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Online distributed optimization is particularly useful for solving optimization problems with streaming data collected by multiple agents over a network. When the solutions lie on a Riemannian manifold, such problems become challenging to solve, particularly when efficiency and continuous adaptation are required. This work tackles these challenges and devises a diffusion adaptation strategy for decentralized optimization over general manifolds. A theoretical analysis shows that the proposed algorithm is able to approach network agreement after sufficient iterations, which allows a non-asymptotic convergence result to be derived. We apply the algorithm to the online decentralized principal component analysis problem and Gaussian mixture model inference. Experimental results with both synthetic and real data illustrate its performance.