Defects of micropolar continua in Riemann-Cartan manifolds and its applications

We derive equations of motion and its solutions in the form of solitons from deformational energy functionals of a coupled system of microscopic and macroscopic deformations. Then criteria in constructing the chiral energy functional is specified to be included to obtain soliton-like solutions. We show various deformational measures, used in deriving the soliton solutions, can be written when both curvature and torsion are allowed, especially by means of microrotations and its derivatives. Classical compatibility conditions are re-interpreted leading to a universal process to derive a distinct set of compatibility conditions signifying a geometrical role of the Einstein tensor in Riemann-Cartan manifolds. Then we consider position-dependent axial configurations of the microrotations to construct intrinsically conserved currents. We show that associated charges can be written as integers under a finite energy requirement in connection with homotopic considerations. This further leads to a notion of topologically stable defects determined by invariant winding numbers for a given solution classification. Nematic liquid crystals are identified as a projective plane from a sphere hinted by the discrete symmetry in its directors. Order parameters are carefully defined to be used both in homotopic considerations and free energy expansion in the language of microcontinua. Micropolar continua are shown to be the general case of nematic liquid crystals in projective geometry, and in formulations of the order parameter, which is also the generalisation of the Higgs isovectors. Lastly we show that defect measures of pion fields description of the Skyrmions are related to the defect measures of the micropolar continua via correspondences between its underlying symmetries and compatibility conditions of vanishing curvature.