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Differential geometry: a natural tool for describing symmetry operations
Philippe Kocian1, Kurt Schenk, Gervais Chapuis
1Laboratoire de Cristallographie, IPMC-FSB, Ecole Polytechnique Fédérale de Lausanne, BSP-Le Cubotron, Dorigny, CH-1015 Lausanne, Switzerland. philippe.kocian@epfl.ch
Summary
Differential geometry offers a unified framework for crystallography, describing crystal symmetries using manifold and tangent spaces. This approach simplifies the analysis of conventional and modulated structures without higher dimensions.
Area of Science:
- Crystallography
- Differential Geometry
- Mathematical Physics
Background:
- Crystallography describes the arrangement of atoms in crystals.
- Symmetry operations are crucial for understanding crystal structures.
- Existing formalisms can be complex, especially for modulated crystals.
Purpose of the Study:
- To apply differential geometry concepts to crystallography.
- To develop a unified formalism for symmetry operations in conventional and modulated crystals.
- To demonstrate the power of manifold theory in this context.
Main Methods:
- Utilizing differential geometry, specifically manifold and tangent space concepts.
- Representing space-group operations as maps on a manifold.
- Representing point-group operations as linear maps between tangent spaces.
Main Results:
- Differential geometry provides a robust framework for crystallographic concepts.
- Manifold theory offers a unified approach for both conventional and modulated crystals.
- Modulated structures exhibit three-dimensional periodicity in tangent spaces.
- Point groups of modulated structures are described as linear applications.
Conclusions:
- Differential geometry provides a powerful and unified mathematical framework for crystallography.
- Manifold theory simplifies the description of symmetry operations, including those in modulated crystals.
- This approach avoids the need for higher-dimensional representations.
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