  In the context of a noncommutative differential calculus on the algebra of
real valued functions of an $n$-dimensional manifold $M$, a commutative and
associative product of 1-forms is naturally defined. Ordinary differential
calculus corresponds to this product being trivially zero. We consider the
minimal generalization in which the algebra of 1-forms with this product is
nilpotent of degree 3. Basic tensor analysis and differential geometry are
developped in this context and applied to the formulation of symplectic
geometry and Hamiltonian dynamics. It is shown that the corresponding Liouville
equation can take the form of a generalized Fokker-Planck equation, well known
in statistical mechanics of open systems, as well as in stochastic calculus.
Specifically it may be the same with that obtained via Ito stochastic calculus,
when solving a linear stochastic differential equation with a Wiener process as
the stochastic term. The close connection between noncommutative differential
structures of this kind, with semimartingale stochastic processes on manifolds
thus suggested, is further explored.
