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Question 2: Is there some nice geometric or algebraic reason why symplectic geometry should lead to a dimension-independent Poisson geometry, and most other "nondegenerate" theories don't without losing some information?

Question 2: Is there some nice geometric or algebraic reason why symplectic geometry should lead to a dimension-independent Poisson geometry, and most other "nondegenerate" theories don't without losing some information? In particular, why no other integrable theories (at least among those checked) give dimension-independent equations?

I've tried to work out "Poisson-Riemann" and "Poisson-Kahler" algebras, and none of them seem to work out as nicely as symplectic geometry. For example, "Poisson-Kahler algebras" can be made, but the relevant equations depend on the dimension in an obvious way (that is, there's an easy way given the dimension to write the equations). "Poisson-Riemann algebras" can be made in multiple ways; the easiest way has no good generalization of the curvature tensor, and the hardest way doesn't seem to work at all. "Flat Poisson-Riemann algebras" seem to make sense, but the equations seem to depend on dimension in a non-obvious way. Is this one reason that Poisson geometry is studied, and other aren't?

EDIT: I guess this comprises a couple of questions.

Question 1: Is there a specific reason other "degeneracy-generalization" theories don't seem to be studied much? Something that makes symplectic nicer than others?

Question 2: Is there some nice geometric or algebraic reason why symplectic geometry should lead to a dimension-independent Poisson geometry, and most other "nondegenerate" theories don't without losing some information?

Or generally: what is it about symplectic geometry that makes Poisson geometry "nice" that other geometries lack?

I've tried to work out "Poisson-Riemann" and "Poisson-Kahler" algebras, and none of them seem to work out as nicely as symplectic geometry. For example, "Poisson-Kahler algebras" can be made, but the relevant equations depend on the dimension in an obvious way (that is, there's an easy way given the dimension to write the equations). "Poisson-Riemann algebras" can be made in multiple ways; the easiest way has no good generalization of the curvature tensor, and the hardest way doesn't seem to work at all. "Flat Poisson-Riemann algebras" seem to make sense, but the equations seem to depend on dimension in a non-obvious way. Is this one reason that Poisson geometry is studied, and other aren't?

$\sum_{\sigma \in S_n} (-1)^{sgn(\sigma)} (\lceil f_1, \lceil f'_1, g_{\sigma(1)}\rceil \rceil - \lceil f'_1, \lceil f_1, g_{\sigma(1)}\rceil \rceil) \prod_{i = 2}^n \{f_i, g_{\sigma(i)}\} = 0$

$\sum_{\sigma \in S_n} (-1)^{sgn(\sigma)} (\lceil f_1, \lceil f'_1, g_{\sigma(1)}\rceil \rceil - \lceil f'_1, \lceil f_1, g_{\sigma(1)}\rceil \rceil) \prod_{i = 2}^n \lceil f_i, g_{\sigma(i)}\rceil = 0$