Li, BinGuo, NingHu, YulinPanayirci, ErdalDong, Zhicheng2025-11-152025-11-1520251536-12841558-0687https://doi.org/10.1109/MWC.2025.3601674https://hdl.handle.net/20.500.12469/7586Reconfigurable Intelligent Surfaces (RIS) and Intelligent Omni-Surfaces (IOS) have emerged as transformative technologies in wireless communications, offering enhanced energy and spectral efficiency. However, the inherent characteristics of RIS/IOS also bring new challenges to the resource allocation design of RIS/IOS-assisted wireless networks, such as the nonconvexity caused by the constant modulus constraint of RIS/IOS phase shift elements and the complexity of jointly designing RIS/IOS phase shifts and base station beamforming. Although existing approaches, such as the max-min algorithm and the alternating optimization algorithm, can address these challenges, they suffer from high computational complexity. This paper systematically analyzes the challenges in the design of RIS/IOS-assisted wireless networks and briefly introduces the principles of Riemannian manifold optimization. To address these complex design challenges, we introduce four pivotal manifolds: the complex circle manifold, sphere manifold, Cholesky manifold, and product manifold, each providing unique solutions for enhancing network performance. Finally, we discuss future research directions for Riemannian manifold methods in the design of RIS/IOS-assisted networks.eninfo:eu-repo/semantics/closedAccessOptimizationManifoldsArray Signal ProcessingWireless NetworksVectorsComputational ComplexityReconfigurable Intelligent SurfacesConvergenceResource ManagementFacesArticle10.1109/MWC.2025.36016742-s2.0-105020054740