Parabolic conformally symplectic structures I; definition and distinguished connections

TL;DR

This paper introduces a new class of geometric structures called parabolic conformally symplectic structures, along with their canonical compatible connections, expanding the understanding of contact gradings and their algebraic properties.

Contribution

It defines a new class of G-structures with underlying almost conformally symplectic structures and constructs canonical connections characterized by torsion normalization, linking geometry and Lie algebra cohomology.

Findings

Existence of canonical compatible connections for each structure type

Explicit descriptions of the geometric structures and normalization conditions

Torsion splits into components, with one indicating the obstruction to conformally symplectic structure

Abstract

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.