  We prove large deviations for $g(t)$-Brownian motion in a complete, evolving
Riemannian manifold $M$ with respect to a collection $\{g(t)\}_{t\in [0,1]}$ of
Riemannian metrics, smoothly depending on $t$. We show how the large deviations
are obtained from the large deviations of the (time-dependent) horizontal lift
of $g(t)$-Brownian motion to the frame bundle $FM$ over $M$. The latter is
proved by embedding the frame bundle into some Euclidean space and applying
Freidlin-Wentzell theory for diffusions with time-dependent coefficients, where
the coefficients are jointly Lipschitz in space and time.
