On Ricci Curvature of a Homogeneous Generalized Matsumoto Finsler Space

Abstract

The characterization of Finsler spaces with Ricci curvature is an ancient and cumbersome one. In this paper, we have derived an expression of Ricci curvature for the homogeneous generalized Matsumoto change. Moreover, we have deduced the expression of Ricci curvature for the aforementioned space with vanishing the S-curvature. These findings contribute significantly to understanding the complex nature of Finsler spaces and their curvature properties.

1. Introduction

In the past several years, we have witnessed a rapid development in Finsler geometry. Various curvatures have been studied and their geometric meanings are better understood. The $(\alpha ,\beta )$-metrics form an important class of Finsler metrics appearing iteratively in formulating Physics, Mechanics and Seismology, Biology, etc. This metric is firstly introduced by M. Matsumoto [1] in 1972. It is a scalar function on the tangent bundle $TM$, expressed in terms of a Riemannian metric $\alpha ={\left(a_{jk}\left(x\right)y^j{y}^k\right)}^{1/2}$ and a 1-form $\beta =b_k{y}^k$. The scalar function can be written as $F=\alpha \varphi \left(s\right)$, where $s=\beta /\alpha$. The Randers metric $(\alpha +\beta )$, exponential metric $\left(e^{\beta /\alpha }\right)$, Kropina metric ($\frac{{\alpha }^2}{\beta }$) and Matsumoto metric ($\frac{{\alpha }^2}{\alpha -\beta }$) are special and significant $(\alpha ,\beta )$-metrics which contribute a lot in the theoy of Finsler spaces. Among them, the Kropina metric and the Matsumoto metric are singular metrics. The generalization of standard Kropina metrics is called m-Kropina metrics ($\frac{{\alpha }^{m+1}}{{\beta }^m}$). It has notorious examples including Bogoslovsky–Kropina metrics, which represent the framework for very special relativity (VSR) and its generalization, very general relativity (VGR) [2]. The generalized Matsumoto metric $(\frac{{\alpha }^{m+1}}{{(\alpha -\beta )}^m})$ is studied by G. Shankar [3] and M. K. Gupta [4], etc. This metric has also numerous applications in physics such as the study of the slope of mountains [5]. In this article, we have also discussed the generalized Matsumoto metric which can be also written as $F=\alpha \varphi \left(s\right)$ with

$$\varphi \left(s\right)=\frac{1}{{(1-s)}^m},$$

(1)

where $m\ne 0,-1$, and $s\ne 1$.

Nowadays, Symmetric Finsler space and Homogeneous Finsler space are the active areas of research in the Finslerian Manifold. Many geometers discussed [6,7,8,9,10] these spaces by using the concept of Lie algebras to provide an algebraic description of the spaces and obtained various types of curvatures.

The concept of Einstein Manifold in the Riemann–Finsler geometry plays an important role. S. Chern has posed an open problem, “Does every smooth manifold admit an Einstein Finsler metric?”. This problem is extremely involved and remains open. However, the problem has motivated great interest from geometers [11,12,13] and has led to many results on Einstein Finsler metrics on manifolds. Up to now, most known Einstein Finsler metrics are either of Randers type or Ricci flat; see, for example [12,14,15]. One effective approach to the above problem is to consider some special Finsler metrics. In this direction, invariant Einstein Finsler metrics on homogeneous manifolds are very interesting; see [16] for some results on homogeneous Einstein–Randers metrics.

Studying the curvature property in Finslerian Manifold has always been a central idea of geometers. In Riemannian–Finsler geometry, Ricci curvature is the trace of the Riemann curvature, which is defined by $Ric=R_m^m$. A Finsler metric F is called an Einstein metric [11] if there is a scalar function $\zeta =\zeta \left(x\right)$ on M such that F satisfies, $Ric(x,y)=(n-1)\zeta \left(x\right)F^2$. In particular, a Finsler metric F is said to be of Ricci constant if $\zeta$ is constant and F is called a Ricci-flat metric if $\zeta =0$.

In 2010, L. Zhou [13] gave the formula of Riemann curvature and Ricci curvature, while studying the Finslerian space with $(\alpha ,\beta )$-metrics. Further, in 2012, X. Cheng et al. [11] obtained that the formulae given by Zhou [13] are wrong, and they have provided the correct version of these formulae. In 2016, by using these formulae, Yan and Deng [12] have expanded an explicit formula for Ricci curvature of the homogeneous $(\alpha ,\beta )$ Finslerian space and also obtained the condition under which the S-curvature has to vanish. Further, many geometers [10,17,18,19,20] studied the property of Ricci curvature for the Homogeneous Finsler space.

The arrangement of this paper is as follows: In Section 2, we have discussed the notion of Finsler space, Lie group theory, homogeneous Finsler space, Ricci curvature and Riemann curvature of $(\alpha ,\beta )$-metrics, and Ricci curvature of the homogeneous Finsler space. In Section 3, we have derived an expression of Ricci curvature for the homogeneous generalized Matsumoto metric and also deduced this expression for the Matsumoto metric by putting $m=1$ in the defined aforesaid space. Next, in Section 4 we have deduced the expression of Ricci curvature for the aforesaid space with the S-curvature having vanished.

2. Preliminaries

Definition 1.

Let V be a real vector space of dimension n endowed with a smooth norm F defined on $V\backslash \left\{0\right\}$ which satisfies the following conditions:

• $F\left(e\right)\ge$ 0, $\forall eϵV$,
• $F\left(\lambda e\right)=\lambda F\left(e\right),\forall \lambda >0,$ i.e., F is positively homogeneous,
• Let $\{e_1,e_2,\dots e_n\}$ be a basis of V such that $y=y^1{e}_1+y^2{e}_2+\dots +y^n{e}_n.$ Then, the Hessian matrix, $g_{ij}=\left({\left[\frac{1}{2}{F}^2\right]}_{y^i{y}^j}\right)$, is positive definite at each point of $V\backslash \left\{0\right\}$.

Then F is called Minkowskian norm and the pair $(V,F)$ is termed Minkowskian space.

Definition 2.

Let M be a connected smooth manifold. A Finsler metric on M is a function $F:TM\to [0,\infty )$, which is smooth on slit tangent bundle $TM\backslash \left\{0\right\}$ and the restriction of F to any $T_{x}M$, $xϵM$ is a Minkowski norm; then, the pair $(M,F)$ is called a Finsler manifold or Finsler space.

Let $F^n=(M,F)$ be an n-dimensional Finsler space. The fundamental function $F(x,y)$ is called an $(\alpha ,\beta )$-metric if F is a homogeneous function of $\alpha$ and $\beta$ of degree one, where ${\alpha }^2=a_{jk}^{}{y}^j{y}^k$, $y=y^j\frac{d}{d{x}^j}{|}_x^{}ϵT_{x}M$ is a Reimannian metric and $\beta =b_k\left(x\right)y^k$ is a 1-form on the tangent bundle $TM\backslash \left\{0\right\}$. Z. Shen states the following condition for a $(\alpha ,\beta )$-metric to be a Finsler metric:

Lemma 1

([21]). For a Riemannian metric α and a 1-form β, let $F=\alpha \varphi \left(s\right)$ with $s=\frac{\beta }{\alpha }$, whose length with respect to the Riemannian metric α is bounded above, i.e., $b:={\parallel \beta \parallel }_{\alpha }<b_0$, where $b_0\in (0,\infty )$. If a smooth positive function ϕ on $(-b_0,b_0)$ satisfies the condition

$$(b^2-s^2){\varphi }^{″}-s{\varphi }^{\prime }+\varphi >0,for\left|s\right|\le b<b_0,$$

then F is a Finsler space and vice-versa.

