Are local smooth sections sufficient to conclude we have a vector bundle?

Question

Suppose we have a submersion $\pi: E \to M$ such that for each $x \in M$ the fibre $E_x$ is a vector space. Moreover, suppose that for every $x \in M$ we have a collection of smooth local section $\{e_i : U \to M\}$ on a neighbourhood of $x$ such that for all $y \in U$, $\{e_i(y)\}$ is a basis for $E_y$. Can we conclude $E$ is a vector bundle?

I have tried proving this statement in two ways: first by looking at the associated transition functions, but I did not see why these would have to be smooth. Furthermore, I tried looking at the induced local trivialisations of the sections. In both cases the problem seems to be the following, suppose I have a smooth section $s$ over $U$ and I expand it with respect to my local frame, do the coefficients vary smoothly with respect to the base.

I also tried thinking of a counter example, but this was not fruitful either.

Answer 1

So, considering a surjective submersion $\pi\colon E\to M$, you assume the following.

1. vector space structure on the fibers: for each $x\in M$, the corresponding fiber $E_x$ is endowed with a structure of $n$-dimensional vector space,
2. local frames: there is an open cover $\{U_\alpha\}_{\alpha\in A}$ of $M$ and, for each $\alpha\in A$, there are local smooth sections $e_1^\alpha,\ldots,e_n^\alpha\colon U_\alpha\to E$ such that $e_1^\alpha(x),\ldots e_n^\alpha(x)$ is a basis of $E_x$, for all $x\in U_\alpha$.

As you already pointed out in the comments, with your counterexample, these two conditions are not sufficient to make $\pi\colon E\to M$ into a vector bundle. Indeed, it is also necessary to require that:

3. the trivializing map determined by the local frame $e_1^\alpha,\ldots,e_n^\alpha$, i.e. the bijection $$\Phi_\alpha\colon\pi^{-1}(U_\alpha)\longrightarrow U_\alpha\times\mathbb{R}^n,\ \sum_{i=1}^nc_ie_i^\alpha(x)\longmapsto (x,(c_1,\ldots,c_n)),$$ is a diffeomorphism, for each $\alpha\in A$.

Actually, the latter condition (which fails in your counterexample) is not only necessary but, together with conditions 1 and 2 above, it is also sufficient. Indeed, by construction, for any $\alpha,\beta\in A$, the transition map $\Phi_\alpha\circ\Phi_\beta^{-1}\colon(U_\alpha\cap U_\beta)\times\mathbb{R}^n\longrightarrow(U_\alpha\cap U_\beta)\times\mathbb{R}^n$ is a diffeomorphism and it acts by linear isomorphisms on the fibers, i.e. $$(\Phi_\alpha\circ\Phi_\beta^{-1})(x,u)=(x,\Phi_{\alpha\beta}(x)u),$$ with the smooth map $U_\alpha\cap U_\beta\longrightarrow\text{GL}(n,\mathbb{R}),\ x\longmapsto\Phi_{\alpha\beta}(x)$ such that $e_j^\beta(x)=\sum_{i=1}^n(\Phi_{\alpha\beta}(x))_{ij}e^\alpha_i(x)$ for all $x\in U_\alpha\cap U_\beta$ and $i=1,\ldots,n$.