# Curvature of a group manifold

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1. Feb 11, 2016

### JorisL

So I'm working with some group manifolds.
The part that's getting to me is the Ricci scalar I'm using to describe the curvature.

I have identified the groups that I'm using but that's not really relevant at the moment.
We're using a left-invariant metric $\mathcal{M}_{ab}$.

Now I've got the expression
$$R = -\frac{1}{4}f_{abc}f^{abc} -\frac{1}{2}f_{cab}f^{abc} = -\frac{1}{4}\mathcal{M}_{ad}\mathcal{M}^{be}\mathcal{M}^{cf} f^a_{\,\,\,\, bc}f^d_{\,\,\,\, ef} - \frac{1}{2}\mathcal{M}^{cd}f^a_{\,\,\,\, bc} f^b_{\,\,\,\, ad}$$

The second expression (with explicit metrics) is the one I'll be using later on because I can normalise structure constants of that kind. That is however not important, going from the first part to the second is comparatively easy.

I'm wondering if there's an easy way to find the Ricci scalar (or tensor for that matter) that doesn't require looking at the sectional curvature. This last part is important because it would take me too far from the topic.
Meanwhile all examples I found do use this notion.

I want to know this because I like to be as complete as possible in my notes.

Thanks,

Joris

2. Feb 16, 2016