Optimal navigation strategies for microswimmers on curved manifolds

Finding the fastest path to a desired destination is a vitally important task for microorganisms moving in a fluid flow. We study this problem by building an analytical formalism for overdamped microswimmers on curved manifolds and arbitrary flows. We show that the solution corresponds to the geodesics of a Randers metric, which is an asymmetric Finsler metric that reflects the irreversible character of the problem. Using the examples of spherical and toroidal surfaces, we demonstrate that the swimmer performance that follows this “Randers policy” always beats a more direct policy. Moreover, our results show that the relative gain grows significantly when specific structures related to either the geometry or the flow are exploited by the swimmer. A study of the shape of isochrones reveals features such as self-intersections, cusps, and abrupt nonlinear effects. Our work provides a link between microswimmer physics and geodesics in generalizations of general relativity