Given a smooth vector field $V$ on a smooth manifold $M$, $\exists$ $!$ smooth maximal flow $\theta: D \rightarrow M$ generated by $V$ such that:
a) For each $p \in M$, the curve $\theta^{(p)}: D^{(p)} \rightarrow M$ is the unique maximal integral curve of $V$ starting at $p$.
b) If $s \in D^{(p)}$, then $D^{(\theta(s,p))}$ is the interval $D^{(p)}-s = \{t-s : t \in D^{(p)} \}$
c) $\forall t \in \mathbb{R}$, the set $M_t$ is open in $M$ and $\theta_t: M_t \rightarrow M_{-t}$ is a diffeomorphism with inverse $\theta_{-t}$
Here, $D$ is the flow domain of the flow and $M_t = \{ p \in M : (t,p) \in D \}$
So, $M_t$ is every value of $t$, which we can think of as time, where the integral curve starting at $p$ is defined after moving along for $t$.
I'm wondering, what is the geometric significance that $M_t \cong M_{-t}$??
$\theta_t: M_t \rightarrow M_{-t}$ is defined by $\theta_t(p) = \theta^{(p)}(t) = \theta(t,p)$.
Best Answer
Your description of $M_t$ is wrong. It's the set of all points in $M$ from which you can flow for time $t$. The point then is that if $p\in M_t$, then $\theta_t(p)\in M_{-t}$ because you can flow backwards from $\theta_t(p)$ for time $t$ and get back to $p$. The mapping is a diffeomorphism, as Lee says, because you can go "backwards" from $M_{-t}$ to $M_t$ by the flow $\theta_{-t}$.