Chen's Inequalities for Quasi Hemi-Slant Submanifolds of Locally Metallic Riemannian Space Form

In this paper, we study the geometry of quasi hemi-slant (QHS) submanifolds immersed in locally metallic Riemannian space forms. To illustrate these geometric structures, we first construct some non-trivial examples of QHS submanifolds. Directly calculating curvature bounds in these ambient spaces is usually complicated due to the heavy algebraic properties of the nativemetallic structure. To overcome this, we use an alternative approach based on the canonically induced almost product structure F, which significantly simplifies the computations. Using this method, we successfully establish Chen’s first inequality and the general Chen-Ricci inequality for QHS submanifolds. Furthermore, we analyze the equality conditions for these bounds and derive specific, diagonalized inequalities for the invariant, proper slant, and anti-invariant distributions of the submanifold.