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1. • CommentRowNumber1.
   • CommentAuthorjim_stasheff
   • CommentTimeSep 15th 2013
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   Author: jim_stasheff
   Format: TextWhere is there anything written on the obstructions to a smooth manifold admitting a Poisson structure?

   Where is there anything written on the obstructions to a smooth manifold admitting a Poisson structure?
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 15th 2013
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   Author: Urs
   Format: MarkdownItexDo you mean a [[Poisson bracket]] on the space of functions? Then there is no obstruction, as you can always take the Poisson bracket to constantly vanish.

   Do you mean a Poisson bracket on the space of functions? Then there is no obstruction, as you can always take the Poisson bracket to constantly vanish.

3. • CommentRowNumber3.
   • CommentAuthorjim_stasheff
   • CommentTimeSep 15th 2013
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   Author: jim_stasheff
   Format: TextI goofed. So what condition should I demand to avoid the trivial bracket?

   I goofed. So what condition should I demand to avoid the trivial bracket?
4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 15th 2013
   • (edited Sep 15th 2013)
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   Author: Urs
   Format: MarkdownItexNot sure what you need. The vanishing [[Poisson tensor]] has maximal kernel regarded as a map $\pi \colon \Gamma(T^\ast X) \longrightarrow \Gamma(T X)$. You could ask for a smaller kernel. If you ask for vanishing kernel, then you may want a [[symplectic manifold]] structure.

   Not sure what you need. The vanishing Poisson tensor has maximal kernel regarded as a map $\pi \colon \Gamma(T^\ast X) \longrightarrow \Gamma(T X)$. You could ask for a smaller kernel. If you ask for vanishing kernel, then you may want a symplectic manifold structure.