Now we discuss the Ricci curvature of $(\alpha ,\beta )$-metric. The concept of Riemann curvature can be extended from the Riemannian space to the Finslerian space. The Riemann curvature for an n-dimensional Finslerian space $(M,F)$, $xϵM,yϵT_x^{}M$, is a linear map $R_y^{}:T_x^{}M\to T_x^{}M$ defined as

$$R_y^{}\left(v\right)=R_k^j\left(y\right)v^k\frac{\partial }{\partial x^j},v=v^j\frac{\partial }{\partial x^j},$$

where

$$R_k^j=2\frac{\partial G^j}{\partial x^k}+2G^i\frac{{\partial }^2{G}^j}{\partial y^i\partial y^k}-\frac{{\partial }^2{G}^j}{\partial x^i\partial y^k}{y}^i-\frac{\partial G^j}{\partial y^i}\frac{\partial G^i}{\partial y^k},$$

and $G^j$ is the spray coefficients defined as

$$G^j=\frac{1}{4}{g}^{jl}\left\{{\left(F^2\right)}_{x^m{y}^l}^{}{y}^m-{\left(F^2\right)}_{x^l}^{}\right\},j=1,2,\dots ,n.$$

Definition 3.

For $xϵM$ and $0\ne yϵT_x^{}M$, a linear map $Ric\left(y\right):TM\to M$ is called Ricci curvature of an n-dimensional Finslerian space $(M,F)$, defined as

$$Ric\left(y\right)=tr\left(R_y^{}\right)\forall yϵT_x^{}M.$$

The following notations will be used in the next theorem

$$\begin{array}{cc}& 2s_{jk}^{}=b_{j;k}^{}-b_{k;j}^{},2r_{jk}^{}=b_{j;k}^{}+b_{k;j}^{},s_k^j=a^{jl}{s}_{kl}^{},r_k^j=a^{jl}{r}_{kl}^{},\hfill \\ \hfill & s_j^{}=b^l{s}_{lj}^{}=b_k^{}{s}_j^k,r_j^{}=b^l{r}_{lj}^{}=b_k^{}{r}_j^k,r_{j0}^{}=r_{jk}^{}{y}^k,r_{00}^{}=r_{jk}^{}{y}^j{y}^k,\hfill \\ & r=r_{jk}^{}{b}^j{b}^k=b^j{r}_j^{}{s}_{j0}^{}=s_{jk}^{}{y}^k,s_0^{}=s_j^{}{y}^j,r_0^{}=r_j^{}{y}^j,\hfill \\ & a^{jk}={\left(a_{jk}^{}\right)}^{-1},b^j=a^{jk}{b}_k^{},\hfill \end{array}$$

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

In 2012, X. Cheng et al. given the correct formula of Ricci curvature for $(\alpha ,\beta )$-metric as follows:

Theorem 1

([11]). Let F be an $(\alpha ,\beta )$-metric on a Finsler space M and $Ri{c}^{\alpha }$ be the Ricci curvature of α. Then, Ricci curvature of F is given by $Ric=Ri{c}^{\alpha }+R{T}_k^k$, with

$$\begin{array}{cc}\hfill & R{T}_k^k=\frac{1}{{\alpha }^2}\left\{(n-1)D_1^{}+D_2^{}\right\}{r}_{00}^2\hfill \\ & +\frac{1}{\alpha }\left[\left((n-1)D_3^{}+D_4^{}\right)r_{00}^{}{s}_0^{}+\left((n-1)D_5^{}+D_6^{}\right)r_{00}^{}{r}_0^{}+\left((n-1)D_7^{}+D_8^{}\right)r_{00;0}^{}\right]\hfill \\ & +\left((n-1)D_9^{}+D_{10}^{}\right)s_0^2+\left(r{r}_{00}^{}-r_{00}^2\right)D_{11}^{}+\left((n-1)D_{12}^{}+D_{13}^{}\right)r_0^{}{s}_0^{}\hfill \\ & +\left(r_{00}^{}{r}_k^k-r_{0k}^{}{r}_0^k+r_{00;k}^{}{b}^k-r_{0k;0}^{}{b}^k\right)D_{14}^{}+\left((n-1)D_{15}^{}+D_{16}^{}\right)r_{0k}^{}{s}_0^k\hfill \\ & +\left((n-1)D_{17}^{}+D_{18}^{}\right)s_{0;0}^{}+s_{0k}^{}{s}_0^k{D}_{19}^{}+\alpha \left[r{s}_0^{}{D}_{20}^{}+\left((n-1)D_{21}^{}+D_{22}^{}\right)s_k^{}{s}_0^k\right]\hfill \\ & +\alpha \left[\left(3s_k^{}{r}_0^k-2s_0^{}{r}_k^k+2r_k^{}{s}_0^k-2s_{0;k}^{}{b}^k+s_{k;0}^{}{b}^k\right)D_{23}^{}+s_{0;k}^k{D}_{24}^{}\right]\hfill \\ & +{\alpha }^2\left(s_k^{}{s}^k{D}_{25}^{}+s_k^i{s}_i^k{D}_{26}^{}\right),\hfill \end{array}$$

where

$$\begin{array}{l}{D}_1^{}=2\psi {\Theta }_s^{}(B-s^2)-2s\psi \Theta +{\Theta }^2-{\Theta }_s^{},\\ D_2^{}=(2\psi {\psi }_{ss}^{}-{\psi }_s^2){(B-s^2)}^2-(6s\psi {\psi }_s^{}+{\psi }_{ss}^{})(B-s^2)+2s{\psi }_s^{},\\ D_3^{}=4Q{\Theta }_s^{}+2Q_s^{}\Theta +4Q\Theta (s\psi -\Theta )-4(2Q{\Theta }_s^{}+Q_s^{}\Theta )\psi (B-s^2)-2{\Theta }_B^{},\\ D_4^{}=[-4\psi (2Q{\psi }_{ss}^{}+Q_s^{}{\psi }_s^{}+Q_{ss}^{}\psi )+4Q{\psi }_s^2]{(B-s^2)}^2+2\psi (Q-s{Q}_s^{}-4{\psi }_s^{}-Q_{ss}^{}\hfill \\ \hspace{1em} -10sQ{\psi }_s^{})+\left(-4{\psi }^2(Q-s{Q}_s^{})+4Q_{ss}^{}\psi +2Q_s^{}{\psi }_s^{}+4Q{\psi }_{ss}^{}-2{\psi }_{sB}^{}+20sQ\psi {\psi }_s^{}\right)(B-s^2),\hfill \\ D_5^{}=4\psi \Theta -2{\Theta }_B^{},D_6^{}=2(2\psi {\psi }_s^{}-{\psi }_{sB}^{})(B-s^2)-2{\psi }_s^{},D_7^{}=-\Theta ,D_8^{}=-{\psi }_s^{}(B-s^2),\\ D_9^{}=8Q\psi (Q{\Theta }_s^{}+Q_s^{}\Theta )(B-s^2)+4Q^2({\Theta }^2-{\Theta }_s^{})+4Q({\Theta }_B^{}-Q_s^{}),\\ D_{10}^{}=\left[4{\psi }^2(2Q{Q}_{ss}^{}-Q_s^2)+8Q\psi (Q{\psi }_{ss}^{}+Q_s^{}{\psi }_s^{})-4Q_{}^2{\psi }_s^2\right]{(B-s^2)}^2+[4Q{\psi }_{sB}^{}+4Q_s^{}{\psi }_B^{}\hfill \\ \hspace{1em} -16sQ\psi (Q{\psi }_s^{}+Q_s^{}\psi )-4\psi (2Q{Q}_{ss}^{}-Q_s^2)-4Q(Q{\psi }_{ss}^{}+Q_s^{}{\psi }_s^{})](B-s^2)\hfill \\ \hspace{1em} -4s^2{Q}^2{\psi }^2+4(2+3sQ)(Q{\psi }_s^{}+Q_s^{}\psi )-8Q^2\psi +2Q{Q}_{ss}^{}-Q_s^2+4sQ{\psi }_B^{},\hfill \\ D_{11}^{}=4{\psi }^2+4{\psi }_B^{},D_{12}^{}=4Q({\Theta }_B^{}-2\psi \Theta ),\hfill \\ D_{13}^{}=\left[4Q{\psi }_{sB}^{}+4Q_s^{}{\psi }_B^{}+8\psi (Q_s^{}\psi -Q{\psi }_s^{})\right](B-s^2)+8sQ{\psi }^2-4(1-sQ){\psi }_B^{}+4Q{\psi }_s^{},\\ D_{14}^{}=2\psi ,D_{15}^{}=4Q\Theta ,D_{16}^{}=2Q_s^{}-2(1+2sQ)\psi +4(Q{\psi }_s^{}-Q_s^{}\psi )(B-s^2),\hfill \\ D_{17}^{}=2\Theta Q,D_{18}^{}=2(Q_s^{}\psi +Q{\psi }_s^{})(B-s^2)+2sQ\psi -Q_s^{}\hfill \end{array}$$

