# Einstein tori and crooked surfaces

## TL;DR

This paper introduces an algebraic invariant for classifying pairs of hyperplanes in the Einstein universe, aiding in understanding the geometry of crooked surfaces in 3D hyperbolic space.

## Contribution

It develops a determinant-based invariant for Einstein hyperplanes using symplectic splittings, extending classification tools from hyperbolic space to Einstein universe.

## Key findings

- Invariant classifies pairs of hyperplanes in Einstein universe
- Provides a disjointness criterion for crooked surfaces
- Links symplectic geometry with Einstein universe geometry

## Abstract

In hyperbolic space, the angle of intersection and distance classify pairs of totally geodesic hyperplanes. A similar algebraic invariant classifies pairs of hyperplanes in the Einstein universe. In dimension 3, symplectic splittings of a 4-dimensional real symplectic vector space model Einstein hyperplanes and the invariant is a determinant. The classification contributes to a complete disjointness criterion for crooked surfaces in the 3-dimensional Einstein universe.