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Math Help - Calculating the Curvature of a Complex Manifold

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    Calculating the Curvature of a Complex Manifold

    Hey everyone,
    (First off, this is not a HW question, but I can move it there if you are unhappy about my choice) I'm trying to figure out how to calculate the curvature scalar of a complex manifold. As a test case, I am using \mathbb{C}P^1 , which has the Fubini-Study metric on it:
    g=\frac{dzd\bar{z}}{(1+z\bar{z})^2} .

    Using Cartan's differential forms, I can get the curvature two-form to be
    \Omega^1_{~1}=\theta\wedge\bar{\theta} ,
    Where my flat coordinate is
    \theta=\frac{dz}{1+z\bar{z}}

    However, I am stuck at the next step. I should be able to use the connection between the curvature two-form and the Riemann curvature in the following way:
    \Omega^{\mu}_{~\nu}=\frac{1}{2}R^{\mu}_{~\nu\rho\e  ta}e^{\rho}\wedge e^{\eta}.

    Then I can get the scalar curvature from that. But what will the components of the Riemann tensor look like since my coordinate is complex? In other words, instead of a (2,0)-form my curvature is actually a (1,1)-form, perhaps best notated by \Omega^1_{~\bar{1}} . Then, will my Riemann tensor have components that look like R^{1}_{~\bar{1}1\bar{1}}? If so, when I calculate the scalar curvature, will my sum be over "complex indices" like
    R=R_{1\bar{1}}+R_{\bar{1}1}+R_{11}+R_{\bar{1}\bar{ 1}} ?
    Things start to look rather strange...can anyone shed some light on this stuff?
    Last edited by cduston; January 21st 2009 at 07:37 AM. Reason: Fixed tex errors
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