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\title{Geodesic orbit and weakly symmetric spray manifolds}
\alttitle{Orbite gÃ©odÃ©sique et variÃ©tÃ©s de spray faiblement symÃ©triques}

\author{\firstname{Xiyun} \lastname{Xu}}
\address[1]{School of Mathematical Sciences, Capital Normal University, Beijing 100048, P.R.~China}
\email[X. Xu]{2210502058@cnu.edu.cn}

\author{\firstname{Ming} \lastname{Xu}\IsCorresp}
\address{School of Mathematical Sciences, Capital Normal University, Beijing 100048, P.R.~China}
\email[M. Xu]{mgmgmgxu@163.com}

\subjclass{22E46, 53C30}

\keywords{\kwd{Homogeneous geodesic} \kwd{homogeneous spray manifold} \kwd{weakly symmetric space} \kwd{geodesic orbit space} \kwd{spray structure} \kwd{spray vector field}}
\altkeywords{\kwd{GÃ©odÃ©sique homogÃ¨ne} \kwd{espace de pulvÃ©risation homogÃ¨ne} \kwd{espace faiblement symÃ©trique} \kwd{espace d'orbite gÃ©odÃ©sique} \kwd{structure de pulvÃ©risation} \kwd{champ de vecteurs de pulvÃ©risation}}

\thanks{This paper is supported by National Natural Science Foundation of China (No.~11821101, No. ~12131012), and Beijing Natural Science Foundation (No.~1222003)}
\CDRGrant[National Natural Science Foundation of China]{11821101}
\CDRGrant[National Natural Science Foundation of China]{12131012}
\CDRGrant[Beijing Natural Science Foundation]{1222003}

\begin{abstract}
In this paper, we introduce the geodesic orbit and weakly symmetric properties in homogeneous spray geometry. When a homogeneous spray manifold is endowed with a reductive decomposition, we use the spray vector field to describe these properties, and prove that a weakly symmetric spray manifold must be geodesic orbit, which generalizes its analog in homogeneous Riemannian and Finsler geometries.
\end{abstract}

\begin{altabstract}
Dans cet article, nous introduisons l'orbite gÃ©odÃ©sique et les propriÃ©tÃ©s faiblement symÃ©triques dans la gÃ©omÃ©trie homogÃ¨ne de spray. Lorsqu'un collecteur homogÃ¨ne de spray est dotÃ© d'une dÃ©composition rÃ©ductrice, nous utilisons le champ de vecteurs de spray pour dÃ©crire ces propriÃ©tÃ©s, et prouvons qu'un collecteur de spray faiblement symÃ©trique doit avoir une orbite gÃ©odÃ©sique, ce qui gÃ©nÃ©ralise son analogue dans les gÃ©omÃ©tries homogÃ¨nes de Riemann et de Finsler.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}

A connected Riemannian homogeneous manifold is called \emph{geodesic orbit} (or \emph{g.o.} in short), if any geodesic is the orbit of a one-parameter subgroup of isometries~\cite{Go1996}. This notion was introduced by O.~Kowalski and L.~Vanhecke in 1991~\cite{KV1991}, as a generalization of the naturally reductive homogeneity. Since then, there have been many research works on this subject (see~\cite{BN2020} and references therein).

G.o.~Riemannian manifolds include many important subclasses. For example, A.~Selberg proposed the notion of \emph{weak symmetry} in 1956~\cite{Se1956}. A connected homogeneous Riemannian manifold $G/H$ is called {weakly symmetric} if and only if any two points on $G/H$ can be exchanged by some $g\in G$, or equivalently, for any geodesic $c(t)$, there exists $g\in G$, such that $g\cdot c(t)\equiv c(-t)$. Weak symmetry has interesting connections with spherical spaces, commutative spaces, Gelfand pairs, etc.\ \cite{Wo2007}. It is a well-known fact that a weakly symmetric Riemannian manifold must be g.o.\ (see~\cite{BN2018,BKV1997} for two different proofs).

