  Interacting agent and particle systems are extensively used to model complex
phenomena in science and engineering. We consider the problem of learning
interaction kernels in these dynamical systems constrained to evolve on
Riemannian manifolds from given trajectory data. The models we consider are
based on interaction kernels depending on pairwise Riemannian distances between
agents, with agents interacting locally along the direction of the shortest
geodesic connecting them. We show that our estimators converge at a rate that
is independent of the dimension of the state space, and derive bounds on the
trajectory estimation error, on the manifold, between the observed and
estimated dynamics. We demonstrate the performance of our estimator on two
classical first order interacting systems: Opinion Dynamics and a
Predator-Swarm system, with each system constrained on two prototypical
manifolds, the $2$-dimensional sphere and the Poincar\'e disk model of
hyperbolic space.
