Stochastic differential equations, and Hoermander form representations
of diffusion operators, can determine a linear connection associated to
the underlying (sub)-Riemannian structure. This is systematically
described, together with its invariants, and then exploited to discuss
qualitative properties of stochastic flows, and analysis on path spaces
of compact manifolds with diffusion measures. This should be useful to
stochastic analysts, especially those with interests in stochastic
flows, infinite dimensional analysis, or geometric analysis, and also to
researchers in sub-Riemannian geometry. A basic background in
differential geometry is assumed, but the construction of the
connections is very direct and itself gives an intuitive and concrete
introduction. Knowledge of stochastic analysis is also assumed for later
chapters.