In Hatcher above, he claims that "This determines orthonormal bases for the $(k-m)$ planes orthogonal to all nearby $m$-frames", I feel confused about this claim (even after checking the preceding example): why is it orthonormal to all nearby $m$-frames, what are the nearby $m$-frames?
In particular, I am not clear what the nearby $m$-frames look like, so can anyone tell me, given an $m$-frame of a Stiefel manifold, what look like its neighbourhoods?