A simple proof of the Pontryagin maximum principle on manifolds

Applying the tubular neighborhood theorem, we give a simple proof of the Pontryagin maximum principle on a smooth manifold. The idea is as follows. Given a control system on a manifold MM, we embed it into some RnRn and extend the control system to RnRn. Then, we apply the Pontryagin maximum principle on RnRn to the extended system and project the consequence to MM.