A Classification of Riemannian manifolds of quasi-constant sectional curvatures

Abstract

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.