Levi flat CR structures on 3D Lie algebras
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Levi flat CR structures on 3D Lie algebras Giovanni Calvaruso1 · Francesco Esposito1 · Domenico Perrone1 Received: 2 February 2020 / Accepted: 29 March 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We completely classify Levi flat CR structures (that is, CR structures with vanishing Levi form) on three-dimensional real Lie algebras, in terms of their algebraic and almost contact properties. Keywords Levi-flat CR structures · Lie groups and Lie algebras · Almost contact structures · Almost 𝛼-coKähler structures Mathematics Subject Classification 32V05 · 53D15 · 22E60
1 Introduction Let M be a (2n + 1)-dimensional manifold. An almost CR structure (of hypersurface type) on M is a pair (H = H(M), J) where H is a smooth real subbundle of rank 2n of the tangent bundle TM (also called the Levi distribution), and J ∶ H → H is an almost complex structure: J 2 = −I . √ Put H1,0 = {X − iJX ∶ X ∈ H} and H0,1 = {X + iJX ∶ X ∈ H} = H1,0 , i = −1 . An almost CR structure (M, H, J) is said to be a CR structure on M if H1,0 (and hence also H0,1 ) is (formally) integrable. If dim M = 3 , then any almost CR structure is integrable. A pseudo-Hermitian structure on an almost CR manifold (M, H, J) is a differential one-form 𝜃 such that ker𝜃 = H and the Levi form L𝜃 , defined by L𝜃 (X, Y) ∶= (d𝜃)(X, JY), X, Y ∈ H, is Hermitian. Let (M, H, 𝜃, J) be a pseudo-Hermitian almost CR manifold. Denote by G(M, 𝜃) the group of all CR automorphisms f ∶ M → M such that f ∗ 𝜃 = 𝜃 . In general, the group G(M, 𝜃) is not a Lie group, but if M is a pseudo-Hermitian CR manifold then G(M, 𝜃) is
* Giovanni Calvaruso [email protected] Francesco Esposito [email protected] Domenico Perrone [email protected] 1
Dipartimento di Matematica e Fisica “E. De Giorgi”, Università del Salento, Via Provinciale Lecce‑Arnesano, 73100 Lecce, Italy
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a Lie group [3, p. 60]. A pseudo-Hermitian CR manifold (M, H, 𝜃, J) is said to be homogeneous if there exists a Lie group G ⊆ G(M, 𝜃) acting transitively on M [3, p. 341]. We note that a Levi non-degenerate pseudo-Hermitian CR manifold is homogeneous if and only if the corresponding contact semi-Riemannian structure is homogeneous (see, for example, [10]). The “opposite” of a Levi non-degenerate CR structure is a Levi flat (or Levi-degenerate) pseudo-Hermitian CR structure, for which the Levi form L𝜃 vanishes. A complete classification is known for homogeneous Levi non-degenerate CR structures (H, J, 𝜃) on simply connected homogeneous three-manifolds [8], [10, Theorem 21]. Up to our knowledge, a corresponding classification result is not known when (H, J, 𝜃) is an arbitrary homogeneous Levi flat pseudo-Hermitian CR structure. This led in [10] to state the following natural
Problem To classify all simply connected three-manifolds which admit a homogeneous Levi flat pseudo-Hermitian CR structure.
Note that left invariant CR structures on a Lie gr
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