MG13 proceedings: Construction of gauge-invariant variables for
  linear-order metric perturbations on an arbitrary background spacetime
  An outline of a proof of the decomposition of the linear metric perturbation
into gauge-invariant and gauge-variant parts on an arbitrary background
spacetime is discussed through an exlicit construction of gauge-invariant and
gauge-variant parts. Although this outline is incomplete, yet, due to our
assumptions, we propose a conjecture which states that the linear metric
perturbation is always decomposed into its gauge-invariant and gageu-variant
parts. If this conjecture is true, we can develop the higher-order
gauge-invariant perturbation theory on an arbitrary background spacetime.
