  An outline of a proof of the local decomposition of linear metric
perturbations into gauge-invariant and gauge-variant parts on an arbitrary
background spacetime is briefly explained. We explicitly construct the
gauge-invariant and gauge-variant parts of the linear metric perturbations
based on some assumptions. We also point out the zero-mode problem is an
essential problem to globalize of this decomposition of linear metric
perturbations. The resolution of this zero-mode problem implies the possibility
of the development of the higher-order gauge-invariant perturbation theory on
an arbitrary background spacetime in a global sense.
