generation of sub bundle

Question

Let $M$ differentiable manifold with $\dim M=n$. If $(TM,\pi,M)$ be the fiber bundle tangent. Consider the family $E=\lbrace E_x\rbrace _{x\in M}$ such that $E_x \subset T_xM$ and $\dim E_x=k$ for all $x\in M$ then $\pi\vert E: E \rightarrow M$ induces subbundle $(E,\pi \vert E,M)$?

I appreciate any suggestions.

Answer 1

I see no reason that this should be a subbundle. Consider $M = \mathbb{R}^2$. Then the tangent bundle is trivial, spanned by $e_1 = \frac{\partial}{\partial x}$ and $e_2 = \frac{\partial}{\partial y}$ Consider now the family $E = \{E_{(a,b)}\}$ where $$ E_{(a,b)} = \begin{cases} span(e_1) & a \text{ is rational} \\ span(e_2) & \text{otherwise} \end{cases} $$ I see no obvious way that you can produce a bundle structure on this object.