Hopf fibrations and frustrated matter

Frustrated order appear in different scientific contexts like complex crystals, amorphous materials, liquid crystals, foams and even biological organizations, with scales ranging from the atomic level up to macroscopic scales. In this article, we shall first review several cases where it is possible to release frustration and allows for an unfrustrated structural model by curving the underlying Euclidean space into a positively curved space, the hypersphere S ³. The real Euclidean structure is then analyzed in terms ordered regions interrupted by topological defects, whose presence and density is directly related to the change of curvature from S ³ to R ³. We then focus on a rather fascinating geometrical structure, the Hopf fibration, which can be defined on the hypersphere, and show how this tool is very well suited to describe the defect geometries (both linear, helicoidal and bidimensional) which arises from the decurving procedure.