$$\begin{array}{l}{D}_{19}^{}=2Q_s^{}(sQ+1)-2Q^2,D_{20}^{}=-8Q({\psi }^2+{\psi }_B^{}),D_{21}^{}=-4Q^2\Theta ,\\ D_{22}^{}=2Q\psi -4Q^2{\psi }_s^{}(B-s^2),D_{23}^{}=2Q\psi ,D_{24}^{}=2Q,\\ D_{25}^{}=-4Q^2\psi ,D_{26}^{}=-Q^2,\hfill \end{array}$$

(2)

and

$$B=b^2,Q=\frac{{\varphi }^{{}^{\prime }}}{\varphi -s{\varphi }^{{}^{\prime }}},\psi =\frac{{\varphi }^{"}}{2(\varphi -s{\varphi }^{{}^{\prime }}+(B-s^2){\varphi }^{"})},$$

(3)

$$\Theta =\frac{\varphi {\varphi }^{{}^{\prime }}-s(\varphi {\varphi }^{"}+{\varphi }^{{}^{\prime }}{\varphi }^{{}^{\prime }})}{2\varphi (\varphi -s{\varphi }^{{}^{\prime }}+(B-s^2){\varphi }^{"})}.$$

Definition 4

([22]). An isometry of a Finsler space is a diffeomorphism ϕ: $M\to M$, and satisfies the following condition

$$F(\varphi \left(x\right),d{\varphi }_{x}y)=F(x,y),\forall xϵM,yϵT_{x}M.$$

A smooth manifold G is called Lie-group if it satisfies the group properties, and the map, $\lambda :G\times G\to G$, is defined by $\lambda (g_1,g_2)=g_1{g}_2^{-1}$, ∀ $g_1,g_2ϵG$ is smooth.

Definition 5.

A Finsler space $(M,F)$ is said to be a homogeneous Finsler space, if the group of isometries $I(M,F)$ acts transitively on the manifold M.

Let $N=I_a(M,F)$ be a closed isotropy subgroup of G, then by using the result given by S. Deng and Z. Hau [22], the subgroup N is compact and Lie-group itself, then a homogeneous Finsler manifold $(G/N,F)$ can be written as a coset space $G/N$, where G is a connected Lie group, N is a compact subgroup of G and F is invariant under the action of G. The action of G on $G/N$ is most effective and the Lie algebra $\mathfrak{g}$ has a reductive decomposition i.e., $\mathfrak{g}=\mathfrak{n}\oplus \mathfrak{q}$, where $\mathfrak{n}$ is the Lie algebra of N and $\mathfrak{q}$ is a subspace of $\mathfrak{g}$ satisfying $Ad\left(\mathfrak{q}\right)\subset \mathfrak{q}$∀ $\mathfrak{n}ϵN$. The tangent space $T_N(G/N)$ of $G/N$ at the origin N can be canonically identified with $\mathfrak{q}$. Then, F is uniquely determined by a pair $(⟨,⟩,U)$, where $⟨,⟩$ is the inner product on $\mathfrak{q}$ induced by the Riemannian metric $\alpha$, and U is an N-fixed vector in $\mathfrak{q}$ with length less than 1.

Let $\left\{p_1^{},p_2^{},\dots p_n^{}=\frac{p}{c}\right\}$ be the orthogonal basis of $\mathfrak{q}$, where p is the G-invariant vector field of length c corresponding to 1-form $\beta$, i.e., $⟨p,U⟩=\beta \left(U\right)$, ∀ $Uϵ\mathfrak{q}$, and $⟨,⟩$ is the Riemannian metric $\alpha$ to $\mathfrak{q}$. Now, we discuss the Ricci curvature for the homogeneous Finsler space given by Yan and Deng as follows:

Lemma 2

([12]). At the origin 0 = eN, we have

$$\begin{array}{cc}\hfill & \left(1\right){\Gamma }_{kt}^m=f(k,t)C_{kt}^m+⟨{\nabla }_{{\widehat{p}}_k^{}}^{}{\widehat{p}}_t^{},{\widehat{p}}_m^{}⟩,\hfill \\ & \left(2\right)⟨{\nabla }_{{\widehat{p}}_k^{}}^{}{\widehat{p}}_t^{},{\widehat{p}}_m^{}⟩=-\frac{1}{2}\left(C_{tm}^k+C_{km}^t+C_{kt}^m\right),\hfill \\ & \left(3\right)⟨{\nabla }_{{\widehat{p}}_k^{}}^{}{\widehat{p}}_t^{},{\widehat{p}}_m^{}⟩{\widehat{p}}_r^{}=\frac{1}{2}\left(C_{rl}^k{C}_{tm}^l+C_{rl}^t{C}_{km}^l+C_{rl}^m{C}_{kt}^l+C_{tm}^l{C}_{rk}^l+C_{km}^l{C}_{rt}^l+C_{kt}^l{C}_{rm}^l\right),\hfill \end{array}$$

where $p_k^{},p_t^{},p_m^{},p_r^{}ϵ\mathfrak{q}$, and the function $f(k,t)$, the Christoffel symbol ${\Gamma }_{kt}^m$, and the structure constants $C_{kt}^m$ of $\mathfrak{q}$ for $1\le k,t,m\le n$ are defined, respectively, as

$$f(k,t)=\left\{\begin{array}{c}0,ifk\ge t,\hfill \\ 1,ifk<t,\hfill \end{array}\right.$$

and

$${\nabla }_{\frac{\partial }{\partial x^k}}^{}\frac{\partial }{\partial x^t}={\Gamma }_{kt}^m\frac{\partial }{\partial x^m},$$

$$C_{kt}^m=⟨{\left[p_k^{},p_t^{}\right]}_{\mathfrak{q}}^{},p_m^{}⟩,C_{kt}^0=⟨{\left[p_k^{},p_t^{}\right]}_{\mathfrak{q}}^{},U⟩\mathrm{for}0\ne Uϵ\mathfrak{q},$$

where ${\left[p_k^{},p_t^{}\right]}_{\mathfrak{q}}^{}$ denotes the projection of $\left[p_k^{},p_t^{}\right]$ to $\mathfrak{q}$.

