Let and be sub-manifolds of . Then and there are charts:
- for about adapted to
- for about adpapted to
Now let be the injection map. Consider the commutative diagram below.
Then which is smooth. Hence is smooth.
Let be integral curves in a manifold then is closed.
Let be have a limit in . Let be a chart then converges to by continuity. But also converges to . Hence and so . Thus is closed.