Can $S^4$ be the cotangent bundle of a manifold?

Question

I am asking the question because in the classical mechanics book by Arnold, he states that there is a distinguished 1-form on $T^*V $. It seems that there is no such distinguished 1-form on a general even-dimensional manifold.

So, not every even-dimensional manifold can be the phase space of a system, or not every even-dimensional manifold can be a cotangent bundle?

Is $S^2$ a cotangent bundle (intuitively it is not)? How about $S^4$?

Ups: Noticed crossposting: http://math.stackexchange.com/q/1402849/11127 Please don't crosspost. Instead ask for migration. — Qmechanic, Aug 21 '15 at 13:12

score 3 · Accepted Answer · answered Aug 20 '15 at 21:31

3

No, the total space of a cotangent bundle $T^*M$ is always non-compact, while a sphere $S^{2n}$ is compact.

answered Aug 20 '15 at 21:31

Qmechanic

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Can $S^4$ be the cotangent bundle of a manifold?

1 Answers1