  In the present paper, we give some theorems representing ridigity of a vacuum
brane in static bulk spacetimes. As an application, we show that a static bulk
spacetime with dimension D>3 and spatial symmetry IO(D-2), O(D-1) or O_+(D-2,1)
does not allow a vacuum brane with a black hole on it. We also show that if a
static bulk spacetime with dimension D>4 satsifying the vacuum Einstein
equations can be foliated by a continuous family of vacuum branes with
asymptotically constant curvature, it is a black string solution.
