  In this paper, we give a new derivation of the incompressible Navier-Stokes
equations on a compact Riemannian manifold $M$ via the Bellman dynamic
programming principle on the infinite dimensional group of volume preserving
diffeomorphisms $G={\rm SDiff}(M)$. In particular, when the viscosity vanishes,
we give a new derivation of the incompressible Euler equation on a compact
Riemannian manifold. The main result of this paper indicates an interesting
relationship between the incompressible Navier-Stokes equations on $M$ and the
Hamilton-Jacobi-Bellman equation on $G={\rm SDiff}(M)$.
