Natural Metrics on (Special) Unitary groups.

[Kreizhn]

So I know that every smooth manifold can be endowed with a Riemannian structure. In particular though, I'm wondering if there is a natural structure for the unitary and special unitary groups.

I often see people using the "trace/Hilbert-Schmidt" inner product on these spaces, where
$$\langle X, Y \rangle = \text{Tr}(X^\dagger Y)$$
but these are often applied directly to elements of the manifold rather than to their tangent spaces. Is this the same inner-product one the Lie-algebra/tangent spaces? Or is there a more natural one?

Edit: I guess another way to phrase the question might be "Is the trace-inner product the natural inner-product to use on (traceless) skew-Hermitian matrices?

 

[zhentil]

I imagine if you didn't take the induced metric from the natural embedding in C^(n^2), you'd have to justify it somehow.

So the answer to your question (in my opinion) is yes.

 

[Kreizhn]

Alternatively, I could probably use the Killing form here no? Since U(n) and SU(n) are compact, the Killing form is negative definite so the negative killing form could define a metric. Further, I think I remember reading something about all simple Lie groups having unique bi-invariant metrics. But U(n) is only semi-simple right?