In recent years, g.o.\ and weakly symmetric properties were generalized by S.~Deng and his coworkers to Finsler geometry~\cite{DH2010,YD2014}. By similar arguments as in~\cite{BKV1997}, they proved that a weakly symmetric Finsler manifold is g.o.~\cite{De2012}. Beyond the Riemann--Finsler geometry, there is a more general spray geometry, where the metrics are ignored and the (properly parametrized) geodesics are concerned~\cite{Sh2001}. Nowadays, spray geometry has caught the attention of many geometers~\cite{LM2021,LS2018}, and it has been studied in the homogeneous context recently~\cite{HM2021,Xu2022-1,Xu2022,Xu2023}. Homogeneous spray geometry generalizes homogeneous Riemannian and Finsler geometries (indeed, it generalizes homogeneous pseudo-Riemannian and pseudo-Finsler geometries as well).

In this paper, we generalize the g.o.\ and weakly symmetric properties to homogeneous spray geometry. To avoid iterations, we automatically assume the homogeneous manifolds are connected and Lie groups have only countable connected components.

\begin{defi}\label{def-spray-g.o. space}
We call a homogeneous spray manifold $(G/H,\mathbf{G})$ \emph{g.o.}\ if for any geodesic $c(t)$, which is defined when $t$ is sufficiently close to\/ $0$, there exists $u\in \mathfrak{g}=\Lie(G)$ such that $c(t)\equiv\exp(tu)\cdot c(0)$.
\end{defi}

\begin{defi}\label{def-spray-weakly symmetric}
We call a homogeneous spray manifold $(G/H,\mathbf{G})$ \emph{weakly symmetric} if for any geodesic $c(t)$, which is defined when $t$ belongs to $(-\epsilon,\epsilon)$, where $\epsilon$ is some positive number, there exists $g\in G$ such that $g\cdot c(t)\equiv c(-t)$.
\end{defi}

\begin{rema}
A homogeneous spray manifold may not be complete (see~\cite[Section~4.3]{Xu2022-1} for an explicit example), so the geodesics $c(t)$ in above definitions may not be defined for all $t\in\mathbb{R}$.
\end{rema}

Next, we use the spray vector field $\eta$~\cite{Xu2022-1,Xu2022} to characterize these properties. Suppose that $G/H$ admits a reductive decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$. Then, $\eta$ is an $\Ad(H)$-equivariant positively 2-homogeneous smooth map from $\mathfrak{m}\backslash\{0\}$ to $\mathfrak{m}$. Conversely, any map satisfying this description is the spray vector field for some homogeneous spray structure on $G/H$ (see~\cite[Theorem~5.5]{Xu2022}). Usually $\eta$ can be viewed as a smooth tangent vector field on $\mathfrak{m}\backslash\{0\}$.

\begin{theo}\label{thm-1}
Let $(G/H,\mathbf{G})$ be a homogeneous spray manifold with a reductive decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ and the spray vector field $\eta$. Then $(G/H,\mathbf{G})$ is g.o.~if and only if $\eta$ is tangent to all $\Ad(H)$-orbits in\/ $\mathfrak{m}\backslash\{0\}$, i.e., $\eta(y)\in T_{y}\parens[\big]{\Ad(H)y}=[\mathfrak{h},y]$, $\forall y\in\mathfrak{m}\backslash\{0\}$.
\end{theo}

The orbit $\Ad(H)y$ in Theorem~\ref{thm-1} is viewed as a submanifold in $\mathfrak{m}\backslash\{0\}$ (see~\cite[Theorem~2.9.7]{Va1984}). Theorem~\ref{thm-1} helps us find g.o.~spray manifolds. For example, any homogeneous manifold $G/H$ with a reductive decomposition can be endowed with a canonical g.o.~spray structure $\mathbf{G}_0$ with a vanishing spray vector field (see~\cite[p.~30]{Xu2022}). Theorem~\ref{thm-1} also reveals some difference between homogeneous Finsler and spray geometries. For example, on $S^{2n-1}=U(n)/U(n-1)$, any homogeneous Finsler metric is g.o.~\cite{Xu2018}, but non-g.o.~homogeneous spray structures can be easily constructed.

