Metallic Riemannian Space Forms

In this study we examine the cases of constant sectionalcurvature in Metallic Riemannian manifolds. We show that classical andholomorphic-like curvatures generally make the manifold at. Based on this, we introduce a new concept called Metallic sectional curvature,and in this setting, we derive the explicit expression of the curvaturetensor. In addition, the Ricci tensor and scalar curvature are computed,and the conditions under which the manifold is Einstein are discussed.Finally, this new curvature notion is applied to semi-invariant and totallyumbilical submanifolds, yielding rich geometric results.