (3.1), we obtain the spray coefficient $G^{j}$ for the cubic metric as

In view of Eqs (2.5) and (3.1) and using Mathematica program, we obtain the S-curvature for the cubic Finsler metric as

Now, we are in the position to prove Theorem 1.1.

Proof of Theorem 1.1. First we prove the converse part.

Let us assume that $\beta$ is a constant Killing form i.e., $s_{0} = 0$ and $r^{}_{00} = 0$; putting this in Eq (3.3) vanishes the S-curvature.

For the if part, let us take $S = 0$; then Eq (3.3) becomes

where

Taking the rational and irrational parts of Eq (3.4), we obtain

From Eqs (3.5) and (3.6), we can say that $\alpha^2$ will divide $t_{0}$ as well as $t_{1}$. In view of Lemma 1.1$, \alpha^2$ is coprime with $\beta$ for $n > 2$. Solving Eqs (3.5) and (3.6), we get, respectively,

and

From the above equations, we obtain

for some scalar function $c = c(x)$ on $M$.

Putting the above values in Eq (3.4) and simplifying, we get

where $(...)$ denotes the polynomial term in $\alpha$ and $\beta$. Here also $\alpha^{2}$ does not divide $\beta^{5}$ and $(c\beta+s^{}_{0})$. Therefore, $c\beta+s^{}_{0} = 0$. Differentiating it with respect to $y^{i}$, we obtain $c b^{}_{i}+s^{}_{i} = 0$, which, on contracting by $b^{i}$, gives $c = 0$, implying $s^{}_{0} = 0$ and $r^{}_{00} = 0$. Which means $\beta$ is a constant Killing form.

This completes, the proof of Theorem 1.1. □

4.   Ricci curvature of cubic metric

In this section we obtain the projective Ricci curvature for the aforesaid metric.

Proof of Theorem 1.2. For this, we first obtain all the values of Eq (2.8) by using Eq (3.1) and the Mathematica program as

Plugging all the values of the above Eq (4.1) into Eq (2.7) and simplifying by the using Mathematica program, we obtain the projective Ricci curvature for the aforesaid metric as

where

Next, we obtain the flatness condition under which the projective Ricci curvature vanishes.

Let the projective Ricci curvature $PRic = 0$, which implies $U(\alpha, \beta) = 0$, where

Using Mathematica, we can see that

Therefore

where $(....)$ are polynomial in $\alpha$ and $\beta$. As $B < \frac{1}{4}$, therefore $((1 + 6 B) \alpha^2-\alpha \beta-8 \beta^2)$ does not divide $(\alpha-2\beta)^3$ or $(\alpha+\beta)^2$ or $(\alpha+16\beta)^2$. Therefore $((1 + 6 B) \alpha^2-\alpha \beta-8 \beta^2)$ will divide $(r^{}_{00}(\alpha-2\beta)-6s^{}_{0}\alpha^{2})^2$; then $((1 + 6 B) \alpha^2-\alpha \beta-8 \beta^2)$ will also divide $(r^{}_{00}(\alpha-2\beta)-6s^{}_{0}\alpha^{2})$, i.e.,

where $c^{}_{1}$ is a $1$-form and $c^{}_{0}$ is a scalar. Taking the rational and irrational parts of the above equation, we obtain

and

Solving the above equations, we get $c^{}_{1} = -\frac{8}{5}\beta c^{}_{0}$, and then (4.5) gives

Substituting the above values into Eq (4.4), we obtain

Differentiating the above equation with respect to $y^{i}$ gives $(4B-1)c^{}_{0}b^{}_{i}+s^{}_{i} = 0$, which, on contracting by $b^{i}$, we obtain $c^{}_{0} = 0$. Then from Eqs (4.6) and (4.7), we obtain

In view of (4.8), Eq (4.3) becomes

which can be rewritten as

Since $(\alpha-2\beta)$ does not divide $\alpha^2$ or $\beta$, therefore $(\alpha-2\beta)$ will divide $s^{}_{0k}s^{k}_{0}$. Thus

where $d^{}_{1}$ is a $1$-form and $d^{}_{0}$ is a scalar. Taking the rational and irrational parts of the above equation and solving, we obtain

If $d^{}_{0} \neq 0$ then one can conclude by the above equation that $\alpha$ is not positive definite, which is not possible. Therefore, $d^{}_{0} = 0$. This implies that

i.e., $\beta$ is closed. In view of Eqs (4.8) and (4.10), we obtain $b^{}_{i; k} = 0$, then $1$-form $\beta$ is parallel with respect to $\alpha$.

This completes the proof of Theorem 1.2. □

Now, we obtain the condition for the weak projective Ricci curvature of a cubic Finsler metric.

Proof of Theorem 1.3. Let $F$ be a cubic Finsler metric with weak projective Ricci curvature. Then from Eq (2.6) we obtain

For the cubic metric, we have $B < \frac{1}{4}$, which implies that $\alpha^4$ does not divide $(\alpha\!-\!2\beta)^3$ or $((1\! +\! 6 B) \alpha^2 \!-\! \alpha \beta$$- 8 \beta^2)^4$ or $3\theta(\alpha\!+\!\beta)^3 \alpha^2$. Consequently, it follows that $\alpha^2$ must divide $\gamma(\alpha\!+\!\beta)^6$. However, such division is only possible if $\gamma = 0$. Combining this result with Eq (4.11), then we deduce that $3\theta(\alpha\!+\!\beta)^3$ is divided by $\alpha^2$. This is impossible unless $\theta = 0$. Then $F$ reduces to a projective Ricci-flat metric.

The converse is obvious. This completes the proof. □

Example 4.1. The Finsler metric $\frac{ 1}{|y|^2}\big(|y|+ < a, y > \big)^3$ for $a =$constant is projectively Ricci flat.

5.   Conclusions

Projective Ricci curvature is a concept in differential geometry that generalizes the notion of Ricci curvature. It has various applications in the fields of general relativity, optimal transformation theory, complex geometry, Weyl geometry, Einstein metrics, and many more. In this article, we have proved that if the cubic metric $F = \frac{(\alpha+\beta)^3}{\alpha^2}$ is projective Ricci flat $(PRic = 0),$ then $\beta$ is parallel with respect to Riemannian metric $\alpha$, and then from Eq (2.3), the S-curvature vanishes. Therefore, from Eq (2.5), we obtain that the Riemannian metric $\alpha$ is also Ricci-flat, which is Corollary 1.1.

Author contributions

M. K. Gupta and S. Sharma wrote the framework and the original draft of this manuscript. Y. Li and Y. Xie reviewed and validated the manuscript. All authors have read and agreed to the final version of the manuscript.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare no conflict of interest.