\begin{theo}\label{thm-2}
Let $(G/H,\mathbf{G})$ be a homogeneous spray manifold with a reductive decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$. Then $(G/H,\mathbf{G})$ is weakly symmetric if and only if for each $y\in\mathfrak{m}\backslash\{0\}$, there exists $g\in H$ such that $\Ad(g)y=-y$. In particular, when $(G/H,\mathbf{G})$ is weakly symmetric, its spray vector field $\eta$ is even, i.e., $\eta(y)=\eta(-y)$, $\forall y\in\mathfrak{m}\backslash\{0\}$.
\end{theo}

The first statement in Theorem~\ref{thm-2} implies that the classification for weakly symmetric spray manifolds is an algebraic problem~\cite{WC2022}. Here is a simple example. On any connected Lie group~$G$, there is a canonical affine bi-invariant spray structure $\mathbf{G}_0$~\cite{Xu2022-1}, whose geodesics are orbits of one-parameter subgroups in $G$. $\mathbf{G}_0$ is weakly symmetric when $G$ is viewed as the homogeneous manifold $\widetilde{G}/\widetilde{H}$, where $\widetilde{G}$ is generated by all left and right translations, and the inverse map $\tau(g)=g^{-1}$. However, $\mathbf{G}_0$ is g.o.~but not weakly symmetric when $G$ is viewed as the homogeneous manifold $G/\{e\}$.

Finally, we use Theorem~\ref{thm-1} and Theorem~\ref{thm-2} to prove:

\begin{theo}\label{main-theorem}
Any weakly symmetric spray manifold $(G/H,\mathbf{G})$ with a reductive decomposition $\mathfrak{g=h+m}$ is g.o.
\end{theo}

Theorem~\ref{main-theorem} generalizes its analogs in homogeneous Riemannian and Finsler geometries. To prove it, we applied a new approach, which is based on the theoretical framework of homogeneous spray geometry~\cite{Xu2022-1,Xu2022,Xu2023}. There is a remaining question:

\begin{enonce*}{Question}
If a weakly symmetric spray manifold does not admit a reductive decomposition, must it still be g.o.?
\end{enonce*}

This paper is organized as follows. In Section~\ref{section_2}, we summarize some preliminary knowledge in spray geometry and introduce the spray vector field for a homogeneous spray manifold. In Section~\ref{section_3}, we prove all the theorems.

\section{Preliminaries}\label{section_2}

\subsection{Spray structure and geodesic}

The \emph{spray structure} on a smooth manifold $M$ is a smooth tangent vector field $\mathbf{G}$ on $TM\backslash0$, which can be locally presented as
\begin{equation*}
\mathbf{G}=y^{i}\partial_{x^{i}}-2\mathbf{G}^{i}\partial_{y^{i}}
\]
for any standard local coordinate $(x,y)=(x^i, y^i)$ (i.e., $x = (x^i) \in M $ and $y = y^j\partial _{x^j} \in T_xM$). Here the spray coefficients $\mathbf{G}^{i}=\mathbf{G}^{i}(x,y)$ are required to be positively 2-homogeneous for their $y$-entries (i.e., $ \mathbf{G}^{i}(x,\lambda y) = \lambda^{2}\mathbf{G}^{i}(x,y), \forall \lambda &gt; 0$). We call $(M,\mathbf{G})$ a \emph{spray manifold}.

A smooth curve $c(t)$ on $(M,\mathbf{G})$ with a nowhere-vanishing $\dot{c}(t)$ is called a \emph{geodesic} if its lifting $\parens[\big]{c(t),\dot{c}(t)}$ in $TM\setminus 0$ is an integral curve of $\mathbf{G}$. Locally, a geodesic $c(t) = \parens[\big]{c^i(t)}$ can be characterized by the following ordinary differential equations,
\begin{equation*}
\ddot{c}(t)+2\mathbf{G}^{i}\parens[\big]{c(t),\dot{c}(t)}=0, \quad\forall i.
\]

For example, a Finsler metric $F$ on $M$ induces a canonical spray structure $\mathbf{G}_F$ with the spray coefficients
\[
\mathbf{G}^i_F=\tfrac14g^{il}\parens[\big]{[F^2]_{x^ky^l}y^k-[F^2]_{x^l}}.
\]
The geodesic and many curvatures (including Riemann curvature, Ricci curvature, S-curvature, etc.) of $(M,F)$ coincide with those of $(M,\mathbf{G}_F)$ respectively. See~\cite{Sh2001} for more details.

