Compact Locally Conformally Pseudo-K\"ahler Manifolds with Essential Conformal Transformations

TL;DR

This paper constructs examples of compact pseudo-Riemannian manifolds with essential conformal transformations that are locally conformally pseudo-K"ahler and not conformally flat, expanding understanding beyond the Riemannian case.

Contribution

It provides the first known compact examples of conformally curved pseudo-Riemannian manifolds with essential conformal transformations that are locally conformally pseudo-K"ahler.

Findings

Examples in signature (4n+2k,4n+2l) with essential conformal transformations.

Manifolds are not conformally flat but are locally conformally pseudo-K"ahler.

Local conformal rescaling yields Ricci-flat pseudo-K"ahler metrics.

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 $(4n+2k,4n+2\ell)$ with essential conformal transformations that are locally conformally pseudo-K\"ahler and not conformally flat, where $n\ge 1$, $k, \ell \ge 0$. The corresponding local pseudo-K\"ahler metrics obtained by a local conformal rescaling are Ricci-flat.