Dynamic programming principle of control systems on manifolds and its relations to maximum principle

We study the dynamic programming principle (DPP for short) on manifolds, obtain the Hamilton-Jacobi-Bellman (HJB for short) equation, and prove that the value function is the only viscosity solution to the HJB equation. Then, we investigate the relation between DPP and Pontryagin's maximum principle (PMP for short), from which we obtain PMP on manifolds.