Coordinate transformation and relationships between Reconstructing the metric on $CP^2$ with special one forms

I have two sets of complex coordinates $(z_1,z_2)$ and $(t_1,t_2)$. I have the following relation between them $$ \begin{split} z_1&=\frac{t_1}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right)\\ z_2&=\frac{t_2}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right) \end{split} $$ I know that $(z_1,z_2)$ are the affineaffine\inhomogeneous coordinates on the complex projective space $CP^2$ Now. 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 $(Y_1,Y_2, Y_3, Y_4)$ . The expressions for the one forms in terms of $(z1,z2)$ are given below \begin{split} &&Y1=\frac{|z_1|^2+|z_2|^2\sqrt{1+|z_1|^2+|z_2|^2}}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz1-\frac{z1\bar{z}_2(-1+\sqrt{1+|z_1|^2+|z_2|^2})}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz2\\ &&Y2=\frac{|z_2|^2+|z_1|^2\sqrt{1+|z_1|^2+|z_2|^2}}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz2-\frac{z2\bar{z}_1(-1+\sqrt{1+|z_1|^2+|z_2|^2})}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz1\\ &&Y_3=-\bar{Y}_1\\ &&Y_4=-\bar{Y}_2 \end{split}

I have been able to workout the following relations $$ \begin{split} \frac{\bar{z}_1 dz_1+\bar{z}_2 dz_2}{1+|z_1|^2+|z_2|^2} &=\bar{z}_1Y_1+\bar{z}_2Y_2\\ \frac{z_1d\bar{z}_1+z_2 d\bar{z}_2}{1+|z_1|^2+|z_2|^2}&=-(z_1Y_3+z_2Y_4) \end{split} $$ I don't know the exact relation between $(z_1,z_2)$ and $(Y_1,Y_2, Y_3, Y_4)$. Also note that $Y_3=-\bar{Y}_1,~Y_4=-\bar{Y}_2$. IsIs 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? Special Case: To make my question clear let us consider the case $z2=0$. Then we see the relationship reduces to the following $$ \begin{split} \frac{ dz_1}{1+|z_1|^2} &=Y_1\\ \frac{d\bar{z}_1}{1+|z_1|^2}&=-Y_3 \end{split} $$ Taking the product of the above equation gives $$ \begin{split} ds^2=\frac{ dz_1d\bar{z}_1}{(1+|z_1|^2)^2}=-Y_1Y_3=Y_1\bar{Y}_1 \end{split} $$ This is the Fubini Study metric on $CP^1$ with inhomogeneous/affine coordinates $(z_1,z_2)$. I am expecting something like this forto rewrite using the case offorms the Fubini-Study metric on $CP^2$ also.

Coordinate transformation and relationships between one forms

I have two sets of complex coordinates $(z_1,z_2)$ and $(t_1,t_2)$. I have the following relation between them $$ \begin{split} z_1&=\frac{t_1}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right)\\ z_2&=\frac{t_2}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right) \end{split} $$ 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 $$ \begin{split} \frac{\bar{z}_1 dz_1+\bar{z}_2 dz_2}{1+|z_1|^2+|z_2|^2} &=\bar{z}_1Y_1+\bar{z}_2Y_2\\ \frac{z_1d\bar{z}_1+z_2 d\bar{z}_2}{1+|z_1|^2+|z_2|^2}&=-(z_1Y_3+z_2Y_4) \end{split} $$ I don't know the exact relation between $(z_1,z_2)$ and $(Y_1,Y_2, Y_3, Y_4)$. Also note that $Y_3=-\bar{Y}_1,~Y_4=-\bar{Y}_2$. 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? Special Case: To make my question clear let us consider the case $z2=0$. Then we see the relationship reduces to the following $$ \begin{split} \frac{ dz_1}{1+|z_1|^2} &=Y_1\\ \frac{d\bar{z}_1}{1+|z_1|^2}&=-Y_3 \end{split} $$ Taking the product of the above equation gives $$ \begin{split} ds^2=\frac{ dz_1d\bar{z}_1}{(1+|z_1|^2)^2}=-Y_1Y_3=Y_1\bar{Y}_1 \end{split} $$ This is the Fubini Study metric on $CP^1$ with inhomogeneous/affine coordinates $(z_1,z_2)$. I am expecting something like this for the case of $CP^2$ also.

