How to define the distance between two points in a conformal transformed space? Consider a particular conformal transformation $x^\mu\rightarrow x'^\mu$, and the metric of a flat space transforms in the following way,
$$\eta_{\mu\nu}\rightarrow g'_{\mu\nu}=\Lambda^2(x)\eta_{\mu\nu}.$$
A special conformal transformation (SCF) takes the following form:
$$x'^\mu=\frac{x^\mu-b^\mu x^2}{1-2b\cdot x+b^2 x^2}\tag{4.15d} $$
Here, vector $b^\mu$ parametrizes SCF, and $x^2$ and $b\cdot x$ are defined using $\eta_{\mu\nu}$. It turns out that $\Lambda(x)$ is the denominator.
OK. According to Di Francesco's Conformal Field Theory, a relation is given (Eq.(4.22)):
$$|x'_i-x'_j|=\frac{|x_i-x_j|}{\sqrt{\Lambda(x_i)\Lambda(x_j)}}. \tag{4.22}$$
Here, $x_i,x_j$ collectively represents the coordinates of two arbitrary points in the original coordinate system. That is, $x_i=(x_i^1, x_i^2, ...)$. And $x'_i,x'_j$ are coordinates related to $x_i,x_j$ via the SCF. ${|x_i-x_j|}$ is apparently the distance. 
The authors also call the LHS of the above equation a distance, too! But the transformed metric $g'_{\mu\nu}$ is no longer flat. So the distance between any pair of  finitely separated points is not well defined, unless a path from one point to the other is specified. So how to understand this distance (the LHS)? 

 A: Comments to the post (v2):


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*Ref. 1 is considering the $d$-dimensional real Euclidean space $(\mathbb{R}^d,|\cdot|^2)$ with the standard norm 
$$|x|^2~:=~\sum_{\mu=1}^d (x^{\mu})^2~=~\sum_{\mu,\nu=1}^d x^{\mu}\eta_{\mu\nu}x^{\nu}, \qquad \eta_{\mu\nu} ~=~{\rm diag}(1,\ldots, 1),\tag{A}$$
and inner product
$$\langle x ,y\rangle~:=~\sum_{\mu,\nu=1}^d x^{\mu}\eta_{\mu\nu}y^{\nu} .\tag{B}$$

*We stress that the metric is fixed and the same, induced from the standard norm (A). (However, as always, it takes different explicit forms in various different coordinate systems.)

*From the SCT (4.15d), it follows that 
$$ |x^{\prime}|^2~=~\frac{|x|^2}{\Lambda(x)^2}. \tag{C} $$
Together with a similar calculation of the inner product $\langle x^{\prime} ,y^{\prime}\rangle$, it is possible to derive eq. (4.22):
$$ |x^{\prime}-y^{\prime}|^2~=~\frac{|x-y|^2}{\Lambda(x)\Lambda(y)}. \tag{4.22} $$

*From eq. (4.22) it follows directly that the SCT (4.15d) is a conformal map
$$ ds^{\prime 2}~=~\frac{ds^2}{\Lambda(x)^2}. \tag{D}$$
References:


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*P. Di Francesco, P. Mathieu and D. Senechal, CFT, 1997.

