Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 084, 12 pages      arXiv:2309.11184      https://doi.org/10.3842/SIGMA.2024.084
Contribution to the Special Issue on Global Analysis on Manifolds in honor of Christian Bär for his 60th birthday

Compact Locally Conformally Pseudo-Kähler Manifolds with Essential Conformal Transformations

Vicente Cortés a and Thomas Leistner b
a) Department Mathematik, University of Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
b) School of Computer & Mathematical Sciences, University of Adelaide, SA 5005, Australia

Received September 21, 2023, in final form September 09, 2024; Published online September 21, 2024

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-Kähler and not conformally flat, where $n\ge 1$, $k, \ell \ge 0$. The corresponding local pseudo-Kähler metrics obtained by a local conformal rescaling are Ricci-flat.

Key words: pseudo-Riemannian manifolds; essential conformal transformations; Kähler metrics; symmetric spaces.

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