Reconstructing the metric on $CP^2$ with special one forms

I know that $(z_1,z_2)$ are the affine\inhomogeneous coordinates on the complex projective space $CP^2$. Now I have four one forms $(Y_1,Y_2, Y_3, Y_4)$. I want to rewrite the Fubini Study metric on $CP^2$ in terms of the forms $(Y_1,Y_2, Y_3, Y_4)$ . The expressions for the one forms in terms of $(z1,z2)$ are given below \begin{split} &&Y1=\frac{|z_1|^2+|z_2|^2\sqrt{1+|z_1|^2+|z_2|^2}}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz1-\frac{z1\bar{z}_2(-1+\sqrt{1+|z_1|^2+|z_2|^2})}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz2\\ &&Y2=\frac{|z_2|^2+|z_1|^2\sqrt{1+|z_1|^2+|z_2|^2}}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz2-\frac{z2\bar{z}_1(-1+\sqrt{1+|z_1|^2+|z_2|^2})}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz1\\ &&Y_3=-\bar{Y}_1\\ &&Y_4=-\bar{Y}_2 \end{split}

I have been able to workout the following relations $$ \begin{split} \frac{\bar{z}_1 dz_1+\bar{z}_2 dz_2}{1+|z_1|^2+|z_2|^2} &=\bar{z}_1Y_1+\bar{z}_2Y_2\\ \frac{z_1d\bar{z}_1+z_2 d\bar{z}_2}{1+|z_1|^2+|z_2|^2}&=-(z_1Y_3+z_2Y_4) \end{split} $$ 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? Special Case: To make my question clear let us consider the case $z2=0$. Then we see the relationship reduces to the following $$ \begin{split} \frac{ dz_1}{1+|z_1|^2} &=Y_1\\ \frac{d\bar{z}_1}{1+|z_1|^2}&=-Y_3 \end{split} $$ Taking the product of the above equation gives $$ \begin{split} ds^2=\frac{ dz_1d\bar{z}_1}{(1+|z_1|^2)^2}=-Y_1Y_3=Y_1\bar{Y}_1 \end{split} $$ This is the Fubini Study metric on $CP^1$ with inhomogeneous/affine coordinates $(z_1,z_2)$. I am expecting to rewrite using the forms the Fubini-Study metric on $CP^2$.

Tried to make my case clear with an example

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edited Sep 17, 2020 at 17:57

m1rohit

69
3

I have two sets of complex coordinates $(z_1,z_2)$ and $(t_1,t_2)$. I have the following relation between them $$ \begin{split} z_1&=\frac{t_1}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right)\\ z_2&=\frac{t_2}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right) \end{split} $$ 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 $$ \begin{split} \frac{\bar{z}_1 dz_1+\bar{z}_2 dz_2}{1+|z_1|^2+|z_2|^2} &=\bar{z}_1Y_1+\bar{z}_2Y_2\\ \frac{z_1d\bar{z}_1+z_2 d\bar{z}_2}{1+|z_1|^2+|z_2|^2}&=-(z_1Y_3+z_2Y_4) \end{split} $$ I don't know the exact relation between $(z_1,z_2)$ and $(Y_1,Y_2, Y_3, Y_4)$. Also note that $Y_3=-\bar{Y}_1,~Y_4=-\bar{Y}_2$. 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? Special Case: To make my question clear let us consider the case $z2=0$. Then we see the relationship reduces to the following $$ \begin{split} \frac{ dz_1}{1+|z_1|^2} &=Y_1\\ \frac{d\bar{z}_1}{1+|z_1|^2}&=-Y_3 \end{split} $$ Taking the product of the above equation gives $$ \begin{split} ds^2=\frac{ dz_1d\bar{z}_1}{(1+|z_1|^2)^2}=-Y_1Y_3=Y_1\bar{Y}_1 \end{split} $$ This is the Fubini Study metric on $CP^1$ with inhomogeneous/affine coordinates $(z_1,z_2)$. I am expecting something like this for the case of $CP^2$ also.

I have two sets of complex coordinates $(z_1,z_2)$ and $(t_1,t_2)$. I have the following relation between them $$ \begin{split} z_1&=\frac{t_1}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right)\\ z_2&=\frac{t_2}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right) \end{split} $$ 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 $$ \begin{split} \frac{\bar{z}_1 dz_1+\bar{z}_2 dz_2}{1+|z_1|^2+|z_2|^2} &=\bar{z}_1Y_1+\bar{z}_2Y_2\\ \frac{z_1d\bar{z}_1+z_2 d\bar{z}_2}{1+|z_1|^2+|z_2|^2}&=-(z_1Y_3+z_2Y_4) \end{split} $$ I don't know the exact relation between $(z_1,z_2)$ and $(Y_1,Y_2, Y_3, Y_4)$. Also note that $Y_3=-\bar{Y}_1,~Y_4=-\bar{Y}_2$. 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?

