I have two sets of complex coordinates $(z_1,z_2)$ and $(t_1,t_2)$. I have the following relation between them $$ z_1=\frac{t1}{\sqrt{|t_1|^2+|t_2|^2}}tan(\sqrt{|t_1|^2+|t_2|^2})\\ z_2=\frac{t2}{\sqrt{|t_1|^2+|t_2|^2}}tan(\sqrt{|t_1|^2+|t_2|^2}) $$ I know that $(z_1,z_2)$ are the affine coordinates on the complex projective space $CP^2$ Now I have four one forms $(Y_1,Y_2, Y_3, Y_4)$ they are complicated functions of $(t_1,t_2)$. I want to rewrite the Fubini Study metric on $CP^2$ in terms of the forms$(Y_1,Y_2, Y_3, Y_4)$ . I have been able to workout the following relations $$ \frac{\bar{z}_1 dz_1+\bar{z}_2 dz_2}{1+|z_1|^2+|z_2|^2}=\bar{z}_1Y_1+\bar{z}_2Y_2\\ \frac{z_1d\bar{z}_1+z_2 d\bar{z}_2}{1+|z_1|^2+|z_2|^2}=-(z_1Y_3+z_2Y_4) $$ I don't know the exact relation between $(z_1,z_2)$ and $(Y_1,Y_2, Y_3, Y_4)$. Is there a way to get the relation using the above condition so that I can rewrite the Fubini Study metric in terms of these forms?