# What is the local trivialization map for quotient vector bundle?

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 $$E'⊂ E$$.

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.
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 $$\eta/\xi$$