(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 , which has the Fubini-Study metric on it:
Using Cartan's differential forms, I can get the curvature two-form to be
Where my flat coordinate is
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:
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 . Then, will my Riemann tensor have components that look like ? If so, when I calculate the scalar curvature, will my sum be over "complex indices" like
Things start to look rather strange...can anyone shed some light on this stuff?