Classification of Hypersurfaces with Two Distinct Principal Curvatures and Closed Moebius Form in $\mathbb{S}^{m+1}$

TL;DR

This paper classifies certain hypersurfaces in spheres that have two distinct principal curvatures and a closed Moebius form, providing explicit descriptions and characterizations of conformally flat hypersurfaces in higher dimensions.

Contribution

It offers a complete classification and explicit expressions for hypersurfaces with two principal curvatures and closed Moebius form in spheres, and characterizes conformally flat hypersurfaces in dimensions greater than three.

Findings

Explicit classification of hypersurfaces with two principal curvatures and closed Moebius form.

Characterization of conformally flat hypersurfaces in higher dimensions.

Provision of explicit formulas for the classified hypersurfaces.

Abstract

Let $x$ be an $m$-dimensional umbilic-free hypersurface in an $(m+1)$-dimensional unit sphere $\mathbb{S}^{m+1}(m\geq3)$. One of important questions is to classify hypersurfaces with two distinct principal curvatures. In this paper, we classify and explicitly express the hypersurfaces with two distinct principal curvatures and closed Moebius form, and then we characterize and classify conformally flat hypersurfaces of dimension larger than 3.