Chirality in Affine Spaces and in Spacetime

TL;DR

This paper classifies chiral objects and isometries in affine spaces and spacetime models, using algebraic group structures to distinguish direct and indirect symmetries in various geometric contexts.

Contribution

It introduces a systematic classification of isometries as direct or indirect in affine and spacetime geometries using outer semidirect products of isometry groups.

Findings

Classified isometries in affine spaces over real quadratic spaces.

Distinguished direct and indirect isometries in Lorentz-Minkowski spacetime.

Analyzed isometries in Newton-Cartan classical spacetime.

Abstract

An object is chiral when its symmetry group contains no indirect isometry. It can be difficult to classify isometries as direct or indirect, except in the Euclidean case. We classify them with the help of outer semidirect products of isometry groups, in particular in the case of an affine space defined over a finite-dimensional real quadratic space. We also classify as direct or indirect the isometries of the real Lorentz-Minkowski spacetime and those of the classical spacetime defined by the Newton-Cartan theory.