Local flow (${\Phi }^t$)

A local flow of a smooth vector field $V$ is a smooth map $\Phi :D\to M$, defined on an open neighborhood of $\{0\}\times M$, such that ${\Phi }^0(x)=x$, ${\partial }_t{\Phi }^t(x)=V({\Phi }^t(x))$, and ${\Phi }^{s+t}={\Phi }^s\circ {\Phi }^t$ whenever both sides are defined.