Manifold optimization is instrumental in control and engineering applications, ranging from trajectory optimization in robotics and safe reinforcement learning to subspace estimation, geometric deep learning, and adaptive fine-tuning of large language models (LLMs). Despite recent advances, most of the research on manifold optimization is limited to centralized approaches not implementable in multi-agent systems. This CAREER project takes a substantial step towards the development and adoption of decentralized manifold optimization (DMO) in large-scale, multi-agent optimization by overcoming three fundamental challenges: scalability, efficiency, and adaptation to dynamic environments. The PI proposes three integrated research thrusts to address these challenges. (i) Thrust 1 will fundamentally advance the scalability of DMO by innovating decentralized retraction-free methods that are computationally efficient, and it will also extend the theory of retraction-free methods to bilevel DMO. (ii) Thrust 2 will provide a systematic understanding of acceleration to improve the iteration complexity of DMO algorithms. This research will establish provably fast convergence guarantees for accelerated DMO and further expand this theory to the zero-order setting using smoothing techniques to generate high-fidelity gradient estimators. (iii) Thrust 3 will pioneer online DMO to contextualize multi-agent Riemannian optimization in dynamic, unpredictable environments. It will investigate proj