All Questions

Although vector fields (which are sections of the tangent bundle) form Lie algebras, the bundle itself, as far as I know, almost never carries Lie algebra structure; that is, in general, I believe, ...

Is the following fact well known? Let $X$ be a manifold and $V$ be a vector space. Let $E_1$ be a sub-bundle of the constant bundle $X \times V$. Let $E_2$ be its orthogonal compliment in $X \...