# Can a sphere be a phase space?

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

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The sphere is compact and $T^*M$ is compact only for zero-dimensional $M$. – HenrikRüping Nov 27 '12 at 9:46
$T^*M$ is not compact. – ThiKu Nov 27 '12 at 9:47
One may still ask whether a sphere can arise as the unit tangent bundle of some manifold. But then of course ist should have odd dimension. – ThiKu Nov 27 '12 at 9:49
In physics, a phase space is no more and no less than a symplectic manifold. It only happens that the most common examples are cotangent bundles. In this sense, to make $S^{2k}$ a phase space, you need only find a symplectic form on it. I think that considering it as a coadjoint orbit of $SO(2k+1)$ might do the trick. – Igor Khavkine Nov 27 '12 at 9:55
@Igor, I don't think so. For $k > 1$, $H^2(S^{2k}) = 0$. – Oliver Nash Nov 27 '12 at 10:02

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.
To answer the question in the title: if by phase space we mean a symplectic manifold, then only for $k=1$ is there a symplectic structure. This is the phase space of a classical spin.
It is not necessary for a manifold to be identified with $T^*M$ for some $M$ to qualify as a phase space. This is the first place we encounter the idea, with $M$ being the configuration space of a system, but the concept is more general