(para)-Kähler Weyl Structures

Abstract

Abstract. We work in both the complex and in the para-complex categories and examine (para)-Kähler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kähler Weyl algebraic curvature tensor is in fact Riemannian in dimension m ≥ 6; this yields as a geometric consequence that any (para)-Kähler Weyl geometric structure is trivial for m ≥ 6. By contrast, the 4-dimensional setting is, as always, rather special as it turns out that there are (para)-Kähler Weyl algebraic curvature tensors which are not Riemannian if m = 4. Since every (para)-Kähler Weyl algebraic curvature tensor is geometrically realizable and since every 4-dimensional Hermitian manifold admits a unique (para)-Kähler Weyl structure, there are also non-trivial 4-dimensional Hermitian (para)-Kähler Weyl manifolds. MSC: 53B05, 15A72, 53A15, 53B10, 53C07, 53C25. IntroductionLet ∇ be a torsion free connection on a pseudo-Riemannian manifold (M, g) of even dimension m = 2m ≥ 4. The triple (M, g, ∇) is said to be a Weyl structure if there exists a smooth 1-form φ so that ∇g = −2φ ⊗ g. Such a geometric structure was introduced by Weyl [37] in an attempt to unify gravity with electromagnetism. Although this approach failed for physical reasons, these geometries are still studied for their intrinsic interest [2,10,21,27,28]; they also appear in the mathematical physics literature [12,20,26]. Weyl geometry is relevant to submanifold geometry [25] and to contact geometry [15]. The pseudo-Riemannian setting also is important [1,24,32] as are para-complex geometries [11,13]. See also [9,22,30,31] for related results. The literature in the field is vast and we can only give a flavor of it for reasons of brevity. We shall be primarily interested in the Hermitian setting. However since there are applications to higher signature geometry, we include the pseudo-Hermitian context as well; similarly we treat para-Hermitian geometries as they can be studied with little additional effort. Section 1.1 of the Introduction deals with the real setting. In Theorem 1.1, we recall the basic theorems of geometric realizability for affine, Riemannian, and Weyl curvature models and in Theorem 1.2 provide various characterizations of the notion of a trivial Weyl structure. Section 1.2 treats the (para)-Kähler setting. In Theorem 1.3 we recall geometric realizibility results for (para)-Kähler affine and (para)-Kähler Riemannian curvature models. Theorem 1.4 presents results in the geometric setting for (para)-Kähler Weyl manifolds. Theorem 1.5 is one of the two main results of this paper: every (para)-Kähler curvature model is geometrically realizable. The proof of Theorem 1.5 relies on a curvature decomposition result; the second main result of the paper, Theorem 1.6, discusses the space of (para)-Kähler Weyl algebraic curvature tensors.1.1. Riemannian, Affine, and Weyl geometry. Let (V, ·, · ) be an inner product space of signature (p, q) and dimension m = p+ q; an inner product of signatur...

Homogeneous 4-Dimensional Kähler–Weyl Structures

“…We refer to Section 3.1 for further details. The following result was established [16] in the Riemannian setting; the proof extends without change to this more general context -we also refer to [8,9] for another treatment and to [12] for related material.…”

“…The space of algebraic Kähler-Weyl curvature tensors is given by K W ⊂ ⊗ 4 V * is defined by imposing the symmetries of Equation (1.a) and Equation (1.d). There is [2,8] an orthogonal direct sum decomposition of K W into inequivalent and irreducible U ⋆ -modules:…”

“…We have W 3 = ker(ρ) ∩ K W and, furthermore, that ρ s and ρ a define U ⋆ module isomorphisms [2,7,8]:…”

4-dimensional (para)-Kähler-Weyl structures

“…Assertion (3) of Theorem 1.2, which deals with the Hermitian setting, is well known -see, for example, the discussion in [8]. Subsequently, Theorem 1.2 was established full generality (see [3,4]) by extending the Higa curvature decomposition [6,7] from the real to the Kähler-Weyl and to the para-Kähler Weyl contexts.…”

Transplanting geometrical structures

“…One then wants to show these identities hold more generally and this involves a transplanting problem. Similarly, when establishing geometric realization results, one often constructs examples which are only defined on a small neighborhood of the origin -the discussion in [4] provides a nice summary of these problems and we refer to other results in [5,6,15,16]. And one wants to then deduce these geometric realization results also hold in the compact setting.…”