Lemma 3

([12]). At the origin $0=eN$, we have

$$b_k^{}=c{\delta }_{nk}^{},s_{kt}^{}=\frac{c}{2}{C}_{kt}^n,s_t^{}=\frac{c^2}{2}{C}_{nt}^n,r_{kt}^{}=\frac{c}{2}\left(C_{tn}^k+C_{kn}^t\right),$$

$$s_{kt;m}^{}=\frac{c}{2}\left\{-C_{kt}^l{C}_{ml}^n+\frac{1}{2}{C}_{kl}^n(-C_{tm}^l+C_{tl}^m+C_{ml}^t)+\frac{1}{2}{C}_{lt}^n(-C_{km}^l+C_{kl}^m+C_{ml}^k)\right\},$$

$$b_{k;t;m}^{}=c\left\{C_{mn}^l⟨{\nabla }_{{\widehat{p}}_l^{}}^{}{\widehat{p}}_t^{},{\widehat{p}}_k^{}⟩-{\Gamma }_{nl}^k⟨{\nabla }_{{\widehat{p}}_n^{}}^{}{\widehat{p}}_t^{},{\widehat{p}}_l^{}⟩-{\Gamma }_{nt}^l⟨{\nabla }_{{\widehat{p}}_m^{}}^{}{\widehat{p}}_k^{},{\widehat{p}}_l^{}⟩+{\widehat{p}}_m^{}⟨{\nabla }_{{\widehat{p}}_n^{}}^{}{\widehat{p}}_t^{},{\widehat{p}}_k^{}⟩\right\},$$

$$s_{k;t}^{}=c\left(\frac{c}{2}{C}_{lk}^n{\Gamma }_{nt}^l+s_{nk;t}^{}\right),r_{kt;m}^{}=b_{t;k;m}^{}+s_{kt;m}^{},$$

By using the above Lemmas 2 and 3, they have derived the expression of the Ricci curvature with $(\alpha ,\beta )$-metric in the homogeneous Finsler space, as follows:

Theorem 2

([12]). The Ricci curvature with $(\alpha ,\beta )$-metric in the homogeneous Finsler space is given as

$$\begin{array}{cc}\hfill & Ric\left(U\right)=Ri{c}^{\alpha }\left(U\right)+\frac{c^2{\left(C_{0n}^0\right)}^2}{{\alpha }^2\left(U\right)}\left\{(n-1)D_1^{}+D_2^{}\right\}+\frac{c^3{C}_{0n}^0{C}_{n0}^n}{2\alpha \left(U\right)}\left\{(n-1)(D_3^{}-D_5^{})+D_4^{}-D_6^{}\right\}\hfill \\ & -\frac{c{C}_{0l}^0(C_{nl}^0+C_{n0}^l)}{\alpha \left(U\right)}\left\{(n-1)D_7^{}+D_8^{}\right\}+\frac{c^4{\left(C_{n0}^n\right)}^2}{4}\left\{(n-1)(D_9^{}-D_{12}^{})+D_{10}^{}-D_{11}^{}-D_{13}^{}\right\}\hfill \\ & -\frac{c^2}{4}\left\{4C_{0n}^0{C}_{nl}^l+(C_{nl}^0+C_{n0}^l)(2C_{nl}^0+C_{0l}^n+2C_{0n}^l)+2C_{nl}^n{C}_{l0}^0\right\}{D}_{14}^{}\hfill \\ & +\frac{c^2}{4}{C}_{0l}^n(C_{n0}^l+C_{nl}^0)\left\{(n-1)D_{15}^{}+D_{16}^{}\right\}+\frac{c^2}{2}{C}_{nl}^n{C}_{0l}^0\left\{(n-1)D_{17}^{}+D_{18}^{}\right\}\hfill \\ & -\frac{c^2}{4}{\left(C_{l0}^n\right)}^2{D}_{19}^{}+\frac{c^3}{4}\alpha \left(U\right)C_{l0}^n{C}_{nl}^n\left\{(n-1)D_{21}^{}+D_{22}^{}\right\}\hfill \\ & +\frac{c^3}{4}\alpha \left(U\right)\left\{4C_{n0}^n{C}_{nl}^l-C_{nl}^n(4C_{nl}^0-C_{0l}^n)\right\}{D}_{23}^{}+\frac{c}{4}\alpha \left(U\right)\left\{2C_{t0}^n{C}_{lt}^l+C_{lt}^n{C}_{lt}^0\right\}{D}_{24}^{}\hfill \\ & +\frac{c^2}{4}{\alpha }^2\left(U\right)\left\{c^2{\left(C_{nl}^n\right)}^2{D}_{25}^{}-{\left(C_{li}^n\right)}^2{D}_{26}^{}\right\},\hfill \end{array}$$

(4)

where $0\ne Uϵ\mathfrak{q}$, and $D_1^{}$ to $D_{26}^{}$ are given by (2).

3. Ricci Curvature

In this section, we have derived the formula of Ricci curvature for a homogeneous generalized Matsumoto Finsler space. Firstly, we have obtained the basic quantity by using the Equation (3) as follows:

$$\begin{array}{cc}\hfill & \varphi \left(s\right)=\frac{1}{{(1-s)}^m},{\varphi }^{{}^{\prime }}\left(s\right)=\frac{m}{{(1-s)}^{m+1}},{\varphi }^{″}\left(s\right)=\frac{m(m+1)}{{(1-s)}^{m+2}},Q=m{A}_1^{-1},\hfill \\ & Q_s^{}=m(m+1)A_1^{-2},Q_{ss}^{}=2m{(m+1)}^2{A}_1^{-3},\Theta =\frac{1}{2}m[1-2s(m+1)]A_2^{-1},\hfill \\ & {\Theta }_s^{}=\frac{1}{2}m{A}_2^{-2}\left[2(m+1)\left[(1-m^2)s^2-(1-m)s-m(m+1)B)\right]-m\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\Theta }_B^{}=\frac{-1}{2}{m}^2{A}_2^{-2}(m+1)[1-2s(m+1)],\psi =\frac{1}{2}m(m+1)A_2^{-1},\hfill \\ & {\psi }_s^{}=\frac{-1}{2}m(m+1)[2s(1-m^2)-(m+2)]A_2^{-2},\hfill \\ & {\psi }_{ss}^{}=m(m+1)\left\{(1-m^2)\left[3(1-m^2)s^2-3(m+2)s-Bm(m+1)\right]+2m^2+4m+3\right\}{A}_2^{-3},\hfill \\ & {\psi }_B^{}=\frac{-1}{2}{m}^2{(m+1)}^2{A}_2^{-2},{\psi }_{sB}^{}=m^2{(m+1)}^2[2s(1-m^2)-(m+2)]A_2^{-3},\hfill \end{array}$$

(5)

where $B=b^2,A_1^{-1}=[1-s(m+1)],\mathrm{and}{A}_2^{-1}=[s^2(1-m^2)-s(m+2)+1+Bm(m+1)]$.

