# Cyclic Finser metrics on homogeneous spaces

Ju Tan  School of Microelectronics and Data Science, Anhui University of
Technology, Maanshan, 243032, P.R. China<EMAIL_ADDRESS>and  Ming Xu∗
School of Mathematical Sciences, Capital Normal University, Beijing 100048,
P.R. China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we generalize the notion of cyclic metric to homogeneous
Finsler geometry. Firstly, we prove that a homogeneous Finsler space $(G/H,F)$
must be symmetric when it satisfies the naturally reductive and cyclic
conditions simultaneously. Then we prove that a Finsler cyclic Lie group which
is either flat or nilpotent must have an Abelian Lie algebra. Finally, we show
how to induce a cyclic $(\alpha,\beta)$ metric from a cyclic Riemannian
metric. Using this method, we construct a Randers cyclic Lie group.

Mathematics Subject Classification(2010): 53C22, 53C30, 53C60.
Keywords: $(\alpha,\beta)$ metric, cyclic Lie group, cyclic metric,
homogeneous Finsler manifold, curvature

∗Ming Xu is the corresponding author.

##  1\. Introduction

We call a homogeneous Riemannian manifold $(G/H,\mathrm{g})$ cyclic with
respect to a given reductive decomposition
$\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$, if the following property is
satisfied

$\displaystyle\langle[x,y]_{{\mathfrak{m}}},z\rangle+\langle[y,z]_{{\mathfrak{m}}},x\rangle+\langle[z,x]_{{\mathfrak{m}}},y\rangle=0,\quad\forall
x,y,z\in{\mathfrak{m}}.$ (1.1)

Here $\langle\cdot,\cdot\rangle$ is the $\mathrm{Ad}(H)$-invariant inner
product on $\mathfrak{m}=T_{eH}(G/H)$ determined by the $G$-invariant metric
$\mathrm{g}$. When $H$ is trivial (and then the reductive decomposition must
be ${\mathfrak{g}}={\mathfrak{h}}+{\mathfrak{m}}=0+{\mathfrak{g}}$), we call
the cyclic $(G,\mathrm{g})$ a cyclic Lie group. Moreover, if $G$ is
unimodular, we specify the cyclic condition as the traceless cyclic condition.

The motivation for studying cyclic metrics can be traced back to Tricerri-
Vanhecke’s classification for homogeneous structures [18]. In their
classification, there are three basic ones, i.e., $\mathcal{S}_{1}$,
$\mathcal{S}_{2}$ and $\mathcal{S}_{3}$. The cyclic (traceless cyclic)
condition corresponds to the type $\mathcal{S}_{1}\oplus\mathcal{S}_{2}$
($\mathcal{S}_{2}$ respectively). Compared with the naturally reductive
metrics, which have been more extensively studied and correspond to the type
$\mathcal{S}_{3}$, cyclic metrics have very different properties, worthy to be
explored.

Here we briefly survey some related progress. Kowalski and Tricerri studied
the traceless cyclic condition in [14]. Gadea, Gonález-Dávila and Oubiña
concerned traceless cyclic Lie groups in [11], and characterized cyclic and
traceless cyclic metrics in [10]. In above literatures, relatively few
examples have been discovered. Only those with low dimensions have been
classified [4, 14]. Meanwhile, this project was generalized to homogeneous
pseudo-Riemannian geometry in [9]. Calvaruso and López classified Lorentzian
cyclic Lie groups of dimension 3 and 4 [5]. Tohidfa and Zaeim classified
cyclic Lie groups of the signature type $(2,2)$ [19].

In this paper, we explore the generalization of the cyclic property in Finsler
geometry, replacing (1.1) with the following analog,

$\langle[x,y]_{{\mathfrak{m}}},z\rangle_{y}+\langle[y,z]_{{\mathfrak{m}}},x\rangle_{y}+\langle[z,x]_{{\mathfrak{m}}},y\rangle_{y}=0,\quad\forall
x,z\in{\mathfrak{m}},y\in{\mathfrak{m}}\backslash\\{0\\}.$ (1.2)

Here the fundamental tensor $\langle\cdot,\cdot\rangle_{y}$ depends on $y$, so
the Finsler cyclic condition (1.2) is not multi-linear. Though this becomes an
essential obstacle, we can still prove some properties and find some examples
for cyclic Finsler metrics.

Firstly, we prove

######  Theorem 1.1.

If the homogeneous Finsler manifold is cyclic and naturally reductive with
respect to a given reductive decomposition, then that decomposition is a
Cartan decomposition, i.e., that homogeneous Finsler manifold is a symmetric
space.

The phenomenon in Theorem 1.1 was pointed out in [18] when the metric is
Riemannian. It implies that, in homogeneous Finsler geometry, the subclass of
cyclic Finsler metrics have a trivial overlap with that of naturally reductive
metrics, i.e., they generalize symmetric Finsler spaces in totally different
directions.

