  We present a scheme for simulating conditioned semimartingales taking values
in Riemannian manifolds. Extending the guided bridge proposal approach used for
simulating Euclidean bridges, the scheme replaces the drift of the conditioned
process with an approximation in terms of a scaled radial vector field. This
handles the fact that transition densities are generally intractable on
geometric spaces. We prove the validity of the scheme by a change of measure
argument, and we show how the resulting guided processes can be used in
importance sampling and for approximating the density of the unconditioned
process. The scheme is used for numerically simulating bridges on two- and
three-dimensional manifolds, for approximating otherwise intractable transition
densities, and for estimating the diffusion mean of sampled geometric data.
