On the regular space-like hypersurfaces in the de Sitter space ${\mathbb S}^{m+1}_{1}$ with parallel Blaschke tensors

TL;DR

This paper classifies regular space-like hypersurfaces in de Sitter space with parallel Blaschke tensors using conformal geometry techniques and coordinate systems.

Contribution

It introduces a conformal geometric approach with two coordinate systems to classify hypersurfaces with parallel Blaschke tensors in de Sitter space.

Findings

Complete classification of hypersurfaces with parallel Blaschke tensors.

Use of conformal coordinate systems to analyze hypersurface geometry.

Connection between conformal geometry and hypersurface classification.

Abstract

In this paper, we use two conformal non-homogeneous coordinate systems, modeled on the de Sitter space ${\mathbb S}^{m+1}_1$, to cover the conformal space ${\mathbb Q}^{m+1}_1$, so that the conformal geometry of regular space-like hypersurfaces in $\mathbb{Q}^{m+1}_1$ is treated as that of hypersurfaces in ${\mathbb S}^{m+1}_1$. As a result, we give a complete classification of the regular space-like hypersurfaces (represented in the de Sitter space ${\mathbb S}^{m+1}_1$) with parallel Blaschke tensors.