Secondly, we generalize Proposition 5.5 and Proposition 3.4 in [11] as
follows.

######  Theorem 1.2.

Non-Abelian nilpotent Lie groups do not admit cyclic left-invariant Finsler
metrics.

######  Theorem 1.3.

If $F$ is a left invariant cyclic Finsler metric on a Lie group $G$ which has
constant flag curvature $K\equiv 0$, then $G$ has an abelian Lie algebra.

Finally, we consider the construction of cyclic Finsler metrics. The following
theorem implies that cyclic Randers and $(\alpha,\beta)$ metrics may be
induced from those Riemannian ones.

######  Theorem 1.4.

Let $G/H$ be a homogeneous manifold with a given reductive decomposition
${\mathfrak{g}}={\mathfrak{h}}+{\mathfrak{m}}$, $\alpha$ an
$\mathrm{Ad}(H)$-invariant inner product on ${\mathfrak{m}}$, and
$\beta=\alpha(X,\cdot)$ in which $X\in{\mathfrak{m}}\backslash\\{0\\}$ is an
$\mathrm{Ad}(H)$-invariant vector. Suppose that $\alpha$ and
$F=\alpha\phi(\tfrac{\beta}{\alpha})$ induce a cyclic Riemannian metric and a
non-Riemannian homogeneous Douglas metric on $G/H$ respectively. Then
$(G/H,F)$ is cyclic if and only if

$\displaystyle\alpha([y,u]_{{\mathfrak{m}}},y)[\alpha(y,y)\alpha(X,v)-\alpha(X,y)\alpha(v,y)]=\alpha([y,v]_{{\mathfrak{m}}},y)[\alpha(y,y)\alpha(X,u)-\alpha(X,y)\alpha(u,y)]$

is satisfied for any $u,v,y\in{\mathfrak{m}}$.

As the application, we constructed left invariant non-Riemannian cyclic
Randers metrics on some solvable Lie groups.

##  2\. Preliminary in homogeneous Finsler geometry

Let $G/H$ be a homogeneous manifold, which is endowed with a reductive
decomposition ${\mathfrak{g}}={\mathfrak{h}}+{\mathfrak{m}}$. Here the
reductiveness means that the decomposition is $\mathrm{Ad}(H)$-invariant (it
implies that, in the Lie algebra level,
$[\mathfrak{h},\mathfrak{m}]\subset\mathfrak{m}$). Then the tangent space
$T_{o}(G/H)$ at the origin $o=eH$ can be identified as ${\mathfrak{m}}$, so
that the isotropy action coincides with the $\mathrm{Ad}(H)$-action on
${\mathfrak{m}}$.

Any $G$-invariant Finsler metric $F$ on $G/H$ can be one-to-one determined by
$F=F(o,\cdot)$, which is any arbitrary $\mathrm{Ad}(H)$-invariant Minkowski
norm on $\mathfrak{m}$ [6]. We call the pair $(G/H,F)$ a homogeneous Finsler
manifold. For example, a homogeneous $(\alpha,\beta)$ metric can be determined
by a Minkowski norm $F=\alpha\phi(\tfrac{\beta}{\alpha})$ on $\mathfrak{m}$,
in which $\alpha$ is an $\mathrm{Ad}(H)$-invariant Euclidean norm,
$\beta=\alpha(X,\cdot)$ is the $\alpha$-dual of some
$\mathrm{Ad}(H)$-invariant $X\in\mathfrak{m}$, and $\phi(r)$ is some positive
smooth function [8].

A homogeneous metric $F$ on $G/H$ is called naturally reductive with respect
to the given reductive decomposition
${\mathfrak{g}}={\mathfrak{h}}+{\mathfrak{m}}$, if

$\langle[x,u]_{\mathfrak{m}},v\rangle_{y}+\langle[x,v]_{\mathfrak{m}},u\rangle_{y}+2C_{y}([x,y]_{\mathfrak{m}},u,v)=0,$

for any $y\in{\mathfrak{m}}\backslash\\{0\\},x,u,v\in\mathfrak{m}$ [15]. Here

$\langle u,v\rangle_{y}=\tfrac{1}{2}\tfrac{\partial^{2}}{\partial s\partial
t}|_{s=t=0}F^{2}(y+su+tv),\quad
C_{y}(u,v,w)=\tfrac{1}{2}\tfrac{d}{dt}|_{t=0}\langle u,v\rangle_{y+tw}$

are the fundamental tensor and the Cartan tensor of the Minkowski norm $F$ on
$\mathfrak{m}$ respectively. See [7] for the equivalent definition and
description of this natural reductiveness.

