Almost symplectic manifold

This article defines a differential manifold with the following additional structure -- the structure group is reduced to: symplectic group

Definition

An almost symplectic manifold is the following data:

1. A differential manifold $M$ of dimension $n=2m$
2. A reduction of the structure group of $TM$ to the symplectic group, or equivalently, an association of an alternating nondegenerate bilinear form to the tangent space at every point, in a smooth way.

The second datum is termed an almost symplectic structure on the manifold.