(para)-K\"ahler Weyl structures

TL;DR

This paper investigates (para)-K"ahler Weyl structures across different dimensions, revealing that higher dimensions are restrictive and trivial, while 4-dimensional cases allow non-trivial structures with geometric realizability.

Contribution

It establishes the dimensional restrictions of (para)-K"ahler Weyl structures and demonstrates the existence of non-trivial structures specifically in four dimensions.

Findings

Higher-dimensional (para)-K"ahler Weyl algebraic curvature tensors are Riemannian in dimension ≥6.

Any (para)-K"ahler Weyl geometric structure is trivial in dimensions ≥6.

Non-trivial (para)-K"ahler Weyl algebraic curvature tensors exist in 4 dimensions.

Abstract

We work in both the complex and in the para-complex categories and examine (para)-K\"ahler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kaehler Weyl algebraic curvature tensor is in fact Riemannian in dimension at least 6; this yields as a geometric consequence that any (para)-Kaehler Weyl geometric structure is trivial if the dimension is at least 6. By contrast, the 4 dimensional setting is, as always, rather special as it turns out that there are (para)-Kaehler Weyl algebraic curvature tensors which are not Riemannian in dimension 4. Since every (para)-Kaehler Weyl algebraic curvature tensor is geometrically realizable and since every 4 dimensional Hermitian manifold admits a unique (para)-Kaehler Weyl structure, there are also non-trivial 4 dimensional Hermitian (para)-Kaehler Weyl manifolds.