  The decomposition of the linear-order metric perturbation is discussed in the
context of the higher-order gauge-invariant perturbation theory. We show that
the linear order metric perturbation is decomposed into gauge-invariant and
gauge-variant parts on the general background spacetime which admits ADM
decomposition. This decomposition was an important premise of the general
framework of the higher order gauge-invariant perturbation theory proposed in
the papers [K. Nakamura, Prog. Theor. Phys. vol.110 (2003), 723; ibid. vol.113
(2005), 481]. This implies that we can develop the higher-order gauge-invariant
perturbation theory on generic background spacetime. Remaining issues to
complete the general-framework of the higher-order gauge-invariant perturbation
theories are also discussed.
