Geometric curvature of a low-dimensional host material often leads to drastic changes in physical properties of the system living upon it. In this article, I give a bird's-eye review on the theoretical progresses on the geometry-property relations, i.e., intriguing correlation between geometric curvature and physical property of the material considered. Some differential-geometry-based approaches that are aimed at quantitative description of physical systems confined to curved surfaces are presented.KEYWORDS : Gaussian curvature, quantum transport, liquid crystal, phase transition, foam coarsening 1．Introduction Mathematical notions of "geometry" and "curvature" are becoming commonplace in diverse classes of physics including surface sciences. Profound effects of geometric curvature have manifested not only in Einstein's gravity theory, but also in various lowdimensional materials such as nanocarbon layers, 1,2) liquid crystal films, 3) and cell membranes. 4) In all the systems abovementioned, the underlying geometric curvature is found to affect dominantly the physical properties of the systems.From a theoretical perspective, the body of research has relied on differential geometry, the mathematical discipline that gives a quantitative description of geometry-property relations in physics. In fact, differential geometry allows us to formulate explicitly the effective Hamiltonian of quantum excitations in curved nanomaterials. It also enables us to appreciate beautiful interplay between surface curvature and topological defect configuration in orientationally ordered systems in two dimensions. In particular, geometry-property relations become salient in soft matters that are mechanically deformable ; liquid crystal membranes and monolayered aqueous foam are the cases in point.In this article, I review the up-to-date findings on the geometry-property relations in low-dimensional materials endowed with nonzero surface curvature.Special focus is placed on the following three issues : (ⅰ) anomalous quantum transport in curved nanocarbons, (ⅱ) liquid crystal membranes with curved shape, and (ⅲ) time-evolution of aqueous foam confined within the gap of two curves substrates. Throughout the discussions, we will see that geometric curvature causes a drastic alteration in the nature of the systems, which imply the development of curved-shaped functional materials based on geometry-property relations.
2．Quantum Mechanics on Curved Surfaces
1 Curvature-induced potentialRapid advances in nanotechnology have made it possible to manufacture quasi one-and twodimensional nanostructures with non-flat geometries.5∼8) These experimental achievements arouse renewed interest in the effect of structural geometry on the quantum-mechanical properties.Surprisingly, the quantum mechanics of a particle whose motion is constrained to curved surfaces has been an old but new problem of theoretical physics for more than 50 years. The difficulty arises from operator-ordering ambiguities, 9) which permit multiple consistent quantizations for a ...