By using Equation (2), we calculate $D_1^{}$ to $D_{26}^{}$ for the generalized Matsumoto metric as follows:

$$\begin{array}{cc}\hfill D_1^{}& =\frac{1}{4}m[3m+4Bm+9B{m}^2+5B{m}^3+(4-12m-8Bm-13m^2-26B{m}^2-28B{m}^3-10B{m}^4)s\hfill \\ & +(-12+15m+4Bm+48m^2+20B{m}^2+21m^3+36B{m}^3+28B{m}^4+8B{m}^5)s^2+(12-6m\hfill \\ & -54m^2-42m^3-6m^4)s^3+(-4+16m^2+8m^3-12m^4-8m^5)s^4]A_2^{-3},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & D_2=\frac{-1}{4}m(m+1)A_2^{-4}[12B+16Bm+8B{m}^2+4B^2{m}^2(1+m)+5B^2{m}^3(1+m)+(-8-48B\hfill \\ & -4m-60Bm-14B{m}^2+2B{m}^3+4B^2{m}^2(1+m)+6B^2{m}^3(1+m)+2B^2{m}^4(1+m))s+(28\hfill \\ & +72B+16m+84Bm-8m^2-28B{m}^2-48B{m}^3-8B{m}^4-8B^2{m}^2(1+m)-8B^2{m}^3(1+m)\hfill \\ & +8B^2{m}^4(1+m)+8B^2{m}^5(1+m))s^2+(-32-48B-28m-52Bm+14m^2+42B{m}^2\hfill \\ & +10m^3+32B{m}^3-30B{m}^4-16B{m}^5)s^3+(8+12B+28m+12Bm+24m^2-8B{m}^2-m^3\hfill \\ & +8B{m}^3-5m^4+12B{m}^4-20B{m}^5-16B{m}^6)s^4+(8-16m-46m^2+2m^3+38m^4+14m^5)s^5\hfill \\ & +(-4+4m+16m^2-8m^3-20m^4+4m^5+8m^6)s^6],\hfill \end{array}$$

$$\begin{array}{cc}\hfill D_3^{}& =A_1^{-1}{A}_2^{-3}[2-m-2Bm-5B{m}^2-3B{m}^3+(-14-6m+2Bm+5m^2+8B{m}^2+10B{m}^3\hfill \\ & +4B{m}^4)s+(30+29m-14m^2-4B{m}^2-13m^3-12B{m}^3-12B{m}^4-4B{m}^5)s^2\hfill \\ & +(-26-34m+18m^2+34m^3+8m^4)s^3+(8+12m-8m^2-16m^3+4m^5)s^4],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & D_4^{}=m(m+1)A_1^{-1}{A}_2^{-4}[-6-3m+6Bm+7B{m}^2+5B{m}^3+4B^2{m}^3+9B^2{m}^4+5B^2{m}^5\hfill \\ & +(36+21m-24Bm-3m^2-23B{m}^2-5B{m}^3-2B{m}^4)s+v(-90-66m+36Bm+17m^2\hfill \\ & +27B{m}^2+4m^3-33B{m}^3-4B^2{m}^3-27B{m}^4-8B^2{m}^4-3B{m}^5+8B^2{m}^6+4B^2{m}^7)s^2\hfill \\ & +(120+114m-24Bm-43m^2-13B{m}^2-34m^3+45B{m}^{3}v+5m^4+23B{m}^4-21B{m}^5\hfill \\ & -10B{m}^6)s^3+(-v90-111m+6Bm+57m^2+2B{m}^2+92m^3-12B{m}^3+6m^4+4B{m}^4\hfill \\ & -8m^5+14B{m}^5-6B{m}^6-8B{m}^7)s^4+(36+57m-38m^2-90m^3-8m^4+33m^5+10m^6)s^5\hfill \\ & +(-6-12m+10m^2+28m^3-2m^4-20m^5-2m^6+4m^7)s^6],\hfill \end{array}$$

$$D_5^{}=2A_2^{-2}{m}^2(m+1)[1-2s(m+1)],$$

$$\begin{array}{cc}\hfill & D_6^{}=m(m+1)A_2^{-3}[-2-m+4Bm(1+m)+2B{m}^2(1+m)(2-2m^2-4Bm(1+m)\hfill \\ & +4B{m}^3(1+m)+2(2+m)+m(2+m))s+(-2(2+m)+2m^2(2+m)-2(1+3m\hfill \\ & +2m^2)-m(1+3m+2m^2))s^2+(2(1+3m+2m^2)-2m^2(1+3m+2m^2))s^3],\hfill \end{array}$$

$$D_7^{}=\frac{-1}{2}m{A}_2^{-1}\left[1-2s(m+1)\right],$$

$$D_8^{}=\frac{1}{2}m(m+1)A_2^{-2}\left[-2(1-m^2)s^3+(m+2)s^2+2B(1-m^2)s-B(m+2)\right],$$

$$\begin{array}{cc}\hfill & D_9^{}=m^2{A}_1^{-3}{A}_2^{-3}[4+6m+12Bm-m^2+20B{m}^2+12B^2{m}^2+3B{m}^3+34B^2{m}^3+4B^3{m}^3\hfill \\ & -5B{m}^4+30B^2{m}^4+16B^3{m}^4+6B^2{m}^5+24B^3{m}^5-2B^2{m}^6+16B^3{m}^6+4B^3{m}^7+(-24\hfill \\ & -52m-48Bm-21m^2-108B{m}^2-24B^2{m}^2+4m^3-61B{m}^3-80B^2{m}^3+10B{m}^4-92B^2{m}^4\hfill \\ & +11B{m}^5-36B^2{m}^5+4B^2{m}^6+4B^2{m}^7)s+(60+152m+72Bm+109m^2+180B{m}^2\hfill \\ & +12B^2{m}^2+9m^3+118B{m}^3+36B^2{m}^3-8m^4-18B{m}^4+24B^2{m}^4-30B{m}^5-24B^2{m}^5\hfill \\ & -2B{m}^6-36B^2{m}^6-12B^2{m}^7)s^2+(-80-208m-48Bm-167m^2-116B{m}^2-19m^3\hfill \\ & -40B{m}^3+27m^4+88B{m}^4+7m^5+64B{m}^5-4B{m}^6-8B{m}^7)s^3+(60+142m+12Bm\hfill \\ & +84m^2+24B{m}^2-28m^3-12B{m}^3-44m^4-48B{m}^4-18m^5-12B{m}^5-4m^6+24B{m}^6\hfill \\ & +12B{m}^7)s^4+(-24-44m+8m^2+44m^3+8m^4-4m^5+8m^6+4m^7)s^5+(4+4m\hfill \\ & -12m^2-12m^3+12m^4+12m^5-4m^6-4m^7)s^6],\hfill \end{array}$$

