  It is shown that Einstein's 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.
