Metrics and conformal transformations

[FuzzySphere]

Conformal field theory is way over my head at the moment, but I decided to "dip my toes into it," and I watched a little video talking about conformal transformations. Now, I know that in a conformal transformation, $$x^\mu \to x'^\mu ,$$ the metric must satisfy $$\Lambda (x) g_{\mu \nu} = g_{\rho \sigma} \frac {\partial x'^\rho}{\partial x^\mu} \frac {\partial x'^\sigma}{\partial x^\nu}.$$ From this I have been informed that we can derive the condition $$dx'^\mu dx'_\mu = \Lambda (x) dx^\mu dx_\mu .$$ I have tried using the condition on the metric to derive this, only to get to this condition: $$g_{\mu \nu} dx'^\mu dx'^\nu = \Lambda g_{\mu \nu} dx^\mu dx ^\nu ,$$ but I have one query: can we use the metric $$g_{\mu \nu}$$ in the $$x^\mu$$ coordinates to lower the indices of $$dx'^\mu ,$$ seeing as they are from a different coordinate system?

 

[anuttarasammyak]

the metric must satisfy Λ(x)gμν=gρσ∂x′ρ∂xμ∂x′σ∂xν.

I observe ' and without ' are upside down.
\Lambda g_{\mu\nu}=g'_{\mu\nu}=g_{\rho\sigma} \ (\partial x^{\rho}/\partial x'^{\mu}) \ ( \partial x^{\sigma}/ \partial x'^{\nu})

 

[FuzzySphere]

I observe ' and without ' are upside down.
\Lambda g_{\mu\nu}=g'_{\mu\nu}=g_{\rho\sigma} \ (\partial x^{\rho}/\partial x'^{\mu}) \ ( \partial x^{\sigma}/ \partial x'^{\nu})

No, that is the transformation law for the metric, what I have is the coordinate representation of the pull back of the metric by the conformal transformation.