Geometry over algebras

TL;DR

This paper explores geometric structures derived from Hermitian forms over various real algebras, describing spaces of geodesics in different geometries and introducing a model for the hyperbolic bidisc.

Contribution

It introduces new geometric structures over non-division algebras and a projective model for the hyperbolic bidisc, expanding understanding of geometric transitions.

Findings

Describes spaces of oriented geodesics in hyperbolic, Euclidean, and spherical geometries.

Introduces a natural geometric transition between these spaces.

Provides a projective model for the hyperbolic bidisc.

Abstract

We study geometric structures arising from Hermitian forms on linear spaces over real algebras beyond the division ones. Our focus is on the dual numbers, the split-complex numbers, and the split-quaternions. The corresponding geometric structures are employed to describe the spaces of oriented geodesics in the hyperbolic plane, the Euclidean plane, and the round $2$-sphere. We also introduce a simple and natural geometric transition between these spaces. Finally, we present a projective model for the hyperbolic bidisc, that is, the Riemannian product of two hyperbolic discs.