Stochastic Analysis on the Path Space of a Riemannian Manifold: I. Markovian Stochastic Calculus

On the Brownian flow of a compact Riemannian manifold, an intrinsic stochastic calculus is defined. This calculus is Markovian. The stochastic calculus of variation on the bundle of orthonormal frames appears only as an intermediate tool. The Bismut connection plays a paramount role in the final construction.