  An outline of a proof of the decomposition of linear metric perturbations
into gauge-invariant and gauge-variant parts on an arbitrary background
spacetime which admits ADM decomposition is briefly discussed. We explicitly
construct the gauge-invariant and gauge-variant parts of the linear metric
perturbations based on some assumptions. This implies that we can develop the
higher-order gauge-invariant perturbation theory on an arbitrary background
spacetime.
