  In the realm of robotics, numerous downstream robotics tasks leverage machine
learning methods for processing, modeling, or synthesizing data. Often, this
data comprises variables that inherently carry geometric constraints, such as
the unit-norm condition of quaternions representing rigid-body orientations or
the positive definiteness of stiffness and manipulability ellipsoids. Handling
such geometric constraints effectively requires the incorporation of tools from
differential geometry into the formulation of machine learning methods. In this
context, Riemannian manifolds emerge as a powerful mathematical framework to
handle such geometric constraints. Nevertheless, their recent adoption in robot
learning has been largely characterized by a mathematically-flawed
simplification, hereinafter referred to as the "single tangent space fallacy".
This approach involves merely projecting the data of interest onto a single
tangent (Euclidean) space, over which an off-the-shelf learning algorithm is
applied. This paper provides a theoretical elucidation of various
misconceptions surrounding this approach and offers experimental evidence of
its shortcomings. Finally, it presents valuable insights to promote best
practices when employing Riemannian geometry within robot learning
applications.
