Generic conformally flat hypersurfaces in $\mathbb{R}^4$

TL;DR

This paper classifies and analyzes the global properties of generic conformally flat hypersurfaces in four-dimensional Euclidean space using M"{o}bius geometry, revealing their local structures and integral characteristics.

Contribution

It provides a local classification of such hypersurfaces with closed M"obius form and explores their global behavior and integral formulas.

Findings

Examples from cones, cylinders, and rotational hypersurfaces over constant Gaussian curvature surfaces.

Classification of hypersurfaces under M"{o}bius transformations.

Integral formulas describing global properties.

Abstract

In this paper, we study generic conformally flat hypersurfaces in the Euclidean $4$-space $\mathbb{R}^4$ using the framework of M\"{o}bius geometry. First, we classify locally the generic conformally flat hypersurfaces with closed M\"obius form under the M\"obius transformation group of $\mathbb{R}^4$. Such examples come from cones, cylinders, or rotational hypersurfaces over the surfaces with constant Gaussian curvature in $3$-spheres, Euclidean $3$-spaces, or hyperbolic $3$-spaces, respectively. Second, we investigate the global behavior of the generic conformally flat hypersurface and give some integral formulas about these hypersurfaces.