 Solution

For a smooth vector field $X$, solve the initial-value problem $d{\varphi }_t(p)/dt=X({\varphi }_t(p))$, ${\varphi }_0(p)=p$. Local existence, uniqueness and smooth dependence for ordinary differential equations give a local flow. Uniqueness yields ${\varphi }_{t+s}(p)={\varphi }_t({\varphi }_s(p))$ whenever both sides are defined, and ${\varphi }_{-t}$ is the inverse of ${\varphi }_t$ on the corresponding domains. Thus each local flow map is a diffeomorphism between open subsets.

A group of diffeomorphisms of the entire manifold for every real $t$ requires a complete vector field. This follows, for example, on a compact manifold without boundary: local existence times can be made uniform on a finite cover, and repeated continuation prevents finite-time escape. Without completeness, the source's group is only local. The vector field $x^2{\partial }_x$ on $\mathbb{R}$ has flow $x\mapsto x/(1-tx)$ and blows up at $t=1/x$ for $x>0$, so smoothness alone cannot imply a global one-parameter group.