  We consider versions of Malliavin calculus on path spaces of compact
manifolds with diffusion measures, defining Gross-Sobolev spaces of
differentiable functions and proving their intertwining with solution maps, I,
of certain stochastic differential equations. This is shown to shed light on
fundamental uniqueness questions for this calculus including uniqueness of the
closed derivative operator $d$ and Markov uniqueness of the associated
Dirichlet form. A continuity result for the divergence operator by Kree and
Kree is extended to this situation. The regularity of conditional expectations
of smooth functionals of classical Wiener space, given I, is considered and
shown to have strong implications for these questions. A major role is played
by the (possibly sub-Riemannian) connections induced by stochastic differential
equations: Damped Markovian connections are used for the covariant derivatives.