\subsection{Homogeneous spray manifold}

A connected spray manifold $(M,\mathbf{G})$ is called \emph{homogeneous} if there exists a connected Lie transformation group $G$, which acts transitively on $M$, and preserves the set of all geodesics (i.e., for any $g\in G$, its tangent map on $TM$ preserves $\mathbf{G}$) \cite{Xu2022}. In this situation, we may identify $M$ with a homogeneous manifold $G/H$.

We assume that $G/H$ has a \emph{reductive} decomposition $\mathfrak{g=h+m}$ with $\mathfrak{g}=\Lie(G)$ and $\mathfrak{h}=\Lie(H)$. Here the reductiveness means that the given decomposition is $\Ad(H)$-invariant (it implies $[\mathfrak{h},\mathfrak{m}] \subset\mathfrak{m}$). The subspace $\mathfrak{m}$ can be naturally identified with the tangent space $T_o(G/H)$ at $o=eH$ such that the $\Ad(H)$-action on $\mathfrak{m}$ coincides with the isotropic $H$-action on $T_o(G/H)$.

\begin{rema}
A homogeneous Finsler manifold always has a reductive decomposition~\cite{De2012,Ni2017}, but a homogeneous spray manifold may not.
\end{rema}

\subsection{Spray vector field}

Let $(G/H,\mathbf{G})$ be a homogeneous spray manifold with a reductive decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$. The \emph{spray vector field} $\eta:\mathfrak{m}\backslash\{0\}\rightarrow\mathfrak{m}$ for $(G/H,\mathbf{G})$ is defined as follows~\cite{Xu2022}.

Let $c(t)$ be a geodesic on $(G/H,\mathbf{G})$ satisfying $c(0)=o$ and $\dot{c}(0)=y\in T_o(G/H)\backslash\{0\}=\mathfrak{m}\backslash\{0\}$. The following lemma (see~\cite[Lemma~5.1]{Xu2022}) provides a unique lifting for $c(t)$.

\begin{lemm}\label{lemma5.1}
There exists a unique smooth curve $\bar{c}(t)$ on $G$, which satisfies $\bar{c}(0)=e$, $c(t)=\bar{c}(t)\cdot o$ and $(L_{\overline{\!c}(t)^{-1}})_{*}\parens[\big]{\dot{\overline{\!c}}(t)} = \parens[\big]{\bar{c}(t)^{-1}}_{*}\parens[\big]{\dot{c}(t)}\in\mathfrak{m}\setminus\{0\}$, when $t$ is sufficiently close to $0$.
\end{lemm}

\begin{rema}
For simplicity, we will always assume that the parameter $t$ in later discussions is sufficiently close to $0$.
\end{rema}
We denote by $y(t)=(L_{\overline{\!c}_{y}(t)^{-1}})_{*}\parens[\big]{\dot{\overline{\!c}}_{y}(t)}$ a smooth curve on $\mathfrak{m}\setminus\{0\}$ and define $\eta(y)= \restr{-\frac{\dd}{\dd t}}{t=0}y(t)$.

Theorem~5.3 in~\cite{Xu2022} is crucial for proving our theorems. We reformulate it as follows.

\begin{theo}\label{def-eta}
Let $(G/H,\mathbf{G})$ be a connected homogeneous spray manifold with a reductive decomposition $\mathfrak{g=h+m}$ and the spray vector field $\eta$. Then there is a one-to-one correspondence between the following two sets:
\begin{enumerate}
\item the set of all geodesics $c(t)$ with $c(0) = o$;
\item the set of all integral curves $y(t)$ of $-\eta$.
\end{enumerate}
The correspondence is given by $y(t) = (L_{\overline{\!c}(t)^{-1}})_{*}\parens[\big]{\dot{\overline{\!c}}(t)}$, where $\bar{c}(t)$ is the lifting of $c(t)$ provided by Lemma~\ref{lemma5.1}.
\end{theo}

