  In this note, we give a geometric characterization of the compact and totally
umbilical hypersurfaces that carry a non trivial locally static Killing Initial
Data (KID). More precisely, such compact hypersurfaces have constant mean
curvature and are isometric to one of the following manifolds: (i) Sn the
standard sphere, (ii) a finite quotient of a warped product of a circle with a
compact Einstein manifold of positive scalar curvature. In particular, these
hypersurfaces have harmonic curvature and strictly positive constant scalar
curvature.