The spray vector field
$\eta:{\mathfrak{m}}\backslash\\{0\\}\rightarrow{\mathfrak{m}}$ and the
connection operator
$N:({\mathfrak{m}}\backslash\\{0\\})\times{\mathfrak{m}}\rightarrow{\mathfrak{m}}$
for a homogeneous Finsler manifold $(G/H,F)$ with respect to a given reductive
decomposition ${\mathfrak{g}}={\mathfrak{h}}+{\mathfrak{m}}$ are determined by
the following equations respectively [12],

$\displaystyle\langle\eta(y),u\rangle_{y}$ $\displaystyle=$
$\displaystyle\langle y,[u,y]_{\mathfrak{m}}\rangle_{y},\quad\forall
y\in{\mathfrak{m}}\backslash\\{0\\},$ $\displaystyle 2\langle
N(y,v),u\rangle_{y}$ $\displaystyle=$
$\displaystyle\langle[u,v]_{\mathfrak{m}},y\rangle_{y}+\langle[u,y]_{\mathfrak{m}},v\rangle_{y}+\langle[v,y]_{m},u\rangle_{y}$
(2.3) $\displaystyle-2C_{y}(u,v,\eta(y)),\quad\forall
y\in{\mathfrak{m}}\backslash\\{0\\},u,v\in{\mathfrak{m}}.$

They are useful for presenting curvature formulae of homogeneous Finsler
manifolds. For example, we have the following homogeneous Riemann curvature
formulae. [12, 13].

######  Theorem 2.1.

For a left invariant Finsler metric $F$ on the Lie group $G$, the Riemann
curvature $R_{y}:{\mathfrak{g}}=T_{e}G\rightarrow T_{e}G={\mathfrak{g}}$ for
any $y\in{\mathfrak{g}}\backslash\\{0\\}$ can be presented as

$R_{y}(u)=D_{\eta}N(y,u)-N(y,N(y,u))+N(y,[y,u])-[y,N(y,u)].$ (2.4)

More details on the curvatures in general Finsler geometry can be found in
[3].

##  3\. Cyclic Finsler metric and its properties

Now we are ready to present the precise definition for the Finsler cyclic
condition.

######  Definition 3.1.

A homogeneous Finsler manifold $(G/H,F)$ is said to be cyclic with respect to
a given reductive decomposition
${\mathfrak{g}}={\mathfrak{h}}+{\mathfrak{m}}$, if we have

$\langle[x,y]_{{\mathfrak{m}}},z\rangle_{y}+\langle[y,z]_{{\mathfrak{m}}},x\rangle_{y}+\langle[z,x]_{{\mathfrak{m}}},y\rangle_{y}=0,\quad\forall
y\in{\mathfrak{m}}\backslash\\{0\\},x,z\in{\mathfrak{m}}.$

In particular, when $H=\\{e\\}$ (and then the reductive decomposition must be
${\mathfrak{g}}={\mathfrak{h}}+{\mathfrak{m}}=0+{\mathfrak{g}}$), we call the
cyclic $(G,F)$ a (Finsler) cyclic Lie group.

Proof of Theorem 1.1. Suppose the homogeneous Finsler metric $F$ is naturally
reductive and cyclic with respect to the reductive decomposition
${\mathfrak{g}}={\mathfrak{h}}+{\mathfrak{m}}$. Choose any
$y\in{\mathfrak{m}}\backslash\\{0\\}$, $x,z\in{\mathfrak{m}}$. Then the
natural reductiveness provides

$\langle[x,u]_{\mathfrak{m}},v\rangle_{y}+\langle[x,v]_{\mathfrak{m}},u\rangle_{y}+2C_{y}([x,y]_{\mathfrak{m}},u,v)=0,\quad\forall
u,v\in{\mathfrak{m}}.$ (3.5)

Input $u=y$ and $v=z$ into (3.5), we get

$\langle[x,y]_{\mathfrak{m}},z\rangle_{y}+g_{y}([x,z]_{\mathfrak{m}},y)=0.$
(3.6)

Similar argument also provides

$\langle[z,y]_{\mathfrak{m}},x\rangle_{y}+g_{y}([z,x]_{\mathfrak{m}},y)=0.$
(3.7)

The cyclic condition provides

$\langle[x,y]_{\mathfrak{m}},z\rangle_{y}+\langle[y,z]_{\mathfrak{m}},x\rangle_{y}+\langle[z,x]_{\mathfrak{m}},y\rangle_{y}=0.$
(3.8)

The sum of (3.7) and (3.8) minus (3.6) provides
$3\langle[z,x]_{\mathfrak{m}},y\rangle_{y}=0$. So we get
$\langle[\mathfrak{m},\mathfrak{m}]_{\mathfrak{m}},y\rangle_{y}=0$, $\forall
y\in{\mathfrak{m}}\backslash\\{0\\}$. It can only happen when
$[\mathfrak{m},\mathfrak{m}]\subset\mathfrak{h}$, i.e, $G/H$ is symmetric.

To prove Theorem 1.2 and Theorem 1.3, we need the following lemma.

