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

Abstract

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 $\mathcal{LC}^n \times \mathbb{R}$ over the light cone $\mathcal{LC}^n$. 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.