# Differential geometry of Lie groups

In Weinberg's Classical Solutions of Quantum Field Theory, he states whilst introducing homotopy that groups, such as $SU(2)$, may be endowed with the structure of a smooth manifold after which they may be interpreted as Lie groups. My questions are:

• If we formulate a quantum field theory on a manifold which is also a Lie group, does that quantum field theory inherit any special or useful properties?
• Does a choice of metric exist for any Lie group?
• Are there alternative interpretations of the significance of Killing vectors if they preserve a metric on a manifold which is also a Lie group?
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Comments on question (v1). Note that the moment you use the term "Lie group," you must be certain that the object being considered is a smooth manifold. So I'd like to suggest the rephrasing "groups, such as $\mathrm{SU}(2)$, may be endowed with the structure of a smooth manifold after which they may be interpreted as Lie groups." Also, do you have a compelling reason to believe that a "natural" choice of metric exists for an arbitrary Lie group? Perhaps "does there exist a natural choice of metric..." would be a better phrasing? Great question! –  joshphysics Apr 27 at 16:26
@joshphysics: Thank you for the suggested edit, I will incorporate it into the question. I will add the question of the existence of a choice of metric for an arbitrary Lie group, but I would also like to know how to select one if possible, given the appropriate Lie group. –  user45389 Apr 27 at 16:33
This question is extremely similar to the following math.SE question. Coincidence? math.stackexchange.com/q/769080 –  joshphysics Apr 27 at 16:38
@joshphysics: Thanks for the link, that answers part of my question. –  user45389 Apr 27 at 16:54