Let me introduce a proposition from Lee's ISM first. With its aid, I'd like to obtain a corollary that is known as diffeomorphism invariance of flows and is also included in ISM.
Proposition 9.13 (Naturality of Flows). Suppose $M$ and $N$ are smooth manifolds, $F:M\to N$ is a smooth map, $X\in\mathfrak{X}(M)$, and $Y\in\mathfrak{X}(N)$. Let $\theta$ be the flow of $X$ and $\eta$ the flow of $Y$. If $X$ and $Y$ are $F$-related, then $\forall t\in\mathbb{R}$, $F(M_t)\subseteq N_t$ and $\eta_t\circ F=F\circ\theta_t$ on $M_t$.
This proposition can be easily proved by using the fundamental theorem on flows, but somehow I have difficulty arriving at its corollary:
Let $F:M\to N$ be a diffeomorphism. If $X\in\mathfrak{X}(M)$, $\theta:\mathscr{D}\to M$ is the flow of $X$, and $\eta:\mathscr{C}\to N$ is the flow of $F_*X$, then $\forall t\in\mathbb{R}$, $\eta_t=F\circ\theta_t\circ F^{-1}$ with domain $N_t=F(M_t)$.
In view of the preceding proposition, all we need to do is to show that $N_t\subseteq F(M_t)$, which can be done by picking $q\in N_t$ and proving $q\in F(M_t)$. So we must find $p\in M_t$ s.t. $F(p)=q$. In this regard, there's no reason not to consider $p=F^{-1}(q)$. But how do I get to know that $(t,F^{-1}(q))\in\mathscr{D}$? Thank you for your patience.
Best Answer
If $F$ is a diffeomorphisms, then $X$ and $F_*X$ are $F$-related, so from the proposition, you obtain $F(M_t)\subseteq N_t$ and $\eta_t\circ F=F\circ \theta_t$ on $M_t$. Since $F$ is invertible, the last condition is equivalent to $\eta_t=F\circ \theta_t\circ F^{-1}$ on $F(M_t)$. Now $X=(F^{-1})_*(F_*X)$ so applying the proposition to $F^{-1}$, you conclude that $F^{-1}(N_t)\subseteq M_t$ and hence $N_t\subseteq F(M_t)$. Hence you get $N_t=F(M_t)$, which was the only missing bit.