In W.M. Boothby, An introduction to differentiable manifolds and Riemannian geometry , Page 64. We use coordinate correspondences $\phi_j : U_j \to R^{k(n-k)}$ in order to define Grassman manifolds where $U_j'$s are open sets in $F(k, n)$. I can't see :
- Why the map $\phi_j$ are properly defined?
- Why each $\phi_j$ maps $U_j$ onto $R^{k(n-k)}$ homeomorphicaly?
It is mentioned in the book that verification of these questions for Grassman manifold $G(2,4)$, the 2-planes through the origin of $R^4$, is sufficient to show how proceed in general. Could someone guide me how could I prove the above questions for the case $G(2,4)$? and how can we switch to the general case? Is there any reference include a complete proof of the above questions?
Thanks in advance for any help.