Let $n$ be a positive integer. Let $O(n)$ be the orthogonal group. Let $I(k,n)$, for $0\leq k < n$, be the space of linear isometries $\mathbb{R}^k\rightarrow\mathbb{R}^n$. Is the map $O(n)\rightarrow I(k,n)$ given by restriciton a fiber bundle? (Both spaces are topologized as a subspace of the relevant set of linear maps.)
Moreover, if the above is true, is it possible to have that the trivializing open sets satisfy the following property:
If $U\subseteq I(k,n)$ is trivializing, there exists $V\subseteq I(k,n+1)$ trivializing such that $U\subseteq V\cap I(k,n)$. Under the obvious inclusion $I(k,n)\subseteq I(k,n+1)$ given by $\mathbb{R}^n\rightarrow\mathbb{R}^{n+1}$?
Hint: Use the Ehresmann lemma
https://en.wikipedia.org/wiki/Ehresmann%27s_lemma