Fluctuation theorem on a Riemannian manifold

Based on the covariant underdamped and overdamped Langevin equations with Stratonovich coupling to multiplicative noises and the associated Fokker-Planck equations on a Riemannian manifold, we present the first law of stochastic thermodynamics on the trajectory level. The corresponding fluctuation theorems are also established, with the total entropy production of the Brownian particle and the heat reservoir playing the role of dissipation function.