2 gauge-invariant perturbation theory, we first explain the general arguments of the Taylor expansion on a mani- fold without introducing an explicit coordinate system in Sec.II A. Further, we also have to clarify the notion of “gauge” in general relativity to develop the gauge- invariant perturbation theory from general point of view, which is explained in Sec. II B. After clarifying the no- tion of “gauge” in general relativistic perturbations, in Sec. II C, we explain the formulation of the general rela- tivistic gauge-invariant perturbation theory from general point of view. Although our understanding of “gauge” in general relativistic perturbations essentially is different from “degree of freedom of coordinates” as in many lit- erature, “a coordinate transformation” is induced by our understanding of “gauge”. This situation is explained in Sec. II D. To exclude “gauge degree of freedom” which is unphysical degree of freedom in perturbations, we con- struct “gauge-invariant variables” of perturbations as re- viewed in Sec. II E. These “gauge-invariant variables” are regarded as physical quantities. First, we briefly review the issues on the general form of the Taylor expansion of tensors on a manifold M . The gauge issue of general relativistic perturbation theories which we will discuss is related to the coordinate transfor- mation. Therefore, we have to discuss the general form of the Taylor expansion without the explicit introduction of coordinate systems. Although we only consider the Tay- lor expansion of a scalar function f : M ↦→ R , here, the resulting formula is extended to that for any tensor field on a manifold as in Appendix A. We have to emphasize that the general formula of the Taylor expansion shown here is the starting point of our gauge-invariant formula- tion of the second-order general relativistic perturbation theory. The Taylor expansion of a function f is an approxi- mated form of f ( q ) at q ∈ M in terms of the variables at p ∈ M , where q is in the neighborhood of p . To de- rive the formula for the Taylor expansion of f , we have to compare the values of f at the different points on the manifold. To accomplish this, we introduce a one- parameter family of diffeomorphisms Φ$_{λ}$ : M ↦→ M , where Φ$_{λ}$( p ) = q and Φ$_{λ}$$_{=0}$( p ) = p . One example of a diffeomorphisms Φ$_{λ}$ is an exponential map with a gen- erator. However, we consider a more general class of diffeomorphisms. The diffeomorphism Φ$_{λ}$ induces the pull-back Φ ∗ λ of the function f and this pull-back enable us to compare the values of the function f at different points. Further, the Taylor expansion of the function f ( q ) is given by Since this expression hold for an arbitrary smooth func- tion f , the function f in Eq. (2.1) can be regarded as a dummy. Therefore, we may regard the Taylor expansion (2.1) to be the expansion of the pull-back Φ ∗ λ of the diffeo- morphism Φ$_{λ}$, rather than the expansion of the function f . According to this point of view, Sonego and Bruni[18] showed that there exist vector fields ξ a 1 and ξ a 2 such that the expansion (2.1) is given by without loss of generality (see Appendix A). Equation (2.2) is not only the representation of the Taylor ex- pansion of the function f , but also the definitions of the generators ξ a 1 and ξ a $_{2}$. These generators of the one- parameter family of diffeomorphisms Φ$_{λ}$ represent the di- rection along which the Taylor expansion is carried out. The generator ξ a 1 is the first-order approximation of the flow of the diffeomorphism Φ$_{λ}$, and the generator ξ a 2 is the second-order correction to this flow. We should regard the generators ξ a 1 and ξ a 2 to be independent. Further, as shown in Appendix A, the representation of the Taylor expansion of an arbitrary scalar function f is extended to that for an arbitrary tensor field Q just through the replacement f → Q . We must note that, in general, the representation (2.2) of the Taylor expansion is different from an usual ex- ponential map which is generated by a vector field. In general, As noted in Ref. [10], if the second-order generator ξ$_{2}$ in Eq. (2.2) is proportional to the first-order generator ξ$_{1}$ in Eq. (2.2), the diffeomorphism Φ$_{λ}$ is reduced to an exponential map. Therefore, one may reasonably doubt that Φ$_{λ}$ forms a group except under very special con- ditions. However, we have to note that the properties (2.3) does not directly mean that Φ$_{λ}$ does not form a group. There will be possibilities that Φ$_{λ}$ form a group in a different sense from exponential maps, in which the properties (2.3) will be maintained. Now, we give an intuitive explanation of the repre- sentation (2.2) of the Taylor expansion through the case where the scalar function f in Eq. (2.2) is a coordinate function. When two points p, q ∈ M in Eq. (2.2) are in the neighborhood of each other, we can apply a co- ordinate system M ↦→ R n ( n = dim M ), which denoted by { x $^{µ}$} , to an open set which includes these two points. Then, we can measure the relative position of these two points p and q in M in terms of this coordinate system in R n through the Taylor expansion (2.2). In this case, we may regard that the scalar function f in Eq. (2.2) is Φ$_{σ}$ ◦ Φ$_{λ}$ = Φ$_{σ}$$_{+}$$_{λ}$ , Φ − 1 λ = Φ − $_{λ}$. (2.3) f ( q ) = f (Φ$_{λ}$( p )) =: (Φ ∗ $_{λ}$f )( p ) = f ( p ) + ∂ ∂λ (Φ ∗ $_{λ}$f ) ∣ p λ + 1 2 ∂ 2 ∂λ $_{2}$(Φ ∗ $_{λ}$f ) ∣ p λ 2 + O ( λ $^{3}$) . (2.1) f ( q ) = (Φ ∗ $_{λ}$f )( p ) = f ( p ) + ( £$_{ξ}$$_{1}$ f ) |$_{p}$ λ + 1 2 ( £$_{ξ}$$_{2}$ + £ 2 ξ 1 ) f ∣ p λ 2 + O ( λ $^{3}$) , (2.2) A. Taylor expansion of tensors on a manifold