  A Kaluza-Klein decomposition of higher dimensional gravity is performed in
the flexible brane world scenario and the properties of the extra vectors
resulting from this decomposition are explored. These vectors become massive
due to a gravitational Higgs mechanism in which the brane oscillation
Nambu-Goldstone bosons become the longitudinal component of the vector fields.
The vector mass is found to be proportional to the exponential of the vacuum
expectation value of the radion (dilaton) field and as such its magnitude is
model dependent. Using the structure of the embedding geometry, the couplings
of these vectors to the Standard Model, including those resulting from the
extrinsic curvature, are deduced. As an example, we show that for 5D space-time
the geometry of the bulk-brane world, either intrinsic or extrinsic, only
depends on the extra vector and the 4D graviton. The connection between the
embedding geometry and coset construction by non-linear realization is also
presented.
