Large deviations for slow-fast processes on connected complete Riemannian manifolds

We consider a class of slow-fast processes on a connected complete Riemannian manifold $M$.The limiting dynamics as the scale separation goes to $\mathrm{\infty }$ is governed by the averaging principle. Around this limit, we prove large deviation principles with an action-integral rate function for the slow process by nonlinear semigroup methods together with the Hamilton-Jacobi-Bellman equation techniques. The innovation is solving a comparison principle for viscosity solutions on $M$ and the existence of a viscosity solution via a control problem for a non-smooth Hamiltonian.