transformations and gauge invariance in two–parameter perturbation theory we need first to introduce some mathematical tools concerning the two–parameter Taylor expansion of tensor fields. Since Taylor expansions are aimed to provide the value of a quantity at some point in terms of its value, and the value of its derivatives, at another point, a Taylor expansion of tensorial quantities can only be defined through a mapping between tensors at different points of the manifold under consideration. In this section we consider the cases where such a mapping is given by a two–parameter family of diffeomorphisms of $\cM$, starting from the simplest case in which such a family constitutes a group.

Two–parameter groups of diffeomorphisms {#groupdif}
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Given a differentiable manifold $\cM$, a two–parameter group of diffeomorphisms $\phi$ of $\cM$ can be represented as follows

[cccc]{} :&\^2 & & \^2\
&(p,,)&|& (\_[,]{}(p),,).\

For the purpose of introducing two-parameter perturbation theory, where perturbing first with respect to the parameter $\lambda$ and afterwards with respect to the parameter $\Omega$ should be equivalent to the converse operation, we will assume that $\phi_{\lambda,\Omega}$ is such that it satisfies the following property[^2] \_[\_1,\_1]{}\_[\_2,\_2]{}= \_[\_1+\_2,\_1+\_2]{},    ,. By one hand, this property implies that the two-parameter group is 