Some differential complexes within and beyond parabolic geometry

The gap phenomenon in parabolic geometries

“…Concrete examples are considered in §5, and submaximally symmetric models are given explicitly (in coordinates) in terms of their underlying structure on the base manifold M. This includes conformal geometry, (2, 3, 5)-distributions, (3,6)-distributions, 2nd order ODE systems, projective structures, and (2, m)-Segré structures. Four-dimensional Lorentzian conformal geometry and (2, 3, 5)-distributions are investigated in finer detail: the maximum symmetry in each Petrov type and root type is identified.…”

“…(3, 6)-geometry. Bryant studied (3,6)-distributions in [4,5], and constructed the associated B 3 /P 3 geometry, where B 3 = SO 7 (C) (or SO 3,4 ). This is a 2-graded geometry with g 0 ∼ = gl 3 (C),…”

“…For m = 2, i.e. A 3 /P 1,2 , the underlying structure is a rank 3 distribution D on a 5-manifold with a splitting D = L ⊕ Π, but now Π (rank 2) may not be integrable; instead [Π, Π] ⊆ D [11,6]. Indeed, w = ( 23…”

Extensions of the coeffective complex

“…This article is both an addendum to [4] and a precursor to [7]. In [4], we discussed the construction of differential complexes on manifolds equipped with various geometric structures.…”

“…This article is both an addendum to [4] and a precursor to [7]. In [4], we discussed the construction of differential complexes on manifolds equipped with various geometric structures. Mostly, these geometries were parabolic [5] but there were two exceptions, specifically contact geometry for which there is the Rumin complex [14] and symplectic for which there is a very similar complex [15], which we dubbed the Rumin-Seshadri complex (it was independently discovered by Tseng and Yau [17]).…”