Is the topology on compact connected Lie groups metrizable?

Question

The question is in the title, really.

Suppose $G$ is a compact connected Lie group. Is there a metric on $G$ which induces the underlying topology? (so in particular $G$ is compact and connected wrt this metric).

I am not really from this area (my differential geometry is slack, to say the least). I am purely interested in the existence of such a metric.

Thanks.

Answer 1

From a general topology viewpoint: a Lie group $G$ is locally Euclidean, so locally metrisable. Also $G$ is compact and hence paracompact. It's $T_1$ (from locally Euclidean) and as a topological group this means $G$ is Tychonoff, Hausdorff etc.

So several metrisation theorems do apply: Bing-Nagata-Smirnov, Smirnov, Urysohn even. I don't think connectedness is necessary : $G$ can have only finitely many components (being compact and locally connected) and a finite sum of metrisable spaces still is metrisable. So $G$ is certainly metrisable.

Answer 2

The short answer is yes, I believe. The metric that you would be looking for would be the one induced by the Killing form. There is a really good discussion on the construction of the Killing form and its properties here:

http://scipp.ucsc.edu/~haber/ph251/KillingForm.pdf