Covariant currents and a thermodynamic uncertainty relation on curved manifolds

A framework for defining stochastic currents associated with diffusion processes on curved Riemannian manifolds is presented. This is achieved by introducing an overdamped Stratonovich-Langevin equation that remains fully covariant under non-linear transformations of state variables. The approach leads to a covariant extension of the thermodynamic uncertainty relation, describing a trade-off between the total entropy production rate and thermodynamic precision associated with short-time currents in curved spaces and arbitrary coordinate systems.