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,

  1. If we can interpret, e.g. $SU(2)$ as a manifold, how does one determine the metric?
  2. 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?
  3. 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 $SU(2)$.

  • $\begingroup$ I thought part of the definition of Lie group was that it had to be a manifold? $\endgroup$ – fretty Oct 29 '14 at 17:27

Manifolds don't have metrics on them by default. A metric is extra structure making a manifold a Riemannian manifold. It's interesting to study metrics on Lie groups but they don't need to be there.

In particular it's interesting to study bi-invariant metrics (metrics invariant under both left and right multiplication). For compact semisimple Lie groups there is a particularly nice choice of such a metric coming from the Killing form and one can express various things about this metric in terms of the Lie algebra; see, for example, this MO question. In particular the curvature tensor can be written in terms of the Lie bracket.

  • $\begingroup$ Can you recommend a text which provides an overview of metrics on Lie groups? $\endgroup$ – JamalS Apr 25 '14 at 17:16
  • $\begingroup$ I don't know of any (but that doesn't mean they don't exist; I don't read much in this area). Poke around in some Riemannian geometry textbooks? $\endgroup$ – Qiaochu Yuan Apr 25 '14 at 18:05
  • $\begingroup$ I will, thanks for your answer. I've found some useful lecture notes online thus far. $\endgroup$ – JamalS Apr 25 '14 at 18:25
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    $\begingroup$ do Carmo's "Riemannian geometry" has discussion of biinvariant metrics, connection and curvature calculations, which is sufficient at this level. If your Lie group is also simple, then such metric is unique up to scale. $\endgroup$ – Moishe Kohan Apr 25 '14 at 18:46
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    $\begingroup$ @user: as I said, on a compact semisimple Lie group a natural choice is given by the bi-invariant metric corresponding to the Killing form. On a compact simple Lie group every bi-invariant metric is a multiple of this one. $\endgroup$ – Qiaochu Yuan Apr 25 '14 at 18:46

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