  Riemannian geometry is a mathematical field which has been the cornerstone of
revolutionary scientific discoveries such as the theory of general relativity.
Despite early uses in robot design and recent applications for exploiting data
with specific geometries, it mostly remains overlooked in robotics. With this
blue sky paper, we argue that Riemannian geometry provides the most suitable
tools to analyze and generate well-coordinated, energy-efficient motions of
robots with many degrees of freedom. Via preliminary solutions and novel
research directions, we discuss how Riemannian geometry may be leveraged to
design and combine physically-meaningful synergies for robotics, and how this
theory also opens the door to coupling motion synergies with perceptual inputs.
