A flow \$$\phi^t\$$ on \$$M\$$ is called a _global flow_ for the tangent vector field \$$\alpha\$$ if for every point \$$s \in M\$$, the trajectory function \$$\phi_s(t)\$$ is a flow line of \$$\alpha\$$.

In other words, the condition is that for all \$$s \in M\$$ and \$$t \\in T\$$, we have that \$$\phi_s'(t) = \alpha(\phi^t(s))\$$.