## #003: Characterize the expression \( G_{\alpha\beta\gamma} = \partial_\alpha g_{\beta \gamma} - \partial_\beta g_{\alpha \gamma} \)

*Category: Classical mechanics - Tags: Differential geometry, Riemannian geometry, Symplectic geometry, General relativity*

Characterize the geometry and the physics captured by the expression \(G_{\alpha\beta\gamma} = \partial_\alpha g_{\beta \gamma} - \partial_\beta g_{\alpha \gamma}\). This expression appears when expressing the symplectic form in terms of position and velocity (see here for more details).

*Mathematical problem.* The expression appears with the EM tensor in the spatial-temporal
part of the symplectic form when it is expressed in terms of position and velocity.
It also appears as a “correction” for the raised anti-symmetrized covariant
derivative. This means the expression should have both geometrical and physical significance,
which merit further study.

As an example of the type of things we are looking for:

- Is the expression a tensor? It would seem so as the spatial-temporal part of the symplectic form should transform like a tensor.
- How is it related to curvature? Is it always zero on flat spaces? Can it be zero on curved spaces?
- Can it be interpreted as a gravitational force?
- From calculation, the tensor can be understood as the difference between taking an antisymmetrized covariant derivative and an antisymmetrized partial derivative: what does that say geometrically?

*Physical significance.* Note that the setting is not really that of space-time
(i.e. Riemannian manifold) but that of phase-space (i.e. symplectic manifold).
The general idea is to reunderstand the metric tensor not as a geometrical property of
space-time but as a geometrical property of phase-space.