Killing initial data on totally umbilical & compact hypersurfaces

In this note, we give a geometric characterization of the compact and totally umbilical hypersurfaces that carry non-trivial locally static Killing Initial Data (KID). More precisely, such compact hypersurfaces (Mn,g,cg)(Mn,g,cg) endowed with a Riemannian metric gg and a second fundamental form cgcg (where c∈C∞(M)c∈C∞(M) a priori) have constant mean curvature and are isometric to one of the following manifolds: (i)SnSn the standard sphere,(ii)a finite quotient of a warped product (S1×Y,dt2+h2(t)g0), where (Yn−1,g0)(Yn−1,g0) is Einstein with positive scalar curvature. In particular, they have harmonic curvature and strictly positive constant scalar curvature.