  We study numerical optimisation algorithms that use zeroth-order information
to minimise time-varying geodesically-convex cost functions on Riemannian
manifolds. In the Euclidean setting, zeroth-order algorithms have received a
lot of attention in both the time-varying and time-invariant cases. However,
the extension to Riemannian manifolds is much less developed. We focus on
Hadamard manifolds, which are a special class of Riemannian manifolds with
global nonpositive curvature that offer convenient grounds for the
generalisation of convexity notions. Specifically, we derive bounds on the
expected instantaneous tracking error, and we provide algorithm parameter
values that minimise the algorithm's performance. Our results illustrate how
the manifold geometry in terms of the sectional curvature affects these bounds.
Additionally, we provide dynamic regret bounds for this online optimisation
setting. To the best of our knowledge, these are the first regret bounds even
for the Euclidean version of the problem. Lastly, via numerical simulations, we
demonstrate the applicability of our algorithm on an online Karcher mean
problem.
