Killing 2-forms in dimension 4
Paul Gauduchon, Andrei Moroianu

TL;DR
This paper classifies 4-dimensional Riemannian manifolds with non-parallel Killing 2-forms, revealing three main geometric structures and providing explicit examples on spheres and Hirzebruch surfaces.
Contribution
It provides a complete local classification of such manifolds, detailing their conformal structures and constructing explicit examples.
Findings
Existence of three distinct geometric types for manifolds with Killing 2-forms.
Explicit construction of infinite-dimensional families of examples.
Characterization of manifolds as conformal to product, ambikähler, or ambitoric structures.
Abstract
A Killing -form on a Riemannian manifold is a -form whose covariant derivative is totally anti-symmetric. In this paper we give the complete (local) description of 4-dimensional Riemannian manifolds (M,g) carrying non-parallel Killing 2-forms . If is connected and oriented, we show that there exists a dense open subset of on which one of the three exclusive situations holds: either is everywhere degenerate and is conformal to a product metric, or is conformal to an ambik\"ahler metric obtained via the Calabi construction from a polarized Riemannian surface, or is conformal to an ambitoric structure of hyperbolic type, and depends locally on two functions of one variable. We also give compact examples, by constructing infinite-dimensional families of Riemannian metrics carrying Killing 2-forms of each of the above types on and on Hirzebruch…
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