Do sections determine a smooth manifold structure on $E$?

This is a follow up question from this one.

If I have $E = \bigsqcup_{p \in M}E_p$, where each $E_p$ is $k-$dimensional vector space, and for each $p \in M$ there is $U \ni p$ an open neighborhood and $s_1,\ldots,s_k\colon U \to E$ such that $\{s_i(p)\}$ span $E_p$ for each $p$, then does it follow that $E$ has a smooth manifold structure such that $(E,\pi,M)$ is a smooth vector bundle over $M$, and with these sections and the projection $\pi :E \to M$ being all smooth?

I attempted to use lemma $5.5$ in Lee's Introduction to Smooth Manifolds as pointed in the previous answer. I only have to define bijections $\varphi\colon \pi^{-1}[U] \to U \times \Bbb R^k$ that are linear isomorphisms in each fiber, and check that they glue nicely. Certainly $\varphi\left(\lambda^i s_i(p)\right) = (p,(\lambda^i))$ is the way to go. Assume that we have done this construction for two open sets $U_\alpha$ and $U_\beta$, with sections $s_i^\alpha$ and $s_i^\beta$, and trivialization candidates $\varphi_\alpha$ and $\varphi_\beta$. Write $s_j^\beta(p) = h^i_{\hspace{.5ex}j}(p)s_i^\alpha(p)$ for some convenient coefficients. We have $$\varphi_\alpha \circ \varphi_\beta^{-1}(p, (\lambda^i)) = \varphi_\alpha(\lambda^js_j^\beta(p))= \varphi_{\alpha}\left(\lambda^j h^i_{\hspace{.5ex}j}(p)s_i^\alpha(p)\right) = (p, \lambda^j h^i_{\hspace{.5ex}j}(p)).$$Since $h^i_{\hspace{.5ex}j}(p)$ works as a change of basis matrix, we have that $$g_{\alpha\beta}\colon U_\alpha\cap U_\beta \to {\rm GL}(k,\Bbb R)$$given by $g_{\alpha\beta}(p)((\lambda^i)) = h^i_{\hspace{.5em}j}(p)\lambda^j$ really is non-singular. But I can't check that these $g_{\alpha\beta}$ are smooth on $p$, so I can apply lemma $5.5$. I know that if I can check that each $h^i_{\hspace{.5em}j}$ depends smoothly on $p$ I'm done, because matrix entries are global coordinates in ${\rm GL}(k,\Bbb R)$. Please help me.

• Your question, as stated in your 2nd paragraph, puts no requirement on the smooth manifold structure apart from the one that it turns E into a vector bundle, so the answer is trivially yes: your set E can be turned into a trivial $k$-dimensional bundle over $M$, and that extra condition you mention is completely irrelevant in this. Sep 18, 2016 at 5:49
• How can I fix that then? The whole point of this is to use sections instead of trivializations to induce the structure on $E$. Sep 18, 2016 at 5:50
• This means that you probably want the smooth structure to satisfy some condition or another. For example, that the $s_i$ that you are given around each points be smooth sections of the bundle. Sep 18, 2016 at 5:50
• Ok, I want smooth sections (I should have been more careful writing, sorry) Sep 18, 2016 at 5:52
• In other words: your question should be: if I am given a map $\pi:E\to M$ defined on a set E such that each fiber is a vector space of dimension $k$, and for each point of $M$ I have $k$ sections of $\pi$ which span the fiber at each point of its domain, does $E$ have a smooth structure maiking $\pi$ a smooth vector bundle and such that the sections I had are smooth sections? Sep 18, 2016 at 5:52

Consider $S^1$, and $E_x=\mathbb{R}$ for every $x$. Now, consider the local "sections" $s_1:R\to E$ given by $s(x)=1_x$ for every $x$, and $s_2:L \to E$ given by $s(x)=1_x$ for every $x$, where $R$ and $L$ are the right and left part of the circle (slightly increased as to cover everything).
Summarizing the discussion in the comments: we can't prove smoothness of the $h^i_{\hspace{.5ex}j}$ - we don't have enough information. The equivalence between local trivializations and local frames only holds once $E$ already has a smooth manifold structure. In particular cases such as the quotient bundle, the idea seems to define sections, write the trivializations from them, and check that in the particular case everything glues correctly, and only then apply lemma $5.5$ of Lee's book.