In Wikipedia, they define the Grassmannian manifold by $$Gr(r,n)=O(n)/(O(r)\times O(n-r))$$ where $O(m)$ is the orthogonal group of $m\times m$ matrices. They say that it gives a metric on the Grassmannian by $$d(W,W^\prime)=\|P_W-P_{W^\prime}\|$$ where $P_V$ is the projection operator on a space $V$.
My questions are:
- In what sense does this metric $d$ rise from the formulation of the Grassmannian with the orthogonal group?
- Is $d$ equal to the metric induced by the Riemannian metric on $O(n)$? (and is this Riemannian metric given by $g_p(A,B)=tr(AB^{t})$ like $GL(n)$?)
- Why is this metric $d$ natural from the construction of the Grassmannian by the orthogonal group as was described above, and not from the alternative definition $GL(n)/H$ for $H$ the stabilizer of a $r$-dimensional subspace? Are these constructions the same in the sense that the Riemannian metrics on both $GL(n)$ and $O(n)$ would induce the same metric $d$ described above?