Given a vector field $X$ on the manifold $M$ I know that I can associate to it in a unique way a local flux $\Theta: W \rightarrow M$, where $W \subset\mathbb{R} \times M$.
The curve $\theta_p(t)=\Theta (t,p)$ is the maximal integral curve of $X$ passing through $p$.
Then I know that $\theta^t(p)=\Theta (t,p)$ is a diffeomorphism from $V_t$ to $V_{-t}$, where $V_t=\{p \in M | (t,p)\in W\}$.
My question is this:
What are intuitively this diffeomorphisms? Can I visualize them in some way on the manifold $M$?
Thanks!
The diffeomorphism $\theta^t$ is called a "flow" along the vector field. If you think of the vector field as a velocity field, $\theta^t$ applied to an open set simply "flows" the open set along the vector field for $t$ seconds.