2020/06/24 by João Paulo dos Santos, Ruy Tojeiro, Santos, João Paulo dos +1
Mathematics · Physics and Astronomy · #53B25 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2006.13928
openalex publication_date 2020/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we classify the conformally flat Euclidean hypersurfaces of dimension three with three distinct principal curvatures of ℝ4, \mathbbS3× ℝ and ℍ3× ℝ with the property that the tangent component of the vector field ∂/∂ t is a principal direction at any point. Here ∂/∂ t stands for either a constant unit vector field in ℝ4 or the unit vector field tangent to the factor ℝ in the product spaces \mathbbS3× ℝ and ℍ3× ℝ, respectively. Then we use this result to give a simple proof of an alternative classification of the cyclic conformally flat hypersurfaces of ℝ4, that is, the conformally flat hypersurfaces of ℝ4 with three distinct principal curvatures such that the curvature lines correspondent to one of its principal curvatures are extrinsic circles. We also characterize the cyclic conformally flat hypersurfaces of ℝ4 as those conformally flat hypersurfaces of dimension three with three distinct principal curvatures for which there exists a conformal Killing vector field of ℝ4 whose tangent component is an eigenvector field correspondent to one of its principal curvatures.