  It is well known that the umbilic points of minimal surfaces in spaces of
constant sectional curvature consist only of isolated points unless the surface
is totally umbilic on some connected component, as for example the Hopf form is
holomorphic. In this note, we prove that on Willmore surfaces in codimension
one the umbilic set is locally a one dimensional real-analytic manifold without
boundary or an isolated point.
