Killing 2-forms in dimension 4

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 $p$-form on a Riemannian manifold is a $p$-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 $\varphi$. If $M$ is connected and oriented, we show that there exists a dense open subset of $M$ on which one of the three exclusive situations holds: either $\phi$ is everywhere degenerate and $g$ is conformal to a product metric, or $g$ is conformal to an ambik\"ahler metric obtained via the Calabi construction from a polarized Riemannian surface, or $g$ 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 $S^4$ and on Hirzebruch surfaces.