With the rapid development of microtechnology, the low-dimensional materials
are fabricated with nontrivial topological structures, and then the action of geometric properties on the effective dynamics receives increasing attention. It is an effective theory that the
quantum mechanics of low-dimensional curved systems can be given by using the thin-layer
quantization approach, in which the geometric potential and the geometric momentum have
been demonstrated experimentally. In the present paper, the thin-layer quantization formalism is first recalled, its fundamental calculation framework is first clarified, and the geometric
quantum effects result from the diffeomorphism transformation and the rotation transformation connecting the local frames of different points. The results are helpful to understand the
gravitational gauge and emerging gauge, and to image the geometries implied in the artificial
gauge. In the particular quantum systems, the novel physical phenomena are briefly recalled
that are induced by geometry, such as resonation tunneling, quantum block, quantum Hall
effect, quantum Hall viscosity, quantum spin Hall effect, effective monopole magnetic field
and so on. The results will shed light from a different angle on the relationships between the
geometry and the novel physical phenomenon.