$$\begin{array}{cc}\hfill D_{10}^{}& =-m^2(m+1)A_1^{-3}{A}_2^{-4}[-15-7m-8Bm-10B{m}^2-7B^2{m}^2+2B{m}^3-7B^2{m}^3\hfill \\ & -2B^3{m}^3+8B^2{m}^4-4B^3{m}^4+8B^2{m}^5+4B^3{m}^6+2B^3{m}^7+(105+86m+40Bm\hfill \\ & +7m^2+76B{m}^2+21B^2{m}^2+20B{m}^3+34B^2{m}^3+2B^3{m}^3-12B{m}^4-13B^2{m}^4\hfill \\ & +4B^3{m}^4-44B^2{m}^5-4B^3{m}^5-18B^2{m}^6-16B^3{m}^6-14B^3{m}^7-4B^3{m}^8)s+(-315\hfill \\ & -361m-80Bm-56m^2-190B{m}^2-21B^2{m}^2+14m^3-116B{m}^3-41B^2{m}^3+2B{m}^4\hfill \\ & +6B^2{m}^4+8B{m}^5+54B^2{m}^5+31B^2{m}^6+3B^2{m}^7)s^2+(525+760m+80Bm+149m^2\hfill \\ & +214B{m}^2+7B^2{m}^2-100m^3+182B{m}^3+14B^2{m}^3-8m^4+58B{m}^4-B^2{m}^4+26B{m}^5\hfill \\ & -4B^2{m}^5+16B{m}^6+29B^2{m}^6+38B^2{m}^7+13B^2{m}^8)s^3+(-525-905m-40Bm\hfill \\ & -183m^2-112B{m}^2+283m^3-112B{m}^3+69m^4-70B{m}^4-17m^5-58B{m}^5-34B{m}^6\hfill \\ & -6B{m}^7)s^4+(315+622m+8Bm+110m^2+22B{m}^2-376m^3+24B{m}^3-162m^4\hfill \\ & +22B{m}^4+24m^5+8B{m}^5+7m^6-30B{m}^6-40B{m}^7-14B{m}^8)s^5+(-105-231m\hfill \\ & -29m^2+235m^3+145m^4-5m^5-11m^6+m^7)s^6+(15+36m+2m^2-56m^3\hfill \\ & -44m^4+4m^5+22m^6+16m^7+5m^8)s^7],\hfill \end{array}$$

$$D_{11}^{}=-m^2{(m+1)}^2{A}_2^{-2},D_{12}^{}=-4A_1^{-1}{A}_2^{-2}{m}^3(m+1)[1-2s(m+1)],$$

$$\begin{array}{cc}\hfill D_{13}^{}& =2m^2(m+1)A_1^{-1}{A}_2^{-3}[3+2m-3Bm-4B{m}^2-B{m}^3+(-9-9m+3Bm-m^2\hfill \\ & +B{m}^2-7B{m}^3-5B{m}^4)s+(9+15m+6m^2)s^2+(-3-8m-2m^2+8m^3+5m^4)s^3],\hfill \end{array}$$

$$D_{14}^{}=m(m+1)A_2^{-1},D_{15}^{}=2m^2{A}_1^{-1}{A}_2^{-1}[1-2s(m+1)],$$

$$\begin{array}{cc}\hfill D_{16}^{}& =-m(m+1)A_1^{-1}{A}_2^{-2}[-1-5Bm-3B{m}^2+(3+2m+5Bm+2B{m}^2-3B{m}^3)s\hfill \\ & +(-3+m+2m^2)s^2+(1-3m-m^2+3m^3)s^3],\hfill \end{array}$$

$$D_{17}^{}=m^2{A}_1^{-1}{A}_2^{-1}\left[1-2s(m+1)\right],$$

$$\begin{array}{cc}\hfill & D_{18}^{}=-m(m+1)A_1^{-1}{A}_2^{-2}[1-Bm+(-3-2m-Bm(-1+2m+3m^2))s+(3+5m+m^2)s^2\hfill \\ & +(-1-3m+m^2+3m^3)s^3],\hfill \end{array}$$

$$D_{19}^{}=2m{A}_1^{-3}[1-(1-m^2)s],D_{20}^{}=2m^3{(m+1)}^2{A}_1^{-1}{A}_2^{-2},D_{21}^{}=-2m^3{A}_1^{-2}{A}_2^{-1}[1-2s(m+1)],$$

$$\begin{array}{cc}\hfill & D_{22}^{}=m^2(m+1)A_1^{-2}{A}_2^{-2}[1-3Bm-B{m}^2+(-3-2m+3Bm-2B{m}^2-5B{m}^3)s\hfill \\ & +(3+7m+2m^2)s^2+(-1-5m+m^2+5m^3)s^3],\hfill \end{array}$$

$$D_{23}^{}=m^2(m+1)A_1^{-1}{A}_2^{-1},D_{24}^{}=2m{A}_1^{-1},$$

(6)

$$D_{25}^{}=-2m^3(m+1)A_1^{-2}{A}_2^{-1},D_{26}^{}=-m^2{A}_1^{-2}.$$

Putting all these values of (6) in Equation (4), and after a long calculation by using the Mathematica program, we have obtained the formula of Ricci curvature for the aforesaid space as follows:

Theorem 3.

For the homogeneous generalized Matsumoto change F = $\frac{{\alpha }^{m+1}}{{(\alpha -\beta )}^m}$, the Ricci curvature is given by

$$Ric\left(U\right)=\frac{1}{4{(\alpha -(1+m)\beta )}^3{(-{\alpha }^2(1+Bm(1+m))+\alpha (2+m)\beta +(-1+m^2){\beta }^2)}^4}\sum _{i=0}^{13}{\alpha }^i{t}_i^{}$$

(7)

where $U(\ne 0)ϵ\mathfrak{q}$, and the value of $t_i^{},$$i=0,1,2,\dots ,13$ are calculated as follows

$$\begin{array}{cc}\hfill t_0^{}& =-4{(-1+m)}^2{(1+m)}^7{\beta }^9(c^2{\left(C_{0n}^0\right)}^{2}m(-1+2m)(-2+n)+c{C}_{0k}^0(C_{nk}^0+C_{n0}^k)(-1\hfill \\ & +m)m(-2+n)\beta +Ri{c}^{\alpha }{(-1+m)}^2{\beta }^2),\hfill \\ & ⋮\hfill \\ & t_{13}^{}=m^2{c}^2{(1+Bm(1+m))}^3(-2c^2{\left(C_{nk}^n\right)}^{2}m(1+m)+{\left(C_{ki}^n\right)}^2(1+Bm(1+m))).\hfill \end{array}$$

If we take $m=1$ in Equation (1), then the generalized Matsumoto metric $F=\frac{{\alpha }^{m+1}}{{(\alpha -\beta )}^m}$ reduces to Matsumoto metric $F=\frac{{\alpha }^2}{(\alpha -\beta )}$. Then, from Equations (5) and (6), we obtain the following values

