  In this paper, we consider Riemannian online convex optimization with dynamic
regret. First, we propose two novel algorithms, namely the Riemannian Online
Optimistic Gradient Descent (R-OOGD) and the Riemannian Adaptive Online
Optimistic Gradient Descent (R-AOOGD), which combine the advantages of
classical optimistic algorithms with the rich geometric properties of
Riemannian manifolds. We analyze the dynamic regrets of the R-OOGD and R-AOOGD
in terms of regularity of the sequence of cost functions and comparators. Next,
we apply the R-OOGD to Riemannian zero-sum games, leading to the Riemannian
Optimistic Gradient Descent Ascent algorithm (R-OGDA). We analyze the average
iterate and best-iterate of the R-OGDA in seeking Nash equilibrium for a
two-player, zero-sum, g-convex-concave games. We also prove the last-iterate
convergence of the R-OGDA for g-strongly convex-strongly concave problems. Our
theoretical analysis shows that all proposed algorithms achieve results in
regret and convergence that match their counterparts in Euclidean spaces.
Finally, we conduct several experiments to verify our theoretical findings.
