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Calculating the Curvature of a Complex Manifold

  1. Jan 15, 2009 #1
    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 [tex]\mathbb{C}P^1[/tex], which has the Fubini-Study metric on it:
    [tex]g=\frac{dzd\bar{z}}{(1+z\bar{z})^2}[/tex].

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

    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:
    [tex]\Omega^{\mu}_{~\nu}=\frac{1}{2}R^{\mu}_{~\nu\rho\eta}e^{\rho}\wedge e^{\eta}[/tex]

    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 [tex]\Omega^1_{~\bar{1}}[/tex]. Then, will my Riemann tensor have components that look like [tex]R^{1}_{~\bar{1}1\bar{1}}[/tex]? If so, when I calculate the scalar curvature, will my sum be over "complex indices" like
    [tex]R=R_{1\bar{1}}+R_{\bar{1}1}+R_{11}+R_{\bar{1}\bar{1}}[/tex]?
    Things start to look rather strange...can anyone shed some light on this stuff?
     
  2. jcsd
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