$$\begin{array}{c}\varphi \left(s\right)=\frac{1}{(1-s)},{\varphi }^{{}^{\prime }}\left(s\right)=\frac{1}{{(1-s)}^2},{\varphi }^{{}^{″}}\left(s\right)=\frac{2}{{(1-s)}^2},Q=\frac{1}{1-2s},Q_s=\frac{2}{{(1-2s)}^2},\hfill \\ Q_{ss}=\frac{8}{{(1-2s)}^3},\psi =\frac{1}{1-3s+2B},{\psi }_s=\frac{3}{{(1-3s+2B)}^2},{\psi }_{ss}=\frac{18}{{(1-3s+2B)}^3},\hfill \\ {\psi }_B=\frac{-2}{{(1-3s+2B)}^2},{\psi }_{sB}=\frac{-12}{{(1-3s+2B)}^3},\Theta =\frac{1-4s}{2[1-3s+2B]},\hfill \\ {\Theta }_B=\frac{4s-1}{{[1-3s+2B]}^2},{\Theta }_s=\frac{-(1+8B)}{2{[1-3s+2B]}^2},\hfill \\ D_1=-\frac{3(1-7s+24s^2-32s^3+B(6-24s+32s^2))}{4{(1+2B-3s)}^3},\hfill \\ D_2=\frac{-(3(B^2(3+4s)-2B(-3+10s-6s^2+6s^3)+s(-2+6s-6s^2+9s^3))}{{(1+2B-3s)}^4},\hfill \\ D_3=\frac{-1+15s-32s^2+2B(5-12s+16s^2)}{{(1+2B-3s)}^3(-1+2s)},\hfill \\ D_4=\frac{(18(-1+2B^2+B(2-6s)+6s-15s^2+18s^3-6s^4)}{{(1+2B-3s)}^4(-1+2s)},D_5=\frac{4(1-4s)}{{[1-3s+2B]}^2},\\ D_6=\frac{6(-1+4B+3s-6s^2)}{{[1-3s+2B]}^3},D_7=\frac{4s-1}{2[1-3s+2B]},D_8=\frac{-3(B-s^2)}{{[1-3s+2B]}^2},\hfill \\ D_9=\frac{9+30B+80B^2+64B^3+(-93-196B-4B^2)s+(322+320B)s^2+(-440-64B)s^3+192s^4}{{(1+2B-3s)}^3{(-1+2s)}^3},\hfill \\ D_{10}=\frac{4}{{(1+2B-3s)}^4{(-1+2s)}^3)}[-11-8B+B^2+(99+62B-10B^2-16B^3)s\hfill \\ +(-359-188B+16B^2)s^2+(663+288B+48B^2)s^3+(-639-216B)s^4+270s^5],\hfill \\ D_{11}=\frac{-4}{{[1-3s+2B]}^2},D_{12}=\frac{8(4s-1)}{(1-2s){[1-3s+2B]}^2},\hfill \\ D_{13}=\frac{-4\left[5-8B+(-19-8B)s+30s^2\right]}{(1-2s){[1-3s+2B]}^3},D_{14}=\frac{2}{[1-3s+2B]},\hfill \\ D_{15}=\frac{2(1-4s)}{(1-2s)[1-3s+2B]},D_{16}=\frac{2\left[-1-8B+(5+4B)s\right]}{{(1+2B-3s)}^2(-1+2s)},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & D_{17}=\frac{1-4s}{(1-2s)[1-3s+2B]},D_{18}=\frac{-2\left[-1+B+(5+4B)s-9s^2\right]}{{(1+2B-3s)}^2(-1+2s)},\hfill \\ \hfill & D_{19}=\frac{2}{{(1-2s)}^3},D_{20}=\frac{8}{(1-2s){[1-3s+2B]}^2},\hfill \\ \hfill & D_{21}=\frac{-2(1-4s)}{{(1-2s)}^2[1-3s+2B]},D_{22}=\frac{24s^2-10s-8B(s+1)+2}{{(1-2s)}^2{[1-3s+2B]}^2},\hfill \\ \hfill & D_{23}=\frac{2}{(1-2s)[1-3s+2B]},D_{24}=\frac{2}{1-2s},\hfill \\ \hfill & D_{25}=\frac{-4}{{(1-2s)}^2[1-3s+2B]},D_{26}=\frac{-1}{{(1-2s)}^2}.\hfill \end{array}$$

(8)

Putting these all values of (8) in Equation (4), and calculating by using the Mathematica program, we have obtained the formula of Ricci curvature for the Homogeneous Matsumoto metric as follows:

Proposition 1.

For the Homogeneous Matsumoto metric $F=\frac{{\alpha }^2}{(\alpha -\beta )}$, we obtain the Ricci curvature as

$$\begin{array}{c}\hfill Ric=\frac{1}{{\alpha }^2{(\alpha +2B\alpha -3\beta )}^4{(\alpha -2\beta )}^3}\sum _{i=0}^{11}{\alpha }^i{t}_i^{}\end{array}$$

where $U(\ne 0)ϵ\mathfrak{q}$, and by using Mathematica Program we have obtained the values of $t_i^{}$, $i=0,$$1,2,\dots 11,$ as follows

$$\begin{array}{cc}\hfill & t_0^{}=-288c^2{\left(C_{0n}^0\right)}^2{\beta }^7(-11+8n)\hfill \\ & ⋮\hfill \\ & t_{11}^{}=c^2{(1+2B)}^3(-4c^2{\left(C_{nk}^n\right)}^2+(1+2B){\left(C_{ki}^n\right)}^2)\hfill \end{array}$$

4. Ricci Curvature with Vanishing S-Curvature

Recently, we have obtained the formula of S-curvature for the homogeneous Finsler space with generalized Matsumoto metric [4]. For this space, we have obtained the equivalent condition under which the S-curvature has vanished.

Lemma 4.

Let a compact homogeneous Finsler space $(G/N,F)$ with a G-invariant generalized Matsumoto metric $F=\frac{{\alpha }^{m+1}}{{(\alpha -\beta )}^m}$ on $G/N$. Then, the Finsler space $(G/N,F)$ has vanishing S-curvature if and only if $⟨{[p,U]}_{\mathfrak{q}}^{},U⟩=0$, $\forall Uϵ\mathfrak{q}$ and $pϵ\mathfrak{q}$ corresponds to β.

Proof. 

Z. Shen and X. Cheng [15] proved that the Finsler space F has vanishing the S-curvature if and only if

$$s_k^{}=0,\mathrm{and}{r}_{kt}^{}=0,\forall 1\le k,t\le n.$$

(9)

We must show that the above conditions are same as $⟨{[p,U]}_q^{},U⟩=0$, $\forall Uϵ\mathfrak{q}$ and $pϵ\mathfrak{q}$.

• For the first part, let $s_k^{}=0$, and $r_{kt}^{}=0,\forall t\le n,1\le k$. Then, in view of Lemma (3), we get

  $$\frac{c}{2}\left(C_{tn}^k+C_{kn}^t\right)=0,[-2mm]$$

  (10)

  and

  $$\frac{c^2}{2}{C}_{nk}^n=0.$$
  Equation (10) gives us $C_{kn}^k=⟨{[p_n^{},p_k^{}]}_{\mathfrak{q}}^{},p_k^{}⟩=⟨{[p,p_k^{}]}_{\mathfrak{q}}^{},p_k^{}⟩=0$, for $k=t$.
  Since $\left\{p_1^{},p_2^{},p_3^{},\dots ,p_n^{}=\frac{p}{c}\right\}$ is orthonormal basis of $\mathfrak{q}$, we obtain

  $$⟨{[p,U]}_{\mathfrak{q}}^{},U⟩=0.$$
  For the sufficient part, let

  $$⟨{[p,U]}_{\mathfrak{q}}^{},U⟩=0,\forall Uϵ\mathfrak{q}.$$
  Therefore,

  $$⟨{[p,p_k^{}]}_{\mathfrak{q}}^{},p_k^{}⟩=0,\forall 1\le k\le n,$$

  (11)

  which is equivalent to

  $$⟨{[p,p_k^{}]}_{\mathfrak{q}}^{},p_k^{}⟩+⟨{[p,p_i^{}]}_{\mathfrak{q}}^{},p_i^{}⟩=0$$

  (12)

  which implies

  $$⟨{[p,p_k^{}+p_i^{}]}_{\mathfrak{q}}^{},p_k^{}+p_i^{}⟩=0,\forall 1\le k\le n,i\le n,$$

  (13)

  $$⟨{[p,p+p_i^{}]}_{\mathfrak{q}}^{},p+p_i^{}⟩=0,\forall 1\le k\le n.$$

  (14)
  By using the Lemma 3, $r_{kt}^{}$ is defined as

  $$r_{kt}^{}=\frac{c}{2}\left[C_{tn}^k+C_{kn}^t\right]$$

  which is equivalent to

  $$r_{kt}^{}=\frac{c}{2}\left(⟨{[p_n^{},p_k^{}]}_{\mathfrak{q}}^{},p_t^{}⟩+⟨{[p_n^{},p_t^{}]}_{\mathfrak{q}}^{},p_k^{}⟩\right)$$
  In view of Equations (11) and (13) we get $r_{kt}^{}=0$. Similarly, by using Equations (11) and (14), we easily get $⟨{[p,p_k^{}]}_{\mathfrak{q}}^{},p⟩=0$, i.e., $s_k^{}=0$. □

Theorem 4.

