I have several basic questions about matrix Lie groups. Suppose $G$ is a matrix Lie group, say $\mathrm{GL}(n)$ or $\mathrm{SO}(n)$.
- I understand that elements of $G$ can be looked at points in $\mathbb{R}^{n^2}$. Does this make the study of these matrix Lie groups easier in any sense? Please elaborate.
- What are "natural" choices for charts on such matrix Lie groups (particularly, $\mathrm{GL}(n)$ and $\mathrm{SO}(n)$). Can you provide an example atlases for these two specific matrix Lie groups?
- Is it possible to define valid charts based on the tangent space and exponential/logarithm maps?
Thanks!
(1) It is just the observation that an $n \times n$ matrix has $n^2$ entries, and so you can think of it as a vector in $n^2$-dimensional space. Going along with question #2, this can help find coordinates.
(2) Going along with point (1) above, for $\mathrm{GL}_n$, you can take the $n^2$ matrix entries as a set of global coordinates, since $\mathrm{GL}_n$ is an open subset of $\Bbb{R}^{n^2}$.
(3) Near the identity element of $G$, the exponential map is a diffeomorphism, so yes, the tangent space can be used to get local coordinates.