Parabolic conformally symplectic structures I; definition and distinguished connections

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 {\mathrm{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.

Parabolic conformally symplectic structures II: parabolic contactification

“…Moreover, we discuss the relation between distinguished connections associated to parabolic contact structures and the canonical connections of PCS-structures. Together with Section 4.7 of [4], this allows us to complete the discussion of the relation of PCS-structures to special symplectic connections in the sense of Cahen and Schwachhöfer [3] and thus to exceptional holonomies, see Theorem 7. The contact gradings of Lie algebras of type C n do not give rise to a PACS-structure. However, there is a type of parabolic contact structures associated to these gradings, the so-called contact projective structures.…”

“…Our article builds on [4], where we introduced a family of first-order structures, which all have an underlying almost conformally symplectic structure. There is one such structure for each contact grading of a simple Lie algebra, which is not of type C n .…”

“…These gradings are related to certain parabolic subalgebras and to parabolic contact structures as discussed below, which motivates the name parabolic almost conformally symplectic structures (or PACS-structures for short). The main result ofČap and Salač [4] is that each such structure determines a canonical connection on the tangent bundle, which is characterized by a normalization condition on its torsion. The torsion of this connection is a basic invariant of the structure, which naturally splits into two parts.…”

“…In Theorem 3, we show that the Grassmannian of complex planes in C n+1 can be realized as a global space of leaves for a transversal infinitesimal automorphism of the space of those quaternionic lines in H n+1 , which are isotropic for a quaternionic skew Hermitian form. This induces on the Grassmannian the PCS-structure of quaternionic type from Corollary 3.8 of [4], which is homogeneous under SU (n + 1) and can be viewed as a counterpart of the quaternion-Kähler metric on the Grassmannian of planes.…”

“…In the last part [5] of the series, we show how invariant differential operators for parabolic contact structures induce natural differential operators on PCS-quotients. There is a welldeveloped theory which in particular produces invariant differential complexes on locally flat parabolic contact structures and in certain curved situations.…”