  Let (M,g) be a four or six dimensional compact Riemannian manifold which is
locally conformally flat and assume that its boundary is totally umbilical. In
this note, we prove that if the Euler characteristic of M is equal to 1 and if
its Yamabe invariant is positive, then (M,g) is conformally isometric to the
standard hemisphere. As an application and using a result of Hang-Wang, we
prove a rigidity result for these hemispheres regarding the Min-Oo conjecture.
