Kocian Philippe, Schenk Kurt, Chapuis Gervais
Laboratoire de Cristallographie, IPMC-FSB, Ecole Polytechnique Fédérale de Lausanne, BSP-Le Cubotron, Dorigny, CH-1015 Lausanne, Switzerland.
Acta Crystallogr A. 2009 Sep;65(Pt 5):329-41. doi: 10.1107/S0108767309024660. Epub 2009 Jul 30.
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.
微分几何为描述晶体学中的基本概念提供了一个有用的数学框架。点和相关向量空间的概念对应于流形以及给定点处的切空间的概念。空间群运算为作用于流形的一一映射，而点群运算为作用于流形的两个切空间之间的线性映射。流形理论作为一种统一的形式体系，在描述传统晶体以及调制晶体的对称操作时，无需更高维空间，显示出其特别强大的作用。我们特别表明，调制结构在任何切空间中恢复三维周期性，并且其点群由线性应用组成。
