  Following Le Jan and Watanabe we define a connection associated with a
non-degenrrate diffusion operators. This connection is characterized here and
shown to be the Levi-Civita connection for gradient systems. This both explains
why such systems have useful properties and allows us to extend these
properties to more general systems. Topics described here include: moment
estimates for $T\xi_t$, a Weitzenb\"ock formula for the generator of the
semigroup on p-forms induced by the flow, a Bismut type formula for $d\log p_t$
in terms of an arbitrary metric connection, and a generalized Bochner vanishing
theorem. A comprehensive theory on this, its generalization to semi-elliptic
case and applications is published in the book `On the geometry of diffusion
operators and stochastic flows'. Related to this is also the book `The geometry
of filtering'. This article is easier to read.
