  This article presents an overview of robot learning and adaptive control
applications that can benefit from a joint use of Riemannian geometry and
probabilistic representations. The roles of Riemannian manifolds, geodesics and
parallel transport in robotics are first discussed. Several forms of manifolds
already employed in robotics are then presented, by also listing manifolds that
have been underexploited but that have potentials in future robot learning
applications. A varied range of techniques employing Gaussian distributions on
Riemannian manifolds is then introduced, including clustering, regression,
information fusion, planning and control problems. Two examples of applications
are presented, involving the control of a prosthetic hand from surface
electromyography (sEMG) data, and the teleoperation of a bimanual underwater
robot. Further perspectives are finally discussed, with suggestions of
promising research directions.
