Compact Locally Conformally Pseudo-Kähler Manifolds with Essential Conformal Transformations

Abstract

A conformal transformation of a semi-Riemannian manifold is essential if there is no conformally equivalent metric for which it is an isometry. For Riemannian manifolds, the existence of an essential conformal transformation forces the manifold to be conformally flat. This is false for pseudo-Riemannian manifolds; however, compact examples of conformally curved manifolds with essential conformal transformation are scarce. Here we give examples of compact conformal manifolds in signature (4𝑛 + 2𝑘, 4𝑛 + 2ℓ) with essential conformal transformations that are locally conformally pseudo-Kähler and not conformally flat, where 𝑛 ≥ 1, 𝑘, ℓ ≥ 0. The corresponding local pseudo-Kähler metrics obtained by a local conformal rescaling are Ricci-flat.