I have two sets of complex coordinates $(z_1,z_2)$ and $(t_1,t_2)$. I have the following relation between them $$ \begin{split} z_1&=\frac{t_1}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right)\\ z_2&=\frac{t_2}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right) \end{split} $$ 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 $$ \begin{split} \frac{\bar{z}_1 dz_1+\bar{z}_2 dz_2}{1+|z_1|^2+|z_2|^2} &=\bar{z}_1Y_1+\bar{z}_2Y_2\\ \frac{z_1d\bar{z}_1+z_2 d\bar{z}_2}{1+|z_1|^2+|z_2|^2}&=-(z_1Y_3+z_2Y_4) \end{split} $$ I don't know the exact relation between $(z_1,z_2)$ and $(Y_1,Y_2, Y_3, Y_4)$. Also note that $Y_3=-\bar{Y}_1,~Y_4=-\bar{Y}_2$. 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? Special Case: To make my question clear let us consider the case $z2=0$. Then we see the relationship reduces to the following $$ \begin{split} \frac{ dz_1}{1+|z_1|^2} &=Y_1\\ \frac{d\bar{z}_1}{1+|z_1|^2}&=-Y_3 \end{split} $$ Taking the product of the above equation gives $$ \begin{split} ds^2=\frac{ dz_1d\bar{z}_1}{(1+|z_1|^2)^2}=-Y_1Y_3=Y_1\bar{Y}_1 \end{split} $$ This is the Fubini Study metric on $CP^1$ with inhomogeneous/affine coordinates $(z_1,z_2)$. I am expecting something like this for the case of $CP^2$ also.

Modified the question and added more details

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edited Sep 17, 2020 at 4:45

m1rohit

69
3

I have two sets of complex coordinates $(z_1,z_2)$ and $(t_1,t_2)$. I have the following relation between them $$ \begin{split} z_1&=\frac{t_1}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right)\\ z_2&=\frac{t_2}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right) \end{split} $$ 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 $$ \begin{split} \frac{\bar{z}_1 dz_1+\bar{z}_2 dz_2}{1+|z_1|^2+|z_2|^2} &=\bar{z}_1Y_1+\bar{z}_2Y_2\\ \frac{z_1d\bar{z}_1+z_2 d\bar{z}_2}{1+|z_1|^2+|z_2|^2}&=-(z_1Y_3+z_2Y_4) \end{split} $$ I don't know the exact relation between $(z_1,z_2)$ and $(Y_1,Y_2, Y_3, Y_4)$. Also note that $Y_3=-\bar{Y}_1,~Y_4=-\bar{Y}_2$. 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?

I have two sets of complex coordinates $(z_1,z_2)$ and $(t_1,t_2)$. I have the following relation between them $$ \begin{split} z_1&=\frac{t_1}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right)\\ z_2&=\frac{t_2}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right) \end{split} $$ 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 $$ \begin{split} \frac{\bar{z}_1 dz_1+\bar{z}_2 dz_2}{1+|z_1|^2+|z_2|^2} &=\bar{z}_1Y_1+\bar{z}_2Y_2\\ \frac{z_1d\bar{z}_1+z_2 d\bar{z}_2}{1+|z_1|^2+|z_2|^2}&=-(z_1Y_3+z_2Y_4) \end{split} $$ 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?

I have two sets of complex coordinates $(z_1,z_2)$ and $(t_1,t_2)$. I have the following relation between them $$ \begin{split} z_1&=\frac{t_1}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right)\\ z_2&=\frac{t_2}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right) \end{split} $$ 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 $$ \begin{split} \frac{\bar{z}_1 dz_1+\bar{z}_2 dz_2}{1+|z_1|^2+|z_2|^2} &=\bar{z}_1Y_1+\bar{z}_2Y_2\\ \frac{z_1d\bar{z}_1+z_2 d\bar{z}_2}{1+|z_1|^2+|z_2|^2}&=-(z_1Y_3+z_2Y_4) \end{split} $$ I don't know the exact relation between $(z_1,z_2)$ and $(Y_1,Y_2, Y_3, Y_4)$. Also note that $Y_3=-\bar{Y}_1,~Y_4=-\bar{Y}_2$. 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?