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 $S^{ab}$

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 previously on MSE

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 $S^{ab}$

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 previously on MSE

Book is Roger Penrose's Road to Reality.