In Weinberg's Classical Solutions in Quantum Field Theory, he states Lie groups, such as $SU(2)$ or $SO(3)$, may be viewed as a manifold. My questions are,
- If we can interpret, e.g. $SU(2)$ as a manifold, how does one determine the metric?
- From a differential geometry perspective, the Riemann tensor encodes the curvature of a particular manifold. If our manifold is a Lie group, is there a group theory interpretation of the curvature of that manifold, i.e. a different way of viewing it?
- If a Lie group as a manifold is Ricci flat, what conclusions can we draw from that, regarding the group in question, if any?
Edit: For future math S.E. users reading this question in the future, see http://www.rmki.kfki.hu/~tsbiro/gratis/LieGroups/LieGroups.html for an explicit example of a particular metric for $SU(2)$.