Suppose we have some Lie manifold (ie a Lie group) that is also a Riemannian manifold endowed with a metric tensor How does the Lie group symmetry manifest in the properties of the metric tensor?
I would guess that the tensor locally adheres to the properties of the Lie algebra corresponding to the group in question, However I'm unclear precisely how this would manifest.
first thought is to lead with a decomposition of the gamma matrices. In physics one can decompose a metric $g_{\mu\nu}$ into gamma matrices $\gamma_{\mu}$ such that:
$$Ig_{\mu\nu}=\frac{1}{2}\left(\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}\right)=\frac{1}{2}\left\{ \gamma_{\mu},\gamma_{\nu}\right\} $$
Where I is the identity matrix. For the Minkowski metric $\eta_{\mu\nu}$ the commutator of the gamma matrices
$$\sigma_{\mu\nu}=\frac{1}{2}\left(\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu}\right)=\frac{1}{2}\left[\gamma_{\mu},\gamma_{\nu}\right]$$
turn out to be the infinitesimal generators of the Lorentz transformations (ie elements of the group algebra). These can in turn be exponentiated to form the elements of the Lorentz Group:
$$T_{\nu}=exp(\theta^{\mu}\sigma_{\mu\nu})$$
In this case we can recover the group properties of the manifold from examining the properties of the metric tensor (or rather it's decomposition). What about a SU(2) (represented as the manifold of a 3-sphere) or a general Lie group? Does anyone know if the metric tensor carries these properties in general???
Any answer or direction towards a book that covers this subject would be greatly appreciated. Thank you!