We will also need the following property of $\eta$ (see~\cite[Lemma~5.2 and Theorem~5.5]{Xu2022}).

\begin{lemm}\label{lemma-2}
Let $(G/H,\mathbf{G})$ be a connected homogeneous spray manifold with a reductive decomposition $\mathfrak{g=h+m}$, then its spray vector field is $\Ad(H)$-equivariant, i.e., $\forall g\in H$, $y\in\mathfrak{m}\backslash\{0\}$, $\eta\parens[\big]{\Ad(g)y}=\Ad(g)\eta(y)$.
\end{lemm}

\begin{rema}
Spray vector field was first introduced by L.~Huang for a homogeneous Finsler manifold with a reductive decomposition~\cite{Hu2015}. When we generalize this notion to homogeneous spray geometry, the reductiveness of the decomposition $\mathfrak{g=h+m}$ is not essential and Theorem~\ref{def-eta} is still valid.
\end{rema}

\section{Proofs of the main results}\label{section_3}

\subsection{Proof of Theorem~\ref{thm-1}}

Let $(G/H,\mathbf{G})$ be a homogeneous spray manifold with a reductive decomposition $\mathfrak{g=h+m}$ and the spray vector field $\eta$. First, we assume the g.o.~property and prove that $\eta$ is tangent to all $\Ad(H)$-orbits in $\mathfrak{m}\backslash\{0\}$.

We choose any vector $y\in\mathfrak{m}\backslash\{0\}$ and denote by $c(t)$ be the geodesic on $(G/H,\mathbf{G})$ which satisfies $c(0)=o$ and $\dot{c}(0)=y$. Then the g.o.~property provides a vector $u\in\mathfrak{g}$ such that $c(t)=\exp(tu)\cdot o$. Notice that $u-y\in\mathfrak{h}$. Lemma~\ref{lemma5.1} provides a smooth curve $\bar{c}(t)$ in $G$ such that
\begin{align}
&amp; \bar{c}(0)=e,\quad c(t)=\exp(tu)\cdot o=\bar{c}(t)\cdot o,\label{001} \\
&amp; (L_{\overline{\!c}(t)^{-1}})_*\parens[\big]{\dot{\overline{\!c}} (t)}\in\mathfrak{m}\backslash\{0\}.\label{002}
\end{align}
By~\eqref{001}, $h(t)=\exp (-tu)\bar{c}(t)$ is a smooth curve on $H$ with $h(0)=e$. Because
\[
\dot{\overline{\!c}}(t)=\frac{\dd}{\dd t}\parens[\big]{\exp (tu) h(t)}=(L_{\exp(tu)})_*\dot{h}(t)+(R_{h(t)})_*(L_{\exp(tu)})_*u,
\]
we have
\begin{equation}\label{003}
\begin{split}
(L_{\overline{\!c}(t)^{-1}})_*\parens[\big]{\dot{\overline{\!c}} (t)}	&amp; = (L_{h(t)^{-1}})_*\bracks[\bigg]{(L_{\exp(-tu)})_*(L_{\exp(tu)})_*\dot{h}(t)+(R_{h(t)})_*u} \\
&amp; = (L_{h(t)^{-1}})_*\dot{h}(t)+\Ad\parens[\big]{{h(t)^{-1}}}u \\
&amp; = \bracks[\bigg]{(L_{h(t)^{-1}})_*\dot{h}(t)+\Ad\parens[\big]{{h(t)^{-1}}}(u-y)} + \bracks[\bigg]{\Ad\parens[\big]{{h(t)^{-1}}}y}.
\end{split}
\end{equation}
By the reductiveness of the given decomposition, the two summands in the right side of~\eqref{003} belong to~$\mathfrak{h}$ and $\mathfrak{m}$ respectively. So~\eqref{002} implies $y(t)=(L_{\overline{\!c}(t)^{-1}})_*\parens[\big]{\dot{\overline{\!c}} (t)}=\Ad\parens[\big]{{h(t)^{-1}}}y$. Now we see that $y(t)$ is a smooth curve on $\Ad(H)y$, and then the tangent vector $\restr{\frac{\dd}{\dd t}}{t=0}y(t)=\bracks[\big]{y,\dot{h}(0)}=-\eta(y)$ is contained in $T_{y}\parens[\big]{\Ad(H)y}=[\mathfrak{h},y]$. One side of Theorem~\ref{thm-1} is proved.

