Second Order Gauge Invariant Perturbation Theory -- Perturbative   curvatures in the two-parameter case --
  Based on the gauge invariant variables proposed in our previous paper [K. Nakamura, Prog. Theor. Phys. vol.110 (2003), 723.], some formulae of the perturbative curvatures of each order are derived. We follow the general framework of the second order gauge invariant perturbation theory on arbitrary background spacetime to derive these formulae. These perturbative curvatures do have the same form as the definitions of gauge invariant variables for arbitrary perturbative fields which are previously proposed. As a result, we explicitly see that any perturbative Einstein equations are given in terms of gauge invarinat form. We briefly discuss physical situations to which this framework should be applied. 
Kouji Nakamura