 Smooth flow

A smooth flow is a smooth one-parameter family of diffeomorphisms satisfying ${\varphi }_0=\mathrm{id}$ and ${\varphi }_{t+s}={\varphi }_t\circ {\varphi }_s$. Its generating vector field is $F(x)={\partial }_t{\varphi }_t(x){|}_{t=0}$. On a closed manifold, every smooth vector field has a flow defined for all real times.