  Error bounds are derived for sampling and estimation using a discretization
of an intrinsically defined Langevin diffusion with invariant measure
$\text{d}\mu_\phi \propto e^{-\phi} \mathrm{dvol}_g $ on a compact Riemannian
manifold. Two estimators of linear functionals of $\mu_\phi $ based on the
discretized Markov process are considered: a time-averaging estimator based on
a single trajectory and an ensemble-averaging estimator based on multiple
independent trajectories. Imposing no restrictions beyond a nominal level of
smoothness on $\phi$, first-order error bounds, in discretization step size, on
the bias and variance/mean-square error of both estimators are derived. The
order of error matches the optimal rate in Euclidean and flat spaces, and leads
to a first-order bound on distance between the invariant measure $\mu_\phi$ and
a stationary measure of the discretized Markov process. This order is preserved
even upon using retractions when exponential maps are unavailable in closed
form, thus enhancing practicality of the proposed algorithms. Generality of the
proof techniques, which exploit links between two partial differential
equations and the semigroup of operators corresponding to the Langevin
diffusion, renders them amenable for the study of a more general class of
sampling algorithms related to the Langevin diffusion. Conditions for extending
analysis to the case of non-compact manifolds are discussed. Numerical
illustrations with distributions, log-concave and otherwise, on the manifolds
of positive and negative curvature elucidate on the derived bounds and
demonstrate practical utility of the sampling algorithm.
