Continuity of a map into $G_n(\mathbb R^N)$

Question

Let $\xi:E\to X$ be a $n$-dimensional vector bundle embedded in the trivial bundle $X\times \mathbb R^N\to X$, for some $N\ge n$. Denote also by $\tau:O\to G_n$ the tautological bundle, where $G_n:=G_n(\mathbb R^N)$ is the Grassmanian manifold and $O:=\{(V, v)\in G_n\times\mathbb R^N\mathrm{\ s.t.\ }v\in V\}$.

There is a map (of sets) $f:X\to G_n$ defined by $x\mapsto E_x:=\xi^{-1}(x)$, since the fibers of $\xi$ are $n$-dimensional subspaces of $\mathbb R^N$.

Moreover, given two continuous maps $g_0$, $g_1:X\to G_n$, assume that:

• $g_0(x)$ (resp. $g_1(x)$) is a subspace of $\mathbb R^N$ containing only vectors whose coordinates of even (resp. odd) index are equal to zero, for all $x\in X$;
• there is a vector bundle isomorphism $\phi:g_0^*(\tau)\to g_1^*(\tau)$ between the pullbacks of $\tau$ along $g_0$, $g_1$.

Then on the fibers $\phi_x:g_0^*(\tau)_x\to g_1^*(\tau)_x$ translates (via isomorphisms) to a linear map $g_0(x)\to g_1(x)$. Define a map (of sets) $h:X\times [0,1]\to G_n$ as follows: $$(x,t)\mapsto \{v\in\mathbb R^N\mathrm{\ s.t.\ }(v=(1-t)u+t\phi_x(u)\mathrm{\ for\ }u\in g_0(x))\}.$$ $h$ is well-defined since, fixed $x$, $t$, the linear map $g_0(x)\to \mathbb R^n$ defined by $(1-t)(-)+t\phi_x(-)$ is injective (because of the hypothesis about even / odd coordinates), so its image is a $n$-dimensional subspace of $\mathbb R^N$.

My question is: what to do to prove the continuity of $f$ and $h$? Since the topology on $G_n$ is a quotient topology, maybe I would know how to check the continuity of a map from $G_n$, but I don't have any intuitions for a map into $G_n$. Any clarify is welcome, thanks.

Answer 1

Let $\newcommand{\Fr}{\operatorname{Fr}}\newcommand{\R}{\mathbb{R}}\Fr_n(\R^N) = \{(v_1, \ldots, v_n) \in (\R^N)^n \mid v_1, \ldots, v_n \text{ linearly independent}\}$ be the space of $n$-frames in $\R^N$ and let $\newcommand{\Gr}{\operatorname{Gr}}q\colon \Fr_n(\R^N) \twoheadrightarrow \Gr_n(\R^N)$, $(v_1, \ldots, v_n) \mapsto \operatorname{span}(v_1, \ldots, v_n)$ denote the quotient map. For continuity of $f$, let $U \subseteq X$ be an open neighborhood over which $\xi$ is trivializable and $\varphi\colon \xi^{-1}(U) \to U \times \R^n$ be a choice of trivialization. We obtain a continuous map $U \times \Fr_n(\R^n) \to U \times \Fr_n(\R^N)$ via $(u, (v_1, \ldots, v_n)) \mapsto (u, (\varphi_u^{-1}(v_1), \ldots, \varphi_u^{-1}(v_n))$ where $\varphi_u$ is the restriction of $\varphi$ to the fibre over $u \in U$. Postcomposing with $q \circ \mathrm{pr}_2$ and precomposing with $U \to U \times \Fr_n(\R^n)$, $u \mapsto (u, (e_1, \ldots, e_n))$ where $e_1, \ldots, e_n \in \R^n$ is the standard basis then recovers $f|_U$, and since every point $x \in X$ has a trivializing neighborhood like this, $f$ is continuous around every point and therefore continuous as a whole.

Continuity of $h$ can be proved similarly: One can cover $X \times [0, 1]$ by open sets $U \times [0, 1]$ where $U \subseteq X$ is as above and then argue similarly.