Next, we assume that $\eta$ is tangent to all $\Ad(H)$-orbits in $\mathfrak{m}\backslash\{0\}$ and prove its g.o.~property. Choosing any $y\in\mathfrak{m}\backslash\{0\}$. Then our assumption provides $\eta(y)=[v,y]$ for some $v\in\mathfrak{h}$.

\begin{claim}\label{claim_A}
$y(t)=\Ad\parens[\big]{\exp(-tv)}y$ is the integral curve of $-\eta$ which satisfies $y(0)=y$.
\end{claim}

It is obvious that $y(0)=y$. By Lemma~\ref{lemma-2},
\begin{equation}\label{006}
\eta\parens[\big]{y(t)}=\Ad\parens[\big]{\exp(-tv)}\eta(y)=\Ad\parens[\big]{\exp(-tv)}[v,y] = \bracks[\big]{u,\Ad\parens[\big]{\exp(-tv)}y}.
\end{equation}
On the other hand,
\begin{equation}\label{007}
\dot{y}(t)=\frac{\dd}{\dd t}e^{-t\Ad(v)}y=-\Ad(v)e^{-t\Ad(v)}y = - \bracks[\big]{v,\Ad\parens[\big]{\exp(-tv)}y}.
\end{equation}
Comparing~\eqref{006} and~\eqref{007}, we get $\dot{y}(t) = -\eta\parens[\big]{y(t)}$. Claim~\ref{claim_A} is proved.

Let $c(t)$ be the geodesic on $(G/H,\mathbf{G})$ which satisfies $c(0)=o$ and $\dot{c}(0)=y$. Then Theorem~\ref{def-eta} tells us $y(t)=(L_{\overline{\!c}(t)^{-1}})_*\parens[\big]{\dot{\overline{\!c}}(t)}$. Denote by $g(t)$ the curve on $G$ which is determined by $\bar{c}(t)=g(t)\exp (tv)$. Obviously $g(0)=e$. Similar calculation as in~\eqref{003} or in the proof of~\cite[Lemma~5.1]{Xu2022} shows
\begin{equation}\label{004}
\begin{split}
(L_{\overline{\!c}(t)^{-1}})_*\parens[\big]{\dot{\overline{\!c}}(t)} &amp;= (L_{\exp(-tv)})_*\frac{\dd}{\dd t}\exp tv+\Ad\parens[\big]{\exp(-tv)}\parens[\big]{(L_{g(t)^{-1}})_*\dot{g}(t)} \\
&amp;=v+\Ad\parens[\big]{\exp(-tv)}\parens[\big]{(L_{g(t)^{-1}})_*\dot{g}(t)} \\
&amp;=\bracks[\bigg]{v+\Ad\parens[\big]{\exp(-tv)}\parens[\big]{\parens[\big]{(L_{g(t)^{-1}})_*\dot{g}(t)}_\mathfrak{h}}} + \bracks[\bigg]{\Ad\parens[\big]{\exp(-tv)}\parens[\big]{\parens[\big]{(L_{g(t)^{-1}})_*\dot{g}(t)}_\mathfrak{m}}},
\end{split}
\end{equation}
in which the subscripts $\mathfrak{h}$ and $\mathfrak{m}$ mean the projection according to the given reductive decomposition. The two summands in the right side of~\eqref{004} belong to $\mathfrak{h}$ and $\mathfrak{m}$ respectively. Combine Claim~\ref{claim_A} and~\eqref{004}, we see
\begin{equation*}
v+\Ad\parens[\big]{\exp(-tv)}\parens[\big]{\parens[\big]{(L_{g(t)^{-1}})_{*}\dot{g}(t)}_\mathfrak{h}}=0 \qquad\text{and}\qquad \parens[\big]{(L_{g(t)^{-1}})_*\dot{g}(t)}_\mathfrak{m}=y,
\]
i.e., $(L_{g(t)^{-1}})_*\parens[\big]{\dot{g}(t)}\equiv y-v$. So $g(t)=\exp t (y-v)$ and $c(t)=\bar{c}(t)\cdot o= g(t)\exp tv \cdot o=\exp t(y-v)\cdot o$. We get the g.o.~property for $(G/H,\mathbf{G})$ and end the proof of Theorem~\ref{thm-1}.

