Given a smooth manifold $M$, we define the differentiable structure on $TM$ in the usual way.
I would like to know examples of two smooth manifolds which are non-diffeomorphic, but their tangent bundles are.
Which is the smaller dimension in which one can find such examples?
What if I ask the same question for $k$ pairwise non-diffeomorphic manifolds?
Can we have $k=\infty$?