Calculating the Curvature of a Complex Manifold

**cduston** · January 20th 2009, 12:39 PM

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?

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