We extend to Riemannian manifolds the theory of conditioned stochastic differential equations. We also provide some enlargement formulas for the Brownian filtration in this nonflat setting.
"Conditioning and initial enlargement of filtration on a Riemannian manifold." Ann. Probab. 32 (3) 2286 - 2303, July 2004. https://doi.org/10.1214/009117904000000126