Is there a reasonably natural way to create an atlas (with coordinate charts) for the space of $k$-tuples of orthonormal vectors in $\mathbb{R}^n$? (Obviously $k \le n$.) The dimension of the desired space is $$\sum_{i=1}^k n - i = k(n - \tfrac{1}{2}k - \tfrac{1}{2})$$ if I am not mistaken.
Naïve approach Choose a random matrix and do Gram-Schmidt orthonormalization (or possibly a more numerically stable alternative, like Householder transforms). Then figure out how to fix $n^2 - k(n - \tfrac{1}{2}k - \tfrac{1}{2})$ parameters to get a dense open subset, and find different ways to do this so that the open sets cover the entire space.
I would accept an answer based on the naïve approach, but I hope something better ("more natural") comes along.
Notes
- I said all $k$-tuples, but really I'm okay with just the connected component containing $(\mathbf{e}_1, \dotsc, \mathbf{e}_k)$.
- What I'm actually hoping to do is plug this into a computer. So computational aspects matter too, although as a former mathematician my first concern is for naturality.