2011/05/16 by Georgi Ganchev, Ganchev, Georgi, Vesselka Mihova +1 · 1 citation
Mathematics · #53A35 #53B20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #math.DG #msc:53A35 #msc:53B20
paper · pdf · doi:10.48550/arxiv.1105.3081
18 pages
arxiv created 2011/05/16 · openalex publication_date 2011/05/16 · arxiv updated 2015/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Riemannian manifolds of quasi-constant sectional curvatures (QC-manifolds) are divided into two basic classes: with positive or negative horizontal sectional curvatures. We prove that the Riemannian QC-manifolds with positive horizontal sectional curvatures are locally equivalent to canal hypersurfaces in Euclidean space, while the Riemannian QC-manifolds with negative horizontal sectional curvatures are locally equivalent to canal space-like hypersurfaces in Minkowski space. We prove that the local theory of conformally flat Riemannian manifolds, which can be locally isometrically embedded as hypersurfaces in Euclidean or Minkowski space, is equivalent to the local theory of Riemannian QC-manifolds. These results give a local geometric classification of conformally flat hypersurfaces in Euclidean space and conformally flat space-like hypersurfaces in Minkowski space.