Micropolar continua as projective space of Skyrmions

TL;DR

This paper explores the topological and geometrical aspects of micropolar continua, linking them to projective spaces and topological defects like Skyrmions, with implications for understanding complex material behaviors.

Contribution

It introduces a novel perspective by relating micropolar continua to projective geometry and topological invariants, connecting defect theory with Higgs fields and Skyrmions.

Findings

Micropolar continua are linked to projective spaces of Skyrmions.

Position-dependent rotational fields are solutions of anisotropic Higgs fields.

Topological invariants classify defects and solutions in these continua.

Abstract

Micropolar continua are shown to be the generalisation of the nematic liquid crystals through perspectives of order parameters, topological and geometrical considerations. Micropolar continua and nematic liquid crystals are recognised as the antipodals of $S^3$ and $S^2$ in projective geometry. We show that position-dependent rotational axial fields in kinematic micropolar continua can be considered as solutions of anisotropic Higgs fields, characterised by integers N. We emphasise that the identical integers N are topological invariants through homotopy classifications based on defects of order parameters and a finite energy requirement. Magnetic monopoles and Skyrmions are investigated based on the theories of defects of continua in Riemann-Cartan manifolds.