We know every tangent bundles is orientable and have even dimension. I think these properties are not sufficient to a vector bunble be (isomorphic) to a tangent one. I want to find some geometric/topological property, which is only satisfied by tangent bundle to formulate something like:
Let $E$ a $2n$-dimensional orientable manifold that can be realised as the total space of some vector bundle over a manifold $M$. If (some condition), then $E$ can also be realised as a tangent bundle.
I'm trying to get some feeling about vector bundle based on the tangent one.
Edit: the comments show me it's not a precise question. Ok, let's change a little bit: Fixed a manifold $M$, I want to characterize $TM$ in the follow sense: in what conditions a VB $E$ over $M$ is (isomorphic to) the tangent one and I can formulate:
Let $M$ be a manifold and $E$ be orientable $2n-$dimensional VB of $M$. If (some conditions over $E$), then $E$ is VB-isomorphic to $TM$, $E \simeq TM$.
I want, at least, a thinner necessary condition to $E \simeq TM$.