Elliptic fibrations associated with the Einstein spacetimes

TL;DR

This paper introduces a geometric construction associating elliptic fibrations to conformally nonflat Einstein spacetimes, revealing how the topology of these fibrations relates to the algebraic type of the Weyl tensor.

Contribution

It defines a new fibration over Einstein spacetimes with fibers as elliptic curves, linking fiber topology to Weyl tensor algebraic type, and describes the fibration's structure as a double branched cover.

Findings

Fibration $P$ is a double branched cover of the null direction bundle.

Fibers are elliptic curves or degenerations, depending on Weyl tensor type.

The fibration admits six linearly independent 1-forms satisfying specific equations.

Abstract

Given a conformally nonflat Einstein spacetime we define a fibration $P$ over it. The fibres of this fibration are elliptic curves (2-dimensional tori) or their degenerate counterparts. Their topology depends on the algebraic type of the Weyl tensor of the Einstein metric. The fibration $P$ is a double branched cover of the bundle of null direction over the spacetime and is equipped with six linearly independent 1-forms which satisfy certain relatively simple system of equations.