######  Lemma 3.2.

Let $(G,F)$ be a Finsler cyclic Lie group. Then we have the following:

  1. (1)

If $y\in{\mathfrak{g}}\backslash\\{0\\}$ satisfies
$\langle[\mathfrak{g},{\mathfrak{g}}],y\rangle_{y}=0$, then $\eta(y)=0$ and
$N(y,\cdot)=-\mathrm{ad}(y)$. Moreover, in this situation,
$R_{y}=-\mathrm{ad}(y)^{2}$ and $\mathrm{ad}(y)$ is self-adjoint with respect
to $\langle\cdot,\cdot\rangle_{y}$.

  2. (2)

Any $y\in\mathfrak{c}(\mathfrak{g})\backslash\\{0\\}$ satisfies
$\langle[\mathfrak{g},{\mathfrak{g}}],y\rangle_{y}=0$ and $R_{y}\equiv 0$.

Proof. (1) Since $\langle[{\mathfrak{g}},{\mathfrak{g}}],y\rangle_{y}=0$, we
have $\eta(y)=0$. By the cyclic condition, we have

$\langle[u,y],v\rangle_{y}-\langle[v,y],u\rangle_{y}=\langle[u,y],v\rangle_{y}+\langle[v,u],y\rangle_{y}+\langle[y,v],u\rangle_{y}=0,$
(3.9)

so (2.3) provides

$\displaystyle 2\langle N(y,v),u\rangle_{y}$ $\displaystyle=$
$\displaystyle\langle[u,v],y\rangle_{y}+\langle[u,y],v\rangle_{y}+\langle[v,y],u\rangle_{y}-2C_{y}(u,v,\eta(y))$
$\displaystyle=$
$\displaystyle\langle[u,y],v\rangle_{y}+\langle[v,y],u\rangle_{y}=2\langle[v,y],u\rangle_{y}.$

It implies $N(y,v)=[v,y]$, $\forall v\in{\mathfrak{g}}$, i.e.,
$N(y,\cdot)=-\mathrm{ad}(y)$. Input $\eta(y)=0$ and
$N(y,\cdot)=-\mathrm{ad}(y)$ into (2.4) in Theorem 2.1, we get
$R_{y}=-\mathrm{ad}(y)^{2}$ immediately. Finally, using (3.9), we see that
$N(y,\cdot)$ is self-adjoint with respect to $\langle\cdot,\cdot\rangle_{y}$.

(2) Assume $y\in\mathfrak{c}(\mathfrak{g})\backslash\\{0\\}$. By the cyclic
condition, we have $\forall x,z\in{\mathfrak{g}}$,

$\langle[z,x],y\rangle_{y}=\langle[x,y],z\rangle_{y}+\langle[y,z],x\rangle_{y}+\langle[z,x],y\rangle_{y}=0,$

i.e., $\langle[{\mathfrak{g}},{\mathfrak{g}}],y\rangle_{y}=0$. By (1) of Lemma
3.2, $R_{y}=-\mathrm{ad}(y)^{2}=0$ follows immediately.

Proof of Theorem 1.2. Assume conversely that $G$ admits a left invariant
cyclic Finsler metric and its Lie algebra $\mathfrak{g}$ is non-Abelian. By
the proof of Theorem 5.1 in [12], there exists a nonzero
$y\in\mathfrak{c}(\mathfrak{g})$ with $\mathrm{Ric}(y)=\mathrm{tr}(R_{y})>0$.
This is a contradiction with (2) in Lemma 3.2.

Proof of Theorem 1.3. Assume $(G,F)$ is a Finsler cyclic Lie group, i.e., $F$
is a left invariant Finsler metric on $G$ which is cyclic with respect to the
reductive decomposition
${\mathfrak{g}}={\mathfrak{h}}+{\mathfrak{m}}=0+{\mathfrak{g}}$. We will prove
${\mathfrak{g}}$ is Abelian by the following three claims.

Claim I: $[{\mathfrak{g}},{\mathfrak{g}}]$ is commutative.

The left invariance of $F$ implies that its Cartan tensor and Landsberg tensor
are both bounded. So the argument proving Akbar-Zadeh’s theorem [1] can be
applied here to prove that $F$ is locally Minkowskian. We average the
fundamental tensor of $F$ on the indicatrix in each tangent space, with
respect to the volume form induced by the Hessian metric, then we get a
Riemannian metric $\overline{F}$ on $G$. Because $F$ is locally Minkowskian,
$\overline{F}$ is flat. This averaging process preserves Killing fields and
isometries. So $\overline{F}$ is left $G$-invariant. By Theorem 1.5 in [17],
$\mathfrak{g}=\mathfrak{l}+\mathfrak{u}$, in which $\mathfrak{l}$ is a
commutative subalgebra and $\mathfrak{u}$ is a commutative ideal. So we see
$[{\mathfrak{g}},{\mathfrak{g}}]\subset\mathfrak{u}$ is commutative, which
proves Claim I.

