During my research I came across Riemannian geometry. Specificlaly I am dealing with $SO(3)$.
I searched on the internet and there are multiple references telling me that a simple, compact Lie group has a unique bi-invariant Riemannian metric up to scale, which is applicable for $SO(3)$.
But now I am confused about the relation between 'metric' and 'distance'. On $SO(3)$ we can define distance as angular distance: $d_1(R_1,R_2) = \arccos (\frac{Tr(R_1^TR_2)-1}{2})$, which is definitely bi-invariant.
But also we can defined distance as chordal distance using Frobenius norm: $d_2(R_1,R_2) = ||R_1 - R_2||_F$. This is also bi-invariant...
So now I come up with three questions: (1) Is any distance on a Lie group possibly induced by some Riemannian metric? (2) Why there are at least two bi-invariant distances?(Possibly because the second can't be induced by a Riemannian metric? Not sure) (3) How can I possibly know if a distance is induced by a Riemannian metric if the answer to (1) is negative.
As a side note, can you recommend some books for me, as an beginner to read, about the Lie group and the Riemannian manifolds to have a glimpse of basic ideas and properties such as geodesic, affine connection, etc. ?
Any help will be appreciated, thanks in advance!