Put in other words, given an even-dimensional sphere $S^{2k}$: is there a manifold $M$ such that $T^\* M$ is diffeomorphic to $S^{2k}$? 



Can a sphere be a phase space?
Of course, the spheres are compact while cotangent bundles are noncompact (unless in dimension 0). Nevertheless, a bit more interesting is the question whether the even dimensional spheres can be phase spaces in the sense of symplectic manifolds. There the $\mathbb{S}^2$ is an example: the volume form is non-degenerate and a two-form. Closedness is for free in 2 dimensions. The higher dimensional spheres $\mathbb{S}^{2n}$ are never symplectic as on a compact symplectic manifold, the deRham cohomology has to be sufficiently non-trivial: the class of the symplectic form and all its $\wedge$-powers up to $n$ are non-trivial. For $\mathbb{S}^{2n}$ and $n \ge 2$ this is known to be not true: all cohomologies vanish except for the zeroth and the $2n$-th, which are both one-dimensional.


