Chirality in Affine Spaces and in Spacetime

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.