What is the local trivialization map for quotient vector bundle?

Question

Let $\eta = (E, \pi, M)$ be a vector bundle of rank $k$ with projection map $\pi \colon E \to M$, and $\xi=(E', \pi', M)$ be a subbundle of $\eta$ that has rank $k'$, with a projection map $\pi'= \pi|E'\colon E' \to M$. (Definition of a vector bundle is https://en.wikipedia.org/wiki/Vector_bundle)

Define the quotient bundle $\eta/\xi = \coprod_{p \in M} (E_p /E'_p)$. I know $\pi(\eta/\xi)\colon \coprod (E_p /E'_p) \to M$ is projection map of $\eta/\xi$, but I don't know what is the local trivialization map for $\eta/\xi$.

I know $\phi':(\pi|E')^{-1}(U) \to U × R^{k'}$ which $\phi'(p,e)=(p,\pi_{R^{k'}}(\phi'(p,e)))= (p, \nu_1,..., \nu_{k'})$ is local trivialization of $E'⊂ E$.

Answer 1

I think this is the local trivialization of quotient bundle. If (u,φ) be a chart for η and $\phi:(\pi)^{-1}(U) \to U × R^{k}$ which $\phi(w)=(p,\pi_{R^{k}}(\phi(w)))= (p, \nu_1,..., \nu_{k})$ be local trivialization of E, and $\phi':(\pi|E')^{-1}(U) \to U × R^{k'}$ which $\phi'(w)=(p,\pi_{R^{k'}}(\phi'(w)))= (p, \nu_1,..., \nu_{k'})$ be local trivialization of $E'⊂ E$ we can define $J:E \to (E_p /E'_p)$. So local trivialization of $\eta/\xi$ is $\psi:(\pi)^{-1}(U) \to U × R^{k''}$ which $\psi(w)=(p,j(\pi_{R^{k}}(\phi(w)))= (p, \nu_1,..., \nu_{k''})$ and $k'' =k-k'$ is the rank of $\eta/\xi$

Answer 2

You have to construct a specific type of local trivialization for $E$ in order to get a local trivialiyzation of the quotient bundle. This is easier to understand in the language of local frames (i.e. the sections given by preimages of the basis elements under a local trivialization). In these terms, you have to start with a local frame for the subbundle $E'$ and then extend it to a local frame of the bundle $E$. (Given $x\in M$, choose a basis for $E'_x$ and extend it to a basis of $E_x$. Then extend the first vectors to local smooth sections of $E'$ and the remaining ones to local smooth sections of $E$. On a sufficiently small neighborhood of $x$, this defines a frame as required.) Converting this to a local trivialzation of $E'$ as a subbundle of $E$, i.e. a trivialization $\phi:\pi^{-1}(U)\to U\times \mathbb R^k$, which restricts to a trivializtaion $(\pi|_{E'})^{-1}(U)\to U\times\mathbb R^{k'}$. Passing to quotients in each fiber, one obtains a local trivialization of $E/E'$ as required.