  Under some dimension restrictions, we prove that totally umbilical
hypersurfaces of Spin$^c$ manifolds carrying a parallel, real or imaginary
Killing spinor are of constant mean curvature. This extends to the Spin$^c$
case the result of O. Kowalski stating that, every totally umbilical
hypersurface of an Einstein manifold of dimension greater or equal to $3$ is of
constant mean curvature. As an application, we prove that there are no
extrinsic hypersheres in complete Riemannian Spin manifolds of non-constant
sectional curvature carrying a parallel, Killing or imaginary Killing spinor.
