A Complete Derivation of Complex Circle Manifold (CCM) Riemannian manifold Optimization Equations

After reviewing manifold optimization techniques in applications like MIMO communication systems, phased array beamforming, radar, and control theory, we observed that the Complex Circle Manifold (CCM) is widely employed, yet its foundational relations and equations lack a rigorous, self-contained derivation in the literature. This paper provides a systematic and rigorous proof of CCM's key properties, including its tangent space and Riemannian gradient operations, with explicit connections to real-world optimization problems. Our work aims to serve as a unified reference for researchers and practitioners applying CCM Manifold Optimization.