Inclusion of the first-order vector- and tensor-modes in the
  second-order gauge-invariant cosmological perturbation theory
  Gauge-invariant treatments of the second-order cosmological perturbation in a
four dimensional homogeneous isotropic universe are formulated without any
gauge fixing. We have derived the Einstein equations in the case of the single
perfect fluid without ignoring any modes. These equations imply that any types
of mode-coupling arise due to the second-order effects of the Einstein
equations.
