Brane gravity from bulk vector field

It is shown that Einsteinʹs equations on the brane can be received from the multi-dimensional vector field equations in pseudo-Euclidean space. The idea is based on the observation that the brane geometry can be equivalently described by the intrinsic metric or by the derivatives of its normal. From the other hand the normal to the brane can be constructed with the components of some multi-dimensional vector fields. For the both cases 4-dimensional effective action for gravity appears to be the same.