  We define a complex connection on a real hypersurface of $\C^{n+1}$ which is
naturally inherited from the ambient space. Using a system of Codazzi-type
equations, we classify connected real hypersurfaces in $\C^{n+1}$, $n\ge 2$,
which are Levi umbilical and have non zero constant Levi curvature. It turns
out that such surfaces are contained either in a sphere or in the boundary of a
complex tube domain with spherical section.
