Codimensions-Two Submanifolds Contained in the Light-like Hypercylinder ${\mathcal{L}\mathcal{C}}^n\times \mathbb{R}$

In this paper, we investigate space-like codimension-two submanifolds of the Lorentz-Minkowski space ${\mathbb{E}}_1^{n+2}$ constrained to lie on the light-like hypercylinder ${\mathcal{L}\mathcal{C}}^n\times \mathbb{R}$ over the light cone ${\mathcal{L}\mathcal{C}}^n$. By constructing a geometrically defined global frame field on the submanifold, we analyze the geometric interpretations of the associated extrinsic invariants by providing characterizations of (pseudo-)isotropic and pseudo-umbilical submanifolds, as well as submanifolds with flat normal bundle. In particular, we obtain local classification theorems for these classes of submanifolds.