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

Large deviations for Brownian motion in evolving Riemannian manifolds