A Concrete Variant of the Twistor Theorem

TL;DR

This paper presents a specific version of the twistor theorem that simplifies the construction of hyper-Kähler structures by starting with a known real manifold, eliminating the need for twistor line parameterization.

Contribution

It introduces a concrete variant of the twistor theorem applicable directly to existing real manifolds for hyper-Kähler structure construction.

Findings

Provides a new method for constructing hyper-Kähler structures

Simplifies the process by removing the need for twistor line parameterization

Broadens applicability to known real manifolds

Abstract

In this note, we prove a concrete variant of the twistor theorem of Hitchin--Karlhede--Lindstr\"om--Ro\v{c}ek which applies when one already has the real manifold on which one wishes to construct a hyper-K\"ahler structure, and so one does not need to construct it as a parameter space of twistor lines.