5 manifold N = M × R , where 4 = dim M and λ ∈ R . The background spacetime M$_{0}$ = N|$_{λ}$$_{=0}$ and the physi- cal spacetime M = M$_{λ}$ = N|$_{R}$$_{=}$$_{λ}$ are also submanifolds embedded in the extended manifold N . Each point on N is identified by a pair ( p, λ ), where p ∈ M$_{λ}$ , and each point in M$_{0}$ ⊂ N is identified by λ = 0. Through this construction, the manifold N is foliated by four-dimensional submanifolds M$_{λ}$ of each λ , and these are diffeomorphic to M and M$_{0}$ . The manifold N has a natural differentiable structure consisting of the di- rect product of M and R . Further, the perturbed space- times M$_{λ}$ for each λ must have the same differential structure with this construction. In other words, we re- quire that perturbations be continuous in the sense that M and M$_{0}$ are connected by a continuous curve within the extended manifold N . Hence, the changes of the dif- ferential structure resulting from the perturbation, for example the formation of singularities and singular per- turbations in the sense of fluid mechanics, are excluded from consideration. Let us consider the set of field equations on the physical spacetime M$_{λ}$ for the physical variables Q$_{λ}$ on M$_{λ}$ . The field equation (2.6) formally represents the Einstein equation for the metric on M$_{λ}$ and the equa- tions for matter fields on M$_{λ}$ . If a tensor field Q$_{λ}$ is given on each M$_{λ}$ , Q$_{λ}$ is automatically extended to a tensor field on N by Q ( p, λ ) := Q$_{λ}$ ( p ), where p ∈ M$_{λ}$ . In this extension, the field equation (2.6) is regarded as an equation on the extended manifold N . Thus, we have extended an arbitrary tensor field and the field equations (2.6) on each M$_{λ}$ to those on the extended manifold N . Tensor fields on N obtained through the above con- struction are necessarily “tangent” to each M$_{λ}$ . To con- sider the basis of the tangent space of N , we introduce the normal form and its dual, which are normal to each M$_{λ}$ in N . These are denoted by ( dλ )$_{a}$ and ( ∂/∂λ ) $^{a}$, respectively, and they satisfy ( dλ )$_{a}$( ∂/∂λ ) a = 1. The form ( dλ )$_{a}$ and its dual, ( ∂/∂λ ) $^{a}$, are normal to any ten- sor field extended from the tangent space on each M$_{λ}$ through the above construction. The set consisting of ( dλ )$_{a}$, ( ∂/∂λ ) a and the basis of the tangent space on each M$_{λ}$ is regarded as the basis of the tangent space of N . Now, we define the perturbation of an arbitrary tensor field Q . We compare Q on M$_{λ}$ with Q$_{0}$ on M$_{0}$ , and it is necessary to identify the points of M$_{λ}$ with those of M$_{0}$ as mentioned above. This point identification map is the gauge choice of the second kind as mentioned above. The gauge choice is made by assigning a diffeomorphism X$_{λ}$ : N → N such that X$_{λ}$ : M$_{0}$ → M$_{λ}$ . Following the paper of Bruni et al.[10], we introduce a gauge choice X$_{λ}$ as an one-parameter groups of diffeomorphisms, i.e., an exponential map, for simplicity. We denote the generator of this exponential map by X η $^{a}$. This generator X η a is decomposed by the basis on N which are constructed above. Although the generator X η a should satisfy some appropriate properties[8], the arbitrariness of the gauge choice X$_{λ}$ is represented by the tangential component of the generator X η a to M$_{λ}$ . The pull-back X ∗ $_{λ}$Q , which is induced by the expo- nential map X$_{λ}$ , maps a tensor field Q on the physical manifold M$_{λ}$ to a tensor field X ∗ $_{λ}$Q on the background spacetime. In terms of this generator X η $^{a}$, the pull-back X ∗ $_{λ}$Q is represented by the Taylor expansion where r = X$_{λ}$ ( p ) ∈ M$_{λ}$ . Because p ∈ M$_{0}$ , we may regard the equation as an equation on the background spacetime M$_{0}$ , where Q$_{0}$ = Q |$_{M}$$_{0}$ is the background value of the physical vari- able of Q . Once the definition of the pull-back of the gauge choice X$_{λ}$ is given, the first- and the second-order perturbations (1) $_{X}$Q and (2) $_{X}$Q of a tensor field Q under the gauge choice X$_{λ}$ are simply given by the expansion with respect to the infinitesimal parameter λ . Comparing Eqs. (2.8) and (2.9), we define the first- and the second- order perturbations of a physical variable Q$_{λ}$ under the gauge choice X$_{λ}$ by We note that all variables in Eq. (2.9) are defined on M$_{0}$ . Now, we consider two different gauge choices based on the above understanding of the second kind gauge choice. Suppose that X$_{λ}$ and Y$_{λ}$ are two exponential maps with the generators X η a and Y η a on N , respectively. In other words, X$_{λ}$ and Y$_{λ}$ are two gauge choices (see Fig. 2). Then, the integral curves of each X η a and Y η a in N are the orbits of the actions of the gauge choices X$_{λ}$ and Y$_{λ}$ , respectively. Since we choose the generators X η a and Y η a so that these are transverse to each M$_{λ}$ everywhere on N , the integral curves of these vector fields intersect with each M$_{λ}$ . Therefore, points lying on the same integral curve of either of the two are to be regarded as the same point within the respective gauges. When these curves are not identical, i.e., the tangential components to each M$_{λ}$ of X η a and Y η a are different, these point identifica- tion maps X$_{λ}$ and Y$_{λ}$ are regarded as two different gauge choices . We next introduce the concept of gauge invariance . In particular, in this paper, we consider the concept of order by order gauge invariance [12]. Suppose that X$_{λ}$ and Y$_{λ}$ (1) $_{X}$Q := £$_{X}$$_{η}$ Q |$_{M}$$_{0}$ , (2) $_{X}$Q := £ 2 $_{$_{X}$η}$Q ∣ M$_{0}$ . (2.10) X ∗ $_{λ}$Q$_{λ}$ |$_{M}$$_{0}$ = Q$_{0}$ + λ (1) $_{X}$Q + 1 2 λ 2 (2) $_{X}$Q + O ( λ $^{3}$) (2.9) X ∗ $_{λ}$Q ( p ) = Q$_{0}$ ( p ) + λ £$_{X}$$_{η}$ Q |$_{M}$$_{0}$ ( p ) + 1 2 λ 2 £ 2 $_{$_{X}$η}$Q ∣ M$_{0}$ ( p ) + O ( λ $^{3}$) (2.8) Q ( r ) = Q ( X$_{λ}$ ( p )) = X ∗ $_{λ}$Q ( p ) = Q ( p ) + λ £$_{X}$$_{η}$ Q |$_{p}$ + 1 2 λ 2 £ 2 $_{$_{X}$η}$Q ∣ p + O ( λ $^{3}$) , (2.7) E [ Q$_{λ}$ ] = 0 (2.6)