  We introduce a class of first order G-structures, each of which has an
underlying almost conformally symplectic structure. There is one such structure
for each real simple Lie algebra which is not of type $C_n$ and admits a
contact grading. We show that a structure of each of these types on a smooth
manifold $M$ determines a canonical compatible linear connection on the tangent
bundle $TM$. This connection is characterized by a normalization condition on
its torsion. The algebraic background for this result is proved using Kostant's
theorem on Lie algebra cohomology.
  For each type, we give an explicit description of both the geometric
structure and the normalization condition. In particular, the torsion of the
canonical connection naturally splits into two components, one of which is
exactly the obstruction to the underlying structure being conformally
symplectic. This article is the first in a series aiming at a construction of
differential complexes naturally associated to these geometric structures.

Differential Geometry;Symplectic Geometry