Coordinates on riemann surface

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I am reading Hitchens text on integral systems




A Riemann surface is a one-dimensional complex manifold with a maximal set of coordinate charts $U_alpha,varphi_alpha_alphain I$ where $varphi_alpha:U_alphato Bbb C$ such that $varphi_betacirc varphi_alpha:varphi_alpha(U_alphacap U_beta)to varphi_beta(U_alphacap U_beta)$ is an invertible holomorphic map for every pair $alpha,betain I$




My text then says:




Thus a neighbourhood of any point can be parametrized by a complex number $z$ and on any overlapping neighbourhood with parameter $w$, $w(z)$ is a holomorphic function of one variable.




What does this second paragraph actually mean? Let $M$ be a Riemann surface. Surely if I consider any point $pin M$ I can find a chart ${U_alpha,varphi_alpha)$ where $pin U_alpha$, and then can consider $p$ in local coordinates via $varphi_alpha(p)$. I could then call $varphi_alpha(p)=z$ and then consider $varphi_beta(p)=w$ and abuse notation and write $w(z)$ for the transition. Is that what they intend (in which case I personally think this is stupidly cumbersome, so I doubt it)?



Later they write:




Let us use the coordinate $z$ corresponding to the coordinate chart $varphi_gamma$




and they start taking polynomials in $z$. What exactly is going on?




riemann-surfaces complex-manifolds




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  • I think what they mean is $w(z):phi_alpha(U_alphacap U_beta)to phi_beta(U_betacap U_alpha)$ given by $zmapsto w$
    – daruma
    Aug 1 at 10:13














up vote
0
down vote

favorite












I am reading Hitchens text on integral systems




A Riemann surface is a one-dimensional complex manifold with a maximal set of coordinate charts $U_alpha,varphi_alpha_alphain I$ where $varphi_alpha:U_alphato Bbb C$ such that $varphi_betacirc varphi_alpha:varphi_alpha(U_alphacap U_beta)to varphi_beta(U_alphacap U_beta)$ is an invertible holomorphic map for every pair $alpha,betain I$




My text then says:




Thus a neighbourhood of any point can be parametrized by a complex number $z$ and on any overlapping neighbourhood with parameter $w$, $w(z)$ is a holomorphic function of one variable.




What does this second paragraph actually mean? Let $M$ be a Riemann surface. Surely if I consider any point $pin M$ I can find a chart ${U_alpha,varphi_alpha)$ where $pin U_alpha$, and then can consider $p$ in local coordinates via $varphi_alpha(p)$. I could then call $varphi_alpha(p)=z$ and then consider $varphi_beta(p)=w$ and abuse notation and write $w(z)$ for the transition. Is that what they intend (in which case I personally think this is stupidly cumbersome, so I doubt it)?



Later they write:




Let us use the coordinate $z$ corresponding to the coordinate chart $varphi_gamma$




and they start taking polynomials in $z$. What exactly is going on?




riemann-surfaces complex-manifolds




share|cite|improve this question





















  • I think what they mean is $w(z):phi_alpha(U_alphacap U_beta)to phi_beta(U_betacap U_alpha)$ given by $zmapsto w$
    – daruma
    Aug 1 at 10:13












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I am reading Hitchens text on integral systems




A Riemann surface is a one-dimensional complex manifold with a maximal set of coordinate charts $U_alpha,varphi_alpha_alphain I$ where $varphi_alpha:U_alphato Bbb C$ such that $varphi_betacirc varphi_alpha:varphi_alpha(U_alphacap U_beta)to varphi_beta(U_alphacap U_beta)$ is an invertible holomorphic map for every pair $alpha,betain I$




My text then says:




Thus a neighbourhood of any point can be parametrized by a complex number $z$ and on any overlapping neighbourhood with parameter $w$, $w(z)$ is a holomorphic function of one variable.




What does this second paragraph actually mean? Let $M$ be a Riemann surface. Surely if I consider any point $pin M$ I can find a chart ${U_alpha,varphi_alpha)$ where $pin U_alpha$, and then can consider $p$ in local coordinates via $varphi_alpha(p)$. I could then call $varphi_alpha(p)=z$ and then consider $varphi_beta(p)=w$ and abuse notation and write $w(z)$ for the transition. Is that what they intend (in which case I personally think this is stupidly cumbersome, so I doubt it)?



Later they write:




Let us use the coordinate $z$ corresponding to the coordinate chart $varphi_gamma$




and they start taking polynomials in $z$. What exactly is going on?




riemann-surfaces complex-manifolds




share|cite|improve this question













I am reading Hitchens text on integral systems




A Riemann surface is a one-dimensional complex manifold with a maximal set of coordinate charts $U_alpha,varphi_alpha_alphain I$ where $varphi_alpha:U_alphato Bbb C$ such that $varphi_betacirc varphi_alpha:varphi_alpha(U_alphacap U_beta)to varphi_beta(U_alphacap U_beta)$ is an invertible holomorphic map for every pair $alpha,betain I$




My text then says:




Thus a neighbourhood of any point can be parametrized by a complex number $z$ and on any overlapping neighbourhood with parameter $w$, $w(z)$ is a holomorphic function of one variable.




What does this second paragraph actually mean? Let $M$ be a Riemann surface. Surely if I consider any point $pin M$ I can find a chart ${U_alpha,varphi_alpha)$ where $pin U_alpha$, and then can consider $p$ in local coordinates via $varphi_alpha(p)$. I could then call $varphi_alpha(p)=z$ and then consider $varphi_beta(p)=w$ and abuse notation and write $w(z)$ for the transition. Is that what they intend (in which case I personally think this is stupidly cumbersome, so I doubt it)?



Later they write:




Let us use the coordinate $z$ corresponding to the coordinate chart $varphi_gamma$




and they start taking polynomials in $z$. What exactly is going on?





riemann-surfaces complex-manifolds





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 1 at 10:06
























asked Aug 1 at 9:59









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  • I think what they mean is $w(z):phi_alpha(U_alphacap U_beta)to phi_beta(U_betacap U_alpha)$ given by $zmapsto w$
    – daruma
    Aug 1 at 10:13
















  • I think what they mean is $w(z):phi_alpha(U_alphacap U_beta)to phi_beta(U_betacap U_alpha)$ given by $zmapsto w$
    – daruma
    Aug 1 at 10:13















I think what they mean is $w(z):phi_alpha(U_alphacap U_beta)to phi_beta(U_betacap U_alpha)$ given by $zmapsto w$
– daruma
Aug 1 at 10:13




I think what they mean is $w(z):phi_alpha(U_alphacap U_beta)to phi_beta(U_betacap U_alpha)$ given by $zmapsto w$
– daruma
Aug 1 at 10:13















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