TITLE: What's the difference between time-dependent flow (isotopy) and time-independent flow?
QUESTION [7 upvotes]: Regarding the fact that both time-independent and time-dependent vector fields correspond with family of diffeomorphisms, i.e. $\{\phi_t | t\in\Re, \phi_t: M\to M\}$, what's the difference between these two families, i.e, time-independent and time-dependent flows (isotopy)?

REPLY [7 votes]: I'll use Amitai Yuval's notation.
Notice that if $X_t=X$ does not depend on $t$, then the equation
$$\frac{d}{dt}\varphi_t(p)=X(\varphi_t(p))$$
means that $\varphi$ is the flow of $X$. In particular, $\varphi_t\circ\varphi_s=\varphi_{t+s}$ for all $t,s\in\mathbb{R}$ (which is a defining property of the flow).
Conversely, if $\varphi_t\circ\varphi_s=\varphi_{t+s}$ $\forall t,s\in\mathbb{R}$, in particular $\varphi_t^{-1}=\varphi_{-t}$. By definition of $X_t$ we have
\begin{align*}
X_t(p)&=\left.\frac{d}{ds}\right|_{s=t}\varphi_s(\varphi_t^{-1}(p))\\
&=\left.\frac{d}{ds}\right|_{s=t}\varphi_{s-t}(p)\\
&=\left.\frac{d}{du}\right|_{u=0}\varphi_u(p),\,\,\,\text{where }u=s-t.
\end{align*}
This shows that $X_t$ does not depend on $t$. So we have just proved that

$X_t$ is independent of $t\Leftrightarrow$ $\varphi_t\circ\varphi_s=\varphi_{t+s}$ for all $t,s\in\mathbb{R}$.