  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.
