# Conformal Transformation and angle preserving [duplicate]

For general co-ordinate transformation $$V^\mu$$ transforms as

$$V^\mu \rightarrow V'^\mu= \large\frac{\partial x'^\mu}{\partial x^\alpha}\ V^\alpha$$

therefore the inner product between two vectors transforms as

$$V^\mu W^\nu \large g_{\mu\nu}\rightarrow V'^\alpha W'^\beta \large g'_{\alpha\beta}= \large\frac{\partial x'^\alpha}{\partial x^\mu}\ V^\mu\large\frac{\partial x'^\beta}{\partial x^\nu}\ W^\nu \large\frac{\partial x^\gamma}{\partial x'^\alpha}\ \frac{\partial x^\sigma}{\partial x'^\beta} g_{\gamma\sigma} = V^\mu W^\nu \large g_{\mu\nu}$$
i.e it is invariant therefore it appears that the angles are preserved as they depend on inner product, but i know it is not true for general co-ordinate transformation, only for conformal it is true.
I have refered other texts that prove that conformal transformation leaves angle invariant but they only transform metric $$\large{g_{\mu\nu}}$$ and not the vectors $$V^\mu$$ and $$W^\nu$$ in the inner product

$$cos(\theta)=\large{ \frac{V^\mu W^\nu \large g_{\mu\nu}}{||V|| \space||W||}}\$$

$$||V||=\sqrt{ V^\mu V^\nu \large g_{\mu\nu}}$$ similarly for $$||W||$$