Wiggling organisms use geometry to swim in curved spaces

In euclidean space, deformable objects are able to generate body rotations -- such as a cat always landing on its feet -- but not translations. This restriction is lifted when a space possesses intrinsic curvature. A consequence of transporting vectors in curved space is holonomy. Holonomy measures differences between a vector and its copy after being transported in a closed loop. Holonomic effects and the fact that center of mass is ill-defined in curved space combine to allow for net translations of objects through body deformations. This is analogous to how microorganisms self-propel themselves in low Reynolds number fluids. Using an implicit integrator, we investigate the dynamics produced by the deformation cycles of quasi-rigid deformable bodies for select 3D riemannian manifolds. Particularly, we consider the stroke efficiency for various deformation cycles. The symmetries of the space provide conservation laws to inform the deformations cycles, however when no symmetries are present it is more complicated. For curved spaces lacking symmetry, we exchange extendable rod connectors with hookian springs to try and produce motion generating body deformations.