Maximally homogeneous para-CR manifolds

Abstract

We define the notion of a (weak) almost para-CR structure on a manifold M as a distribution HM ⊂ TM together with a field K ∈ (End(HM)) of involutive endomorphisms of HM. If K satisfies integrability conditions, then (HM, K ) is called a (weak) para-CR structure. Under some regularity conditions, an almost para-CR structure can be identified with a Tanaka structure. The notion of maximally homogeneous almost para-CR structure of a semisimple type is defined. A classification of such maximally homogeneous almost para-CR structures is given in terms of appropriate gradations of real semisimple Lie algebras. All such maximally homogeneous structures of depth two (which correspond to depth two gradations) are listed and the integrability conditions are verified.Mathematics Subject Classifications (1991): 53C15, 53D99, 58A14

Однородные Пара-Кэлеровы Многообразия Эйнштейна

“…Следующее предложение описывает фундаментальные градуировки полу-простых комплексных (соответственно вещественных) алгебр Ли в терминах отмеченных диаграмм Дынкина (соответственно Сатаке), см., например, рабо-ты [101], [93]. Предложение 7.6.…”

“…В [93] дано описание максимально однородных слабых пара-CR-структур полупростого типа в терминах фундаментальных градуировок вещественных полупростых алгебр Ли.…”

Maximally homogeneous para-CR manifolds of semisimple type

“…In [2] we proved that a para-CR structures of semisimple type on a simply connected manifold M with the automorphism group of maximal dimension dim g can be identified with a (real) flag manifold M = G/P where G is the simply connected Lie group with the Lie algebra g and P the parabolic subgroup generated by the parabolic subalgebra…”

“…Such a para-CR structure defines a parabolic geometry and its group of automorphisms Aut(M, HM, K) is a Lie group of dimension ≤ dim g. Recently in [16] P. Nurowski and G. A. J. Sparling consider the natural para-CR structure which arises on the 3-dimensional space M of solutions of a second order ordinary differential equation y ′′ = Q(x, y, y ′ ). Using the Cartan method of prolongation, they construct the full prolongation G → M of M with a sl(3, R)-valued Cartan connection and a quotient line bundle over M with a conformal metric of signature (2,2). This is a para-analogue of the Feffermann bundle of a CR-structure.…”

Onϕ-SymmetricN(k)-Paracontact Metric Manifolds

“…A systematic study of paracontact metric manifolds and their subclasses were started out by Zamkovoy [2]. Since then, several geometers studied paracontact metric manifolds and obtained various important properties of these manifolds ( [3][4][5][6][7][8][9][10], etc.). The geometry of paracontact metric manifolds can be related to the theory of Legendre foliations.…”