Rigidity Theorems in the Braneworld Model

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 ≥ 4 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.