Claim II:
$\mathfrak{c}(\mathfrak{g})+[{\mathfrak{g}},{\mathfrak{g}}]={\mathfrak{g}}$.

Assume conversely that
$\mathfrak{c}(\mathfrak{g})+[{\mathfrak{g}},{\mathfrak{g}}]\neq{\mathfrak{g}}$.
Then there exists a nonzero vector
$y\in{\mathfrak{g}}\backslash(\mathfrak{c}(\mathfrak{g})+[{\mathfrak{g}},{\mathfrak{g}}])$,
such that
$\langle\mathfrak{c}(\mathfrak{g})+[{\mathfrak{g}},{\mathfrak{g}}],y\rangle_{y}=0$.
Because $g_{y}([\mathfrak{g},\mathfrak{g}],y)=0$, we can apply the flatness
condition for $F$ and (1) of Lemma 3.2 to see that $\mathrm{ad}(y)$ is self
adjoint with respect to $\langle\cdot,\cdot\rangle_{y}$ and
$R_{y}(v)=-\mathrm{ad}(y)^{2}v=0$ for any $v\in{\mathfrak{g}}$. Then we have

$\langle\mathrm{ad}(y)v,\mathrm{ad}(y)v\rangle_{y}=\langle\mathrm{ad}(y)^{2}v,v\rangle_{y}=-\langle
R_{y}(v),v\rangle_{y}=0,\quad\forall v\in{\mathfrak{g}},$

i.e., $y\in\mathfrak{c}(\mathfrak{g})$. This is a contradiction, which proves
Claim II.

Claim III:
$[{\mathfrak{g}},{\mathfrak{g}}]\subset\mathfrak{c}({\mathfrak{g}})$.

Choose any $u\in{\mathfrak{g}}$ and $v\in[{\mathfrak{g}},{\mathfrak{g}}]$. By
Claim II, we have the decomposition $u=u_{1}+u_{2}$ with
$u_{1}\in\mathfrak{c}({\mathfrak{g}})$ and
$u_{2}\in[{\mathfrak{g}},{\mathfrak{g}}]$. By Claim I, $[v,u_{2}]=0$,
$[v,u]=[v,u_{1}]+[v,u_{2}]=0$. So we see
$[{\mathfrak{g}},{\mathfrak{g}}]\subset\mathfrak{c}({\mathfrak{g}})$, which
proves Claim III.

Summarizing Claim II and Claim III, we prove
${\mathfrak{g}}=\mathfrak{c}({\mathfrak{g}})$, which ends the proof.

##  4\. Construction of cyclic $(\alpha,\beta)$ metrics

To prove Theorem 1.4, we need the following lemma.

######  Lemma 4.1.

Keeping all assumptions and notations in Theorem 1.4, then $F$ is cyclic if
and only if $\Phi(r)\cdot\Psi(u,v,y)=0$ is satisfied for any
$u,v\in{\mathfrak{m}}$, $y\in{\mathfrak{m}}\backslash\\{0\\}$, and
$r=\tfrac{\alpha(X,y)}{\alpha(y,y)^{1/2}}$, in which

$\displaystyle\Phi(r)=\phi(r)\phi^{\prime}(r)-r\phi^{\prime}(r)^{2}-r\phi(r)\phi^{\prime\prime}(r)$

and

$\displaystyle\Psi(u,v,y)$ $\displaystyle=$
$\displaystyle\alpha(y,[y,u]_{\mathfrak{m}})(\alpha(y,y)\alpha(X,v)-\alpha(y,v)\alpha(X,y))$
$\displaystyle+\alpha(y,[v,y]_{\mathfrak{m}})(\alpha(y,y)\alpha(X,u)-\alpha(y,u)\alpha(X,y)).$

Proof. Direct calculation for the fundamental tensor shows that, for any
$y\in{\mathfrak{m}}\backslash 0$, $u,v\in{\mathfrak{m}}$,

$\displaystyle\langle u,v\rangle_{y}=\tfrac{1}{2}\tfrac{\partial^{2}}{\partial
s\partial t}|_{s=t=0}F^{2}(y+su+tv)$ (4.10) $\displaystyle=$
$\displaystyle\tfrac{1}{2}\tfrac{\partial^{2}}{\partial s\partial
t}|_{s=t=0}\left(\alpha(y+su+tv,y+su+tv)\phi(\tfrac{\alpha(X,y+su+tv)}{\sqrt{\alpha(y+su+tv,y+su+tv)}})^{2}\right)$
$\displaystyle=$
$\displaystyle\phi(r)^{2}\alpha(u,v)-\phi(r)\phi^{\prime}(r)\cdot\tfrac{\alpha(y,u)\alpha(y,v)\alpha(X,y)}{\alpha(y,y)^{3/2}}$
$\displaystyle+\phi(r)\phi^{\prime}(r)\cdot\tfrac{\alpha(y,v)\alpha(X,u)+\alpha(y,u)\alpha(X,v)-\alpha(u,v)\alpha(X,y)}{\alpha(y,y)^{1/2}}$
$\displaystyle+(\phi^{\prime}(r)^{2}+\phi(r)\phi^{\prime\prime}(r))(\alpha(X,u)-\tfrac{\alpha(y,u)\alpha(X,y)}{\alpha(y,y)})(\alpha(X,v)-\tfrac{\alpha(y,v)\alpha(X,y)}{\alpha(y,y)}).$