\subsection{Proof of Theorem~\ref{thm-2}}

Let $(G/H,\mathbf{G})$ be a weakly symmetric spray manifold with a reductive decomposition $\mathfrak{g=h+m}$. We choose any $y\in\mathfrak{m}\backslash\{0\}$, and denote by $c_1(t)$ the geodesic on $(G/H,\mathbf{G})$ which satisfies $c_1(0)=o$ and $\dot{c}_1(0)=y$. By Definition~\ref{def-spray-weakly symmetric}, there exists $g\in H$ such that the geodesic $c_2(t)=g\cdot c(t)$ coincides with $c_1(-t)$. Obviously $\dot{c}_2(0)=\Ad(g)y=-\dot{c}_1(0)=-y$. This argument can be reversed, which proves the first statement in Theorem~\ref{thm-2}.

Let $\bar{c}_{1}(t)$ be the lifting of $c_{1}(t)$ provided by Lemma~\ref{lemma5.1}.

\begin{claim}\label{claim_B}
$\bar{c}_{2}(t)=\bar{c}_{1}(-t)$ is the lifting of $c_2(t)$ provided by Lemma~\ref{lemma5.1}.
\end{claim}

It is obvious that $\bar{c}_{2}(0)=\bar{c}_{1}(0)=e$ and $ c_2(t)=c_1(-t)=\bar{c}_1(-t)\cdot o=\bar{c}_2(t)\cdot o$. Meanwhile, we have $\dot{\overline{\!c}}_2(t)=\frac{\dd}{\dd t}\bar{c}_1(-t)=-\dot{\overline{\!c}}_1(-t)$, and then
\[
(L_{\overline{\!c}_2(t)^{-1}})_*\parens[\big]{\dot{\overline{\!c}}_2(t)}=(L_{\overline{\!c}_1(-t)^{-1}})_*\parens[\big]{-\dot{\overline{\!c}}_1(-t)} =-(L_{\overline{\!c}_1(-t)^{-1}})_*\parens[\big]{\dot{\overline{\!c}}_1(-t)}\in\mathfrak{m}\backslash\{0\}.
\]
So $\bar{c}_2(t)=\bar{c}_1(-t)$ is the lifting of $c_2(t)=c_1(-t)$ provided by Lemma~\ref{lemma5.1}, which proves Claim~\ref{claim_B}.

By the definition of $\eta$, we have
\[
\eta(y) = - \restr{\frac{\dd}{\dd t}}{t=0}\bracks[\big]{(L_{\overline{\!c}_{1}(t)^{-1}})_{*}\dot{\overline{\!c}}_{1}(t)}
\qquad\text{and}\qquad
\eta(-y) = - \restr{\frac{\dd}{\dd t}}{t=0}\bracks[\big]{(L_{\overline{\!c}_{2}(t)^{-1}})_{*}\dot{\overline{\!c}}_{2}(t)}.
\]
Direct calculation shows
\[
\begin{split}
\eta(-y) &amp; = -\restr{\frac{\dd}{\dd t}}{t=0}\bracks[\big]{(L_{\overline{\!c}_{2}(t)^{-1}})_{*}\parens[\big]{\dot{\overline{\!c}}_{2}(t)}} \\
&amp; = \restr{\frac{\dd}{\dd t}}{t=0}\bracks[\big]{(L_{\overline{\!c}_{1}(-t)^{-1}})_{*}\parens[\big]{\dot{\overline{\!c}}_{1}(-t)}} \\
&amp; = - \restr{\frac{\dd}{\dd s}}{s=-t=0}\bracks[\big]{(L_{\overline{\!c}_{1}(s)^{-1}})_{*}\dot{\overline{\!c}}_{1}(s)} \\
&amp; = \eta(y).
\end{split}
\]
Theorem~\ref{thm-2} is proved.

