  Manifold optimization (MO) is a powerful mathematical framework that can be
applied to optimize functions over complex geometric structures, which is
particularly useful in advanced wireless communication systems, such as
reconfigurable intelligent surface (RIS)-aided massive MIMO (mMIMO) and
extra-large scale massive MIMO (XL-MIMO) systems. MO provides a structured
approach to tackling complex optimization problems. By leveraging the geometric
properties of the manifold, more efficient and effective solutions can be found
compared to conventional optimization methods. This paper provides a tutorial
on MO technique and provides some applications of MO in the context of wireless
communications systems. In particular, to corroborate the effectiveness of MO
methodology, we explore five application examples in RIS-aided mMIMO system,
focusing on fairness, energy efficiency (EE) maximization, intracell pilot
reuse interference mitigation, and grant-free (GF) random access (RA).
