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

TL;DR

This paper classifies space-like codimension-two submanifolds in Lorentz-Minkowski space constrained to a light-like hypercylinder, analyzing their geometric invariants and providing local classification theorems.

Contribution

It introduces a global frame field approach to characterize and classify submanifolds with specific geometric properties in a light-like hypercylinder.

Findings

Characterization of pseudo-isotropic and pseudo-umbilical submanifolds

Classification theorems for submanifolds with flat normal bundle

Analysis of extrinsic invariants in the light-like hypercylinder setting

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.