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

Question

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.

Answer 1

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).

Those sections are smooth both when you pass to the Moebius band and to the cylinder. Therefore, the construction cannot yield an unique vector bundle structure. This is consonant to the fact that Mariano alludes to in the comments: the inexistence of a compatibility assumption of some sort in the sections. Note that in the Moebius band, the sections will twist "wrongly".

Answer 2

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.