I am interested in a rather basic question, as I am studying Lie groups.
Let $G$ be a matrix Lie group, say $O(n)$. Consider the Lie algebra $\mathfrak{g}=\mathfrak{o}(n)$.
On the one hand, $\mathfrak{g}$ has some smooth structure (which I don't quite understand yet), and has some topology on it. On the other hand, $\mathfrak{g}$ is just a set of matrices and so it has a natural notion of a metric on it (say as the inherited metric as a subspace of $\mathbb{R}^{n^2}$).
My questions are:
- is the topology on $\mathfrak{g}$ compatible with some metric?
- In particular, can I interpret the statement that $\exp:\mathfrak{g}\to G$ is a local diffeomorphism, as the fact that the restriction to some ball in $\mathfrak{g}$ around $0$ (rather than just a neighborhood) is a diffeomoprhism on its image?
The topology on $\mathfrak g$ is the one induced by the distance that you will get from any norm that you define on $\mathfrak g$. Since $\dim\mathfrak g<\infty$, all those norms are equivalent (that is, they all induce the same topology). And, yes, what you wrote about the exponential map is correct.