Lie groups as manifolds

Solution 1:

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.

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