Möbius geometry of locally conformal flat hypersurfaces

In this paper, we consider the Möbius geometry of locally conformal flat hypersurfaces. First, we prove that any locally conformal flat hypersurface in the sphere S n + 1 ( n ≥ 4 ) is a canal hypersurface, which is the envelope of a one-parameter family of principal curvature hyperspheres. Conversely, canal hypersurfaces in S n + 1 ( n ≥ 3 ) without umbilical points are locally conformal flat. Finally, by applying the maximum principle, we show that a compact umbilical points hypersurface with isotropic Blaschke tensor in S n + 1 ( n ≥ 3 ) is Möbius equivalent to a compact minimal hypersurface with constant scalar curvature in S n + 1 .