“…Examples are the polarization vector for ferroelectrics, the degrees of freedom exploited at low scales by the atomic rearrangements in quasicrystals (what is collected in the so-called phason field), the peculiar direction of stick molecules with head-to-tail symmetry in liquid crystals, etc. A unified view on the matter foresees that, besides the deformation through which we reach a placement taken as reference-say B, and other macroscopic shapes B a -we have a field, defined over B itself, which takes values on a differentiable manifold M (see [6,37,39,42]) that we call the manifold of microstructural shapes. The common assumption that M is finite-dimensional is sufficient to include in the framework, as special cases, models that we know in solid-state physics (example those for ferroelectrics, magnetoelastic materials, quasicrystals, elastomers) and also more abstract schemes such as the Cosserat one [10] (called also micropolar), used for models of beams and shells (among many, see the basic papers [21] and [62] on the matter, the first one being that opening the application of Cosserat ideas to the description of the elastic structural elements) or liquid crystals in smectic order, and the micromorphic one (either considering microstrain or deformable directors, see [22,29,45]), adopted for polymers or models of strain-gradient plasticity, which have also been the playground for several analytical and geometrical investigations (see, example, [11,30,47,51,64]).…”