Setting

$\displaystyle Q_{1}=\alpha(X,[y,u]_{\mathfrak{m}})\alpha(y,v),\quad
Q_{2}=\alpha(X,[y,u]_{\mathfrak{m}}),\quad
Q_{3}=\alpha(X,[u,v]_{\mathfrak{m}})\alpha(y,y),$ $\displaystyle
Q_{4}=\alpha(X,[v,y]_{\mathfrak{m}})\alpha(y,u),\quad
Q_{5}=\alpha(X,[v,y]_{\mathfrak{m}}),$

then we can get the following from (4.10),

$\displaystyle\langle[y,u]_{\mathfrak{m}},v\rangle_{y}$ $\displaystyle=$
$\displaystyle\alpha([y,u]_{\mathfrak{m}},v)\phi(r)^{2}-\phi(r)\phi^{\prime}(r)\cdot\tfrac{\alpha(y,[y,u]_{\mathfrak{m}})\alpha(y,v)\alpha(X,y)}{\alpha(y,y)^{3/2}}$
(4.11)
$\displaystyle+\phi(r)\phi^{\prime}(r)\cdot\tfrac{Q_{1}+\alpha(y,[y,u]_{{\mathfrak{m}}})\alpha(X,v)-\alpha(v,[y,u]_{\mathfrak{m}})\alpha(X,y)}{\alpha(y,y)^{1/2}}$
$\displaystyle+(\phi(r)\phi^{\prime\prime}(r)+\phi^{\prime}(r)^{2})\cdot(Q_{2}-\tfrac{\alpha(y,[y,u]_{\mathfrak{m}})\alpha(X,y)}{\alpha(y,y)})$
$\displaystyle\cdot(\alpha(X,v)-\tfrac{\alpha(y,v)\alpha(X,y)}{\alpha(y,y)}),$
$\displaystyle\langle[u,v]_{\mathfrak{m}},y\rangle_{y}$ $\displaystyle=$
$\displaystyle\alpha([u,v]_{\mathfrak{m}},y)\phi(r)^{2}-\phi(r)\phi^{\prime}(r)\cdot\tfrac{\alpha(y,[u,v]_{\mathfrak{m}})\alpha(X,y)}{\alpha(y,y)^{1/2}}$
(4.12)
$\displaystyle+\phi(r)\phi^{\prime}(r)\cdot\tfrac{Q_{3}}{\alpha(y,y)^{1/2}},$
$\displaystyle\langle[v,y]_{\mathfrak{m}},u\rangle_{y}$ $\displaystyle=$
$\displaystyle\alpha([v,y]_{\mathfrak{m}},u)\phi(r)^{2}-\phi(r)\phi^{\prime}(r)\cdot\tfrac{\alpha(y,[v,y]_{\mathfrak{m}})\alpha(y,u)\alpha(X,y)}{\alpha(y,y)^{3/2}}$
(4.13)
$\displaystyle+\phi(r)\phi^{\prime}(r)\cdot\tfrac{Q_{4}+\alpha(y,[v,y]_{\mathfrak{m}})\alpha(X,u)-\alpha(u,[v,y]_{\mathfrak{m}})\alpha(X,y)}{\alpha(y,y)^{1/2}}$
$\displaystyle+(\phi(r)\phi^{\prime\prime}(r)+\phi^{\prime}(r)^{2})\cdot(Q_{5}-\tfrac{\alpha(y,[v,y]_{\mathfrak{m}})\alpha(X,y)}{\alpha(y,y)})$
$\displaystyle\cdot(\alpha(X,u)-\tfrac{\alpha(y,u)\alpha(X,y)}{\alpha(y,y)}).$

The cyclic condition for $\alpha$ implies

$\alpha([y,u]_{\mathfrak{m}},v)+\alpha([u,v]_{\mathfrak{m}},y)+\alpha([v,y]_{\mathfrak{m}},u)=0.$

Since $F$ is a non-Riemannian Douglas metric, we have
$\alpha([{\mathfrak{m}},{\mathfrak{m}}]_{{\mathfrak{m}}},X)=0$ (see [2] or
[16]), so $Q_{1}=Q_{2}=Q_{3}=Q_{4}=Q_{5}=0$. Using these observations, the sum
of (4.11)-(4.13) provides