Let a compact homogeneous Finsler space $(G/N,F)$ with a G-invariant generalized Matsumoto metric $F=\frac{{\alpha }^{m+1}}{{(\alpha -\beta )}^m}$ on $G/N$. If the space $(G/N,F)$ has vanishing S-curvature, then the Ricci curvature is given as

$$\begin{array}{cc}\hfill Ric\left(U\right)& =\frac{1}{4{(\alpha -(1+m)\beta )}^3}[4Ri{c}^{\alpha }{(\alpha -(1+m)\beta )}^3+{\alpha }^{2}m(-2c^2{\left(C_{k0}^n\right)}^2(\alpha +(-1+m^2)\beta )\hfill \\ & +(\alpha -(1+m)\beta )(c^2{\left(C_{ki}^n\right)}^2{\alpha }^{2}m+\frac{c}{2}(2C_{ao}^n{C}_{ka}^k+C_{ka}^n{C}_{ka}^0)(\alpha -(1+m)\beta ))\left)\right].\hfill \end{array}$$

(15)

Proof. 

By using the above Lemma 4, $(G/N,F)$ has vanishing S-curvature if and only if

$$⟨{[p,U]}_{\mathfrak{q}}^{},U⟩=0,\forall Uϵ\mathfrak{q}.$$

Therefore,

$$\begin{array}{cc}\hfill & C_{nk}^n=⟨{[p_n^{},p_k^{}]}_{\mathfrak{q}}^{},p_n^{}⟩,\hfill \\ & =⟨{[p_n^{},p_k^{}]}_{\mathfrak{q}}^{},p_n^{}⟩+⟨{[p_n^{},p_k^{}]}_{\mathfrak{q}}^{},p_k^{}⟩,\hfill \\ & =\frac{1}{c^2}\left(⟨{[p,p_k^{}]}_{\mathfrak{q}}^{},p⟩+⟨{[p,p_k^{}]}_{\mathfrak{q}}^{},p_k^{}⟩\right)\hfill \\ & =\frac{1}{c^2}⟨{[p,p_k^{}]}_{\mathfrak{q}}^{},p+p_k^{}⟩,\hfill \\ & =\frac{1}{c^2}⟨[p,p]+{[p,p_k^{}]}_{\mathfrak{q}}^{},p+p_k^{}⟩,\hfill \\ & =\frac{1}{c^2}⟨{[p,p+p_k^{}]}_{\mathfrak{q}}^{},p+p_k^{}⟩=0.\hfill \end{array}$$

(16)

Further for $0\ne Uϵ\mathfrak{q}$, we get

$$C_{0n}^0=⟨{[U,p_n^{}]}_{\mathfrak{q}}^{},U⟩=⟨{[U,\frac{p}{c}]}_{\mathfrak{q}}^{},U⟩=0,$$

(17)

$$\begin{array}{cc}\hfill & C_{n0}^n=⟨{[p_n^{},U]}_{\mathfrak{q}}^{},p_n^{}⟩\hfill \\ & =\frac{1}{c^2}\left\{⟨{[p,U]}_{\mathfrak{q}}^{},p⟩+⟨{[p,U]}_{\mathfrak{q}}^{},U⟩\right\}\hfill \\ & =\frac{1}{c^2}⟨{[p,U]}_{\mathfrak{q}}^{},p+U⟩\hfill \\ & =\frac{1}{c^2}⟨[p,p]+{[p,U]}_{\mathfrak{q}}^{},p+U⟩\hfill \\ & =\frac{1}{c^2}⟨{[p,p+U]}_{\mathfrak{q}}^{},+U⟩=0,\hfill \end{array}$$

(18)

and

$$\begin{array}{cc}\hfill & C_{nt}^0+C_{n0}^t=⟨{[p_n^{},p_t^{}]}_{\mathfrak{q}}^{},U⟩+⟨{[p_n^{},U]}_{\mathfrak{q}}^{},p_t^{}⟩\hfill \\ & =\frac{1}{c}\left\{⟨{[p,p_t^{}]}_{\mathfrak{q}}^{},U⟩+⟨{[p,U]}_{\mathfrak{q}}^{},p_t^{}⟩\right\}\hfill \\ & =\frac{1}{c}\left\{⟨{[p,p_t^{}]}_{\mathfrak{q}}^{},U⟩+⟨{[p,U]}_{\mathfrak{q}}^{},p_t^{}⟩+⟨{[p,p_t^{}]}_{\mathfrak{q}}^{},p_t^{}⟩+⟨{[p,U]}_{\mathfrak{q}}^{},U⟩\right\}\hfill \\ & =\frac{1}{c}\left\{⟨{[p,p_t^{}]}_{\mathfrak{q}}^{},U+p_t^{}⟩+⟨{[p,U]}_{\mathfrak{q}}^{},p_t^{}+U⟩\right\}\hfill \\ & =\frac{1}{c}\{⟨{[p,p_t^{}]}_{\mathfrak{q}}^{}+{[p,U]}_{\mathfrak{q}}^{},p_t^{}+U⟩\hfill \\ & =\frac{1}{c}\{⟨{[p,p_t^{}+U]}_{\mathfrak{q}}^{},p_t^{}+U⟩=0.\hfill \end{array}$$

(19)

Plugging the value of (16)–(19) in (4), we get

$$Ric\left(U\right)=Ri{c}^{\alpha }\left(U\right)-\frac{c^2}{4}{\left(C_{k0}^n\right)}^2{D}_{19}^{}+\frac{c}{4}\alpha \left(U\right)\left\{2C_{a0}^n{C}_{ka}^k+C_{ka}^n{C}_{ka}^0\right\}{D}_{24}^{}-\frac{c^2}{4}{\alpha }^2\left(U\right){\left(C_{ik}^n\right)}^2{D}_{26}^{},$$

(20)

on putting the values of $D_{19}^{},D_{24}^{},D_{26}^{},$ which are given in Equation (6) and calculating by using Mathematica program, we get Equation (15). □

5. Conclusions

In order to characterize the Einstein metric and reversible Einstein metric, it is necessary to compute Ricci curvature. In this article, we have demonstrated an expression of Ricci curvature for the homogeneous generalized Matsumoto metric and also obtained the expression of Ricci curvature for the homogeneous Matsumoto metric. We have also deduced the expression of Ricci curvature for the homogeneous generalized Matsumoto metric with vanishing the S-curvature. It has several significant applications in cosmological models, general relativity, string theory, quantum gravity, geometry of spacetime and curvature analysis. We can explore the applications combined with singularity theory and submanifold theory, etc., as discussed in [23,24,25,26,27,28,29,30], intending to obtain additional new results.