The motion of a quantum particle constrained to a two-dimensional non-compact Riemannian manifold with nontrivial metric can be described by a flat-space Schrödinger-type equation at the cost of introducing local mass and metric and geometry-induced effective potential with no classical counterpart. For a metric tensor periodically modulated along one dimension, the formation of bands is demonstrated and transport-related quantities are derived. Using S-matrix approach, the quantum conductance along the manifold is calculated and contrasted with conventional quantum transport methods in flat spaces. The topology, e.g., whether the manifold is simply connected, compact or non-compact shows up in global, non-local properties such as the Aharonov–Bohm phase. The results vividly demonstrate emergent phenomena due to the interplay of reduced-dimensionality, particles quantum nature, geometry, and topology.
Schwager et al. (Mon,) studied this question.