Suppose we are working in the hyperbolic plane, that is the set $$ \mathbb{H}^2 = \{ u + iv \colon v > 0\} \quad \text{with metric} \quad \frac{du^2 + dv^2}{v^2}\,. $$ Now, formally, I can rewrite the metric in complex variables as $$ \frac{du^2 + dv^2}{v^2} = \frac{1}{\Im(z)^2}dzd\bar z $$ (where $\Im(w)$ denotes the imaginary part). But here I am running into doubt whether this makes sense, because complex conjugation on $\mathbb{C}$ does not preserve the upper half plane, so I am worried that I am producing nonsense. On the other hand, in the unit disc model I would still be able to take the complex conjugate of a number.
How can I best understand the complex conjugate in Hyperbolic Geometry, in particular, can I use the metric representation above on the right hand side?
Many thanks for your help!
You can view $\overline z$ as a function from the half plane to $\mathbb{C}$, instead of as a map from the half plane to itself. This is similar to viewing $e^{it}$ as a function from the real line to the complex plane, even though complex values don't exist in the real line.