  Locomotion by shape changes (spermatozoon swimming, snake slithering, bird
flapping) or gas expulsion (rocket firing) is assumed to require environmental
interaction, due to conservation of momentum. As first noted in (Wisdom, 2003)
and later in (Gu\'eron, 2009) and (Avron et al, 2006), in curved space or
spacetime the non-commutativity of translations permits translation without
momentum exchange, just as falling cats and lizards can self-deform to reorient
in flat space without environmental interaction. Translation in curved space
can occur not only in gravitationally induced curved spacetime (where
translation is predicted to be on the order of $10^{-23}$ m per gait cycle) but
also in the curved surfaces encountered by locomotors in real-world
environments. Here we show that a precision robophysical apparatus consisting
of motors driven on curved tracks (and thereby confined to a spherical surface
without a solid substrate) can self-propel without environmental momentum
exchange (impulse) via shape changes that can generate gauge potentials that
manifest as translations. Our system produces shape changes comparable to the
environment's inverse curvatures and generates from zero momentum forward
movement of $10^{-1}$ cm per gait cycle even while resisted by weak
gravitational and frictional forces. Dissipation via friction eventually
arrests the robot but also imbues it with momentum which can be released upon a
cessation of shape changes. This work demonstrates how the interaction between
environmental curvature, active driving and geometric phases yields rich,
exotic phenomena.
