I know that we can define the killing form on a lie algebra. However, when going to the group manifold, does this give rise to a metric on the manifold? I thought that would be the case, but I cant find any useful literature how this works. Especially, I am confused since the Lie algebra seems to always be defined around the unity, but I guess on a group manifold one can use that we can always map any point to unity and thus also map the tangent space?


So Let's say I have a Lie group manifold parametrized by coordinates x0,x1,...,xn. How can I obtain the metric induced from the killing form?



cartan killing metric
Any inner product on $\mathfrak{g}$ will give rise to a metric on $G$. More specifically, you can use a left trivialization of $TG$ to get a vector bundle isomorphism $TG\cong G\times\mathfrak{g}$. The latter bundle has a metric by virtue of the inner product on $\mathfrak{g}$.


The same construction will work if one instead uses the right trivialization.


