Let $M$ be a manifold of dimension $n$. Let $p:E\rightarrow M$ be a locally trivial smooth fibration. Does this give us a way to construct a subbundle of the tangent bundle $TE$ of the manifold $E$, moreover should the dimension of the subbundle be $n$?
It seems that locally trivial smooth fibrations should be able to give us a non vanishing section. Non vanishing sections allow us to create subbundles of the tangent bundle using frames. But I'm not sure how to get such sections since I don't know anything about the fibres of $E$ as a fibration over $M$.