Tri-Slant Submanifolds of Metallic Riemannian Manifolds

In this paper, we introduce and systematically investigate the notion of tri-slant submanifolds within the framework of metallic Riemannian manifolds. We establish the fundamental decompositions of the tangent and normal bundles and characterize the behavior of the associated tangential and normal endomorphisms. Furthermore, we derive the necessary and sufficient conditions for the integrability of the tri-slant distributions. We provide a rigorous geometric analysis by establishing characterization theorems for totally geodesic and totally umbilical properties, both for the integral manifolds (leaves) and the distributions themselves. The conditions under which a tri-slant submanifold reduces to a local Riemannian product are also determined. Finally, we construct non-trivial, explicit examples in Euclidean space equipped with generalized metallic structures to validate the existence of these newly defined submanifolds.