Curvature of a group manifold

[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

 

[jvicens]

Dear Joris,

Thank you for sharing your work with us. It seems like you are dealing with some interesting and complex mathematics. The Ricci scalar is a very useful tool for describing the curvature of manifolds, and it's great that you are incorporating it into your work.

To answer your question, there are indeed other ways to find the Ricci scalar without looking at the sectional curvature. One approach is to use the Einstein field equations, which relate the curvature of a manifold to the energy and matter content of the space. Another approach is to use the Ricci flow, which is a mathematical tool for studying the evolution of the curvature of a manifold.

I would also recommend looking into the work of other mathematicians and physicists who have studied group manifolds and their curvature, such as Élie Cartan and Hermann Weyl. They may have developed other methods for finding the Ricci scalar that could be useful to your work.

I hope this helps and good luck with your research!