$\displaystyle\langle[y,u]_{\mathfrak{m}},v\rangle_{y}+\langle[u,v]_{\mathfrak{m}},y\rangle_{y}+\langle[v,y]_{\mathfrak{m}},u\rangle_{y}$
$\displaystyle=$
$\displaystyle\phi(r)\phi^{\prime}(r)\cdot\tfrac{\alpha(y,[y,u]_{\mathfrak{m}})(\alpha(y,y)\alpha(X,v)-\alpha(y,v)\alpha(X,y))+\alpha(y,[v,y]_{\mathfrak{m}})(\alpha(y,y)\alpha(X,u)-\alpha(y,u)\alpha(X,y))}{\alpha(y,y)^{3/2}}$
$\displaystyle-(\phi(r)\phi^{\prime\prime}(r)+\phi^{\prime}(r)^{2})\tfrac{\alpha(X,y)}{\alpha(y,y)^{1/2}}\cdot$
$\displaystyle\tfrac{\alpha(y,[y,u]_{\mathfrak{m}})(\alpha(y,y)\alpha(X,v)-\alpha(y,v)\alpha(X,y))+\alpha(y,[v,y]_{\mathfrak{m}})(\alpha(y,y)\alpha(X,u)-\alpha(y,u)\alpha(X,y))}{\alpha(y,y)^{3/2}}$
$\displaystyle=$
$\displaystyle\tfrac{1}{\alpha(y,y)^{3/2}}\cdot\Phi(r)\cdot\Psi(u,v,y).$

So $F$ is cyclic, i.e.,
$\langle[y,u]_{\mathfrak{m}},v\rangle_{y}+\langle[u,v]_{\mathfrak{m}},y\rangle_{y}+\langle[v,y]_{\mathfrak{m}},u\rangle_{y}\equiv
0$, if and only if $\Phi(r)\cdot\Psi(u,v,y)\equiv 0$, which proves the lemma.

Now we prove Theorem 1.4, i.e., $F$ is cyclic if and only if
$\Psi(u,v,y)\equiv 0$.

Proof of Theorem 1.4. Assume $\Psi(u,v,y)\equiv 0$, then Lemma 4.1 indicates
$F$ is cyclic. This proves one side of the theorem.

Assume $F$ is cyclic. By Lemma 4.1, we have $\Phi(r)\cdot\Psi(u,v,y)\equiv 0$.
Notice that $\Phi(r)$ can not be constantly zero when
$|r|\leq\alpha(X,X)^{1/2}$, otherwise
$\Phi(r)=(\phi(r)(\phi(r)-r\phi^{\prime}(r)))^{\prime}\equiv 0$, i.e.,
$\phi(r)^{2}-r\phi\phi^{\prime}(r)\equiv c$ for some constant $c$, which can
be easily solved and provides the solutions
$\phi(r)\equiv\sqrt{c_{1}r^{2}+c_{2}}$ for some constants $c_{1}$ and $c_{2}$.
Then the metric $F=\alpha\phi(\tfrac{\beta}{\alpha})$ is Riemannian, which
contradicts our assumption. To summarize, there exists a nonempty open subset
$\mathcal{U}\subset{\mathfrak{m}}\backslash\\{0\\}$, such that $\Psi(u,v,y)=0$
when $y\in\mathcal{U}$. Since $\Psi(u,v,y)$ is a polynomial, it must vanish
identically. This ends the proof for the other side of Theorem 1.4.

Now we use Theorem 1.4 to construct a left invariant cyclic Randers metric.

Let $G$ be a solvable Lie group. Its Lie algebra ${\mathfrak{g}}$ has a chain
of ideals,

${\mathfrak{g}}={\mathfrak{g}}_{0}\supset{\mathfrak{g}}_{1}\supset\cdots\supset{\mathfrak{g}}_{n}=0,$

in which $\dim{\mathfrak{g}}_{i}=n-i$, $\forall 0\leq i\leq n$. We can find a
basis $\\{e_{1},\cdots,e_{n}\\}$ of ${\mathfrak{g}}$, such that
${\mathfrak{g}}_{n-i}=\mathrm{span}\\{e_{1},\cdots,e_{i}\\}$ for each $i$, and
then an inner product $\alpha(\cdot,\cdot)$ on ${\mathfrak{g}}$, such that
$\langle e_{i},e_{j}\rangle=\delta_{ij}$. The following proposition in [11]
answers exactly when this $\alpha$ induces a left invariant cyclic metric on
$G$.

######  Proposition 4.2.

$(G,\alpha)$ is cyclic if and only if $\mathrm{ad}(e_{i})$ is self adjoint on
${\mathfrak{g}}_{n-i}$ for each $i$, or equivalently, the coefficients
$c_{ij}^{k}$ in $[e_{i},e_{j}]=c_{ij}^{k}e_{k}$ satisfy
$c_{ik}^{j}=c_{jk}^{i}$, $\forall 1\leq i<j<k\leq n$.

