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)$(M^n,g,cg)$ endowed with a Riemannian metric g$g$ and a second fundamental form cg$cg$ (where c∈C^∞(M)$c\in C^{\infty }\left(M\right)$ a priori) have constant mean curvature and are isometric to one of the following manifolds:

(i)

Sⁿ${\mathbb{S}}^n$ the standard sphere,