Show that the submersions $F: V_k(\mathbb R^n) \to G_k (\mathbb R^n)$ and $G: O(n) \to G_k (\mathbb R^n)$ are fibrations. Deduce that $\text{Mat}^k_{n \times k}(\mathbb R) \to G_k(\mathbb R^n)$ is a fibration.
Here, $V_k(\mathbb R^n)$ is the Stiefel manifold of $k$ orthonormal vectors in $\mathbb R^n$ and $G_k(\mathbb R^n)$ is the Grassmannian of $k$-dimensional vector spaces in $\mathbb R^n$. Here, $F$ is the map that sends a $n \times k$ matrix in $V_k(\mathbb R^n)$ to the subspace spanned by its columns.
I am really lost on this problem. We are to show that if $p := [v_1 \cdots v_k]/ \sim$ with $v_i \in \mathbb R^n$ with $v_i$ independent, with $\sim$ denoting the equivalence class $[v_1 \cdots v_k] \sim [w_1 \cdots w_k]$ iff they represent the same $k$-dimensional vector space, then there is an open neighborhood $U$ of $p$ such that $\varphi: U \times F^{-1}(p) \cong F^{-1}(U)$ and such that for all $(x,y) \in U \times U \times F^{-1}(p)$, then $\varphi(x,y) \in F^{-1}(x)$, and similarly for $G$. I also am lost on how to prove the second part.
My lecture notes are confusing to me, especially it does not have crucial definitions and theorems that seems to be assumed to be known by students. I am wondering where I can learn these materials, i.e. in what chapter of a differential topology/geometry book can I find these information? What book and chapter do you recommend?