  We integrate in closed implicit form the Navier-Stokes equations for an
incompressible fluid and the kinematical dynamo equation, in smooth manifolds
and Euclidean space. This integration is carried out by applying Stochastic
Differential Geometry, i.e. the gauge-theoretical formulation of Brownian
motions. Non-Riemannian geometries with torsion of the trace-type are found to
have a fundamental role. We prove that in any dimension other than 1, the
Navier-Stokes equations can be represented as a purely diffusive process, while
we can also give a random lagrangian representation for the diffusion of
vorticity and velocity in terms of the non-Riemannian geometry.
