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 M , we embed it into some R n and extend the control system to R n . Then, we apply the Pontryagin maximum principle on R n to the extended system and project the consequence to M .