  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.
