## Context (pg-321):
We have a manifold with an anti symmetric metric tensor/sympletic form $S$ with components in a basis $S_{ab}$ satisfying the property that 

$$dS=0$$

Where $d$ is the exterior derivative. 

The components of the inverse of metric tensor  in the same basis is given as $S^{ab}$


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## Question :

 In page-322 , the following equation is given as the Poisson bracket of two scalar field:

$$ \{\Phi, \Psi \} = - \frac12 S^{ab} \nabla_a \Phi \nabla_b \Psi$$

From this, we find a Jacobi identity for three Scalar fields:

$$ \{ \Theta, \{ \Phi, \Psi \} \} + \{ \Phi , \{ \Psi , \Theta \} \} + \{ \Psi , \{ \Theta, \Phi\} \}=0$$

What does the above expression mean actually? Could perhaps a geometric explanation be given?

I have asked [a related question][1] previously on MSE

Book is Roger Penrose's Road to Reality.


  [1]: https://math.stackexchange.com/questions/4456176/proving-jacobi-identity-in-tensor-notation