Topology and Geometry of Smectic Order on Compact Curved Substrates

Topology and Geometry of Smectic Order on Compact Curved Substrates

We show that smectic order on arbitrary curved substrate can be described by differential form of rank one (1-form), whose geometric meaning is the differential of the local phase field of the density modulation. The exterior derivative of 1-form is shown to be the local dislocation density. Elastic deformations are described by superposition of exact differential forms. More importantly, we us...

Topology of smectic order on compact substrates.

Smectic orders on curved substrates can be described by differential forms of rank one (1-forms), whose geometric meaning is the differential of the local phase field of density modulation. The exterior derivative of the 1-form is the local dislocation density. Elastic deformations are described by superposition of exact differential forms. Applying this formalism to study smectic order on a to...

Extrinsic curvature, geometric optics, and lamellar order on curved substrates.

When thermal energies are weak, two-dimensional lamellar structures confined on a curved substrate display complex patterns arising from the competition between layer bending and compression in the presence of geometric constraints. We present broad design principles to engineer the geometry of the underlying substrate so that a desired lamellar pattern can be obtained by self-assembly. Two dis...

Curved Spacetime Geometry for Strings and Affine Non-compact Algebras

Compact Clifford–Klein Forms – Geometry, Topology and Dynamics

We survey results on compact Clifford–Klein forms of homogeneous spaces, with a focus on recent contributions and organized around approaches via topology, geometry and dynamics. In addition, we survey results on moduli spaces of compact forms.