\subsection{Proof of Theorem~\ref{main-theorem}}

Let $(G/H,\mathbf{G})$ be a weakly symmetric spray manifold with a reductive decomposition $\mathfrak{g=h+m}$ and the spray vector field $\eta$. We choose any vector $y\in\mathfrak{m}\backslash\{0\}$ and denote by $y_1(t)$ the integral curve of $-\eta$ which satisfies $y_1(0)=y$. By Theorem~\ref{thm-1}, we only need to prove that, for each value $t_1$, $y_1(t_1)$ is contained in $\Ad(H)y$, then Theorem~\ref{main-theorem} follows immediately. It should be notified that here we have used the quasi regularity of the submanifold $\Ad(H)y$ in $\mathfrak{m}\backslash\{0\}$ (see~\cite[p.~117]{GHV1973}).

By the first statement in Theorem~\ref{thm-2}, there exist $g_1,g_2\in H$ satisfying $\Ad(g_1)y=-y$ and $\Ad(g_2)y_1\parens[\big]{\tfrac{t_1}2}=-y_1\parens[\big]{\tfrac{t_1}2}$. Denote by $y_2(t)=
\Ad(g_2) y_1\parens[\big]{t+\tfrac{t_1}2}$ another curve on $\mathfrak{m}\backslash\{0\}$. Calculation shows
\[
\begin{split}
\dot{y}_2(t) &amp; = \frac{\dd}{\dd t}\parens[\big]{\Ad(g_2)y_1\parens[\big]{t+\tfrac{t_1}2}} \\
&amp; = \Ad(g_2)\dot{y}_1\parens[\big]{t+\tfrac{t_1}2} \\
&amp; = \Ad(g_2)\parens[\big]{-\eta\parens[\big]{y_1\parens[\big]{t+\tfrac{t_1}2}}} \\
&amp; = -\eta\parens[\big]{\Ad(g_2)y_1\parens[\big]{t+\tfrac{t_1}2}} \\
&amp; = -\eta\parens[\big]{y_2(t)},
\end{split}
\]
in which Lemma~\ref{lemma-2} has been used for the fourth equality. It indicates that $y_2(t)$ is the integral curve of $-\eta$ which passes $-y_1\parens[\big]{\tfrac{t_1}2}$ at $t=0$. On the other hand, we have
\[
\frac{\dd}{\dd t}\parens[\big]{-y_1\parens[\big]{-t+\tfrac{t_1}2}} = \restr{\frac{\dd}{\dd s}}{s=-t+\tfrac{t_1}2}y_1(s)=-\eta\parens[\big]{y_1\parens[\big]{-t+\tfrac{t_1}2}}=-\eta\parens[\big]{-y_1\parens[\big]{-t+\tfrac{t_1}2}},
\]
where the second statement in Theorem~\ref{thm-2} has been used for the last equality. It tells us that $-y_1\parens[\big]{-t+\tfrac{t_1}2}$ is also the integral curve of $-\eta$ which passes $-y_1\parens[\big]{\tfrac{t_1}2}$ at $t=0$. So we must have
\begin{equation}\label{011}
\Ad(g_2)y_1\parens[\big]{t+\tfrac{t_1}2}\equiv-y_1\parens[\big]{-t+\tfrac{t_1}2}.
\end{equation}
Input $t=\tfrac{t_1}2$ into~\eqref{011}, then we get $\Ad(g_2)y_1(t_1)=-y_1(0)=-y=\Ad(g_1)y$, i.e., $y_1(t_1)\in\Ad(H)y$. The proof of Theorem~\ref{main-theorem} is finished.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

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