Differential geometry: a natural tool for describing symmetry operations

Differential geometry provides a useful mathematical framework for describing the fundamental concepts in crystallography. The notions of point and associated vector spaces correspond to those of manifold and tangent space at a given point. A space-group operation is a one-to-one map acting on the manifold, whereas a point-group operation is a linear map acting between two tangent spaces of the manifold. Manifold theory proves particularly powerful as a unified formalism describing symmetry operations of conventional as well as modulated crystals without requiring a higher-dimensional space. We show, in particular, that a modulated structure recovers a three-dimensional periodicity in any tangent space and that its point group consists of linear applications.