# Thread: Calculating the Curvature of a Complex Manifold

1. ## 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 $\displaystyle \mathbb{C}P^1$ , which has the Fubini-Study metric on it:
$\displaystyle g=\frac{dzd\bar{z}}{(1+z\bar{z})^2}$ .

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

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:
$\displaystyle \Omega^{\mu}_{~\nu}=\frac{1}{2}R^{\mu}_{~\nu\rho\e ta}e^{\rho}\wedge e^{\eta}$.

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