According to Proposition 4.2, $(G,\alpha)$ is cyclic when

$\displaystyle[e_{i},e_{j}]=0,1\leq i<j\leq
n-1,\quad[e_{n},e_{i}]=ae_{i},1\leq i\leq n-1,$

in which $a$ is a positive number. We choose $X=ce_{n}$ for some
$c\in(-1,1)\backslash\\{0\\}$ and $\beta=\langle X,\cdot\rangle$. Then
$F=\alpha+\beta$ induces a left invariant Randers metric on $G$. It is a non-
Riemannian Douglas metric because
$\alpha(X,[{\mathfrak{g}},{\mathfrak{g}}])=0$ [2, 6]. Calculation shows that
for any generic $u,v,y$,

$\tfrac{\alpha(y,y)\alpha(X,u)-\alpha(X,y)\alpha(u,y)}{\alpha([y,u],y)}=-\tfrac{c}{a}=\tfrac{\alpha(y,y)\alpha(X,v)-\alpha(X,y)\alpha(v,y)}{\alpha([y,v],y)},$

i.e., $\alpha$ satisfies the equation in Theorem 1.4, so $F$ is cyclic. Using
the same $\alpha$ and $\beta$, left invariant cyclic $(\alpha,\beta)$ metrics
can be similarly constructed.

Acknowledgement. This paper is supported by National Natural Science
Foundation of China (No. 12001007, No. 12131012, No. 11821101), Beijing
Natural Science Foundation (No. 1222003), Natural Science Foundation of Anhui
province (No. 1908085QA03).

## References

  * [1] H. Akbar-Zadeh, Sur les espaces de Finsler á courbures sectionnelles constantes, Bull. Acad. Roy. Bel. Cl, Sci, 5e Série-Tome LXXXIV (1988), 281-322.
  * [2] H. An and S. Deng, Invariant $(\alpha,\beta)$-metrics on homogeneous manifolds, Monatsh. Math. 154 (2008), 89-102.
  * [3] D. Bao, S.S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer, New York, 2000.
  * [4] L. Bieszk, Classification of five-dimensional Lie algebras of class $\mathcal{F}_{2}$, Demonstratio Math. 30 (2) (1997) 403-424.
  * [5] G. Calvaruso and M. Castrillón-López, Cyclic Lorentzian Lie groups, Geom. Dedicata 181 (1) (2016) 119-136.
  * [6] S. Deng, Homogeneous Finsler spaces, Springer, New York, 2012.
  * [7] S. Deng and Z. Hou, Naturally reductive homogeneous Finsler spaces. Manuscripta Math. 131 (1-2) (2010) 215-229.
  * [8] S. Deng and Z. Hu, Curvatures of homogeneous Randers spaces, Adv. Math. 240 (2013) 194-226.
  * [9] P.M. Gadea and J.A. Oubiña: Homogeneous pseudo-Riemannian structures and homogeneous almost para-Hermitian structures, Houston J. Math. 18 (3) (1992) 449-465.
  * [10] P.M. Gadea, J.C. González-Dávila and J.A. Oubiña, Cyclic homogeneous Riemannian manifolds, An. Mat. Pura Appl. 195 (2016) 1619-1637.
  * [11] P.M. Gadea, J.C. González-Dávila and J.A. Oubiña, Cyclic metric Lie groups. Monatsh. Math. 176(2),(2015) 219-239.
  * [12] L. Huang, Ricci curvatures of left invariant Finsler metrics on Lie groups. Israel J. Math. 207 (2015), no. 2, 783-792.
  * [13] L. Huang, On the fundamental equations of homogeneous Finsler spaces. Differential Geom. Appl. 40 (2015), 187-208.
  * [14] O. Kowalski and F. Tricerri: Riemannian manifolds of dimension $n\leq 4$ admitting a homogeneous structure of class $\mathcal{F}_{2}$, in: Conferenze del Seminario di Matematica, Universita di Bari, vol. 222, Laterza, Bari (1987) 24pp.
  * [15] D. Latifi, Homogeneous geodesic on homogeneous Finsler spaces. J.Geom. Phys. 57, (2007), 1421-1433.
  * [16] H. Liu and S. Deng, Homogeneous $(\alpha,\beta)$-metrics of Douglas type. Forum Math. 27 (5) (2015), 3149-3165.
  * [17] J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976), 293-329.
  * [18] F. Tricerri and L. Vanhecke, Homogeneous structures on Riemannian manifolds. London Mathematical Society Lecture Note Series 83, Cambridge University Press, Cambridge (1983)
  * [19] A. Tohidfar and A. Zaeim, On pseudo-Riemannian cyclic homogeneous manifolds of dimension four, J. Lie Theory 31 (2021), No. 1, 169-187.