Homogeneous 4-Dimensional Kähler–Weyl Structures

Abstract

Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kähler-Weyl structure; this structure is locally conformally Kähler if and only if the alternating Ricci tensor ρa vanishes. The tensor ρa takes values in a certain representation space. In this paper, we show that any algebraic possibility Ξ in this representation space can in fact be geometrically realized by a left-invariant Kähler-Weyl structure on a 4-dimensional Lie group in either the pseudo-Hermitian or the para-Hermitian setting. MSC 2010: 53A15, 53C15, 15A72.

4-dimensional (para)-Kähler-Weyl structures

“…Let {e 1 , e 2 , e 3 , e 4 } be a local frame for T M so that Equating Θ ei J + e j with J + Θ ei e j then implies a 1 = a 2 = a 3 = a 4 = 0 so φ = 0 and φ 1 = φ 2 . This establishes Assertion (1).…”

“…By Theorem 1.1, only the 4-dimensional setting is relevant. The following is the main result of this short note; it plays a central role in the discussion of [1].…”

“…3.3. The proof of Theorem 1.2 (1). Let V = R 4 , let S 2 − be the vector space of symmetric 2-cotensors ω so that J * + ω = −ω, and let ε ∈ C ∞ (S 2 ) satisfy ε(0) = 0.…”

Hamilton Equation on Almost Pseudo Kähler-Weyl Manifolds for Mechanical Systems

“…, reduces to the semisymmetry [3]. Antoniou the Hermitian or the para-Hermitian setting [7]. Jelonek…”