  An outline of a proof of the decomposition of linear metric perturbations
into gauge-invariant and gauge-variant parts on the an arbitrary background
spacetime which admits ADM decomposition is discussed. We explicitly construct
the gauge-invariant and gauge-variant parts of the linear metric perturbations
through the assumption of the existence of some Green functions. We also
confirm the result through another approach. This implies that we can develop
the higher-order gauge-invariant perturbation theory on an arbitrary background
spacetime. Remaining issues to complete the general-framework of the
general-relativistic higher-order gauge-invariant perturbation theories are
also discussed.
