Topological states of matter are at present one of the most challenging and active fields in condensed matter physics due to: i) the richness of physical phenomena they display, and ii) the foundational topological concepts on which they are built-in, which drive a broad cross-fertilization with different physics research areas. Low-dimensional semiconducting nanomaterials play a relevant role in the area of topological states of matter. However, apart from these conventional material geometries the most recent advances in nanotechnology have made it possible to have at hand an entirely novel family of low-dimensional nanostructures: flexible semiconductor nanomaterials which are bent into curved, deformable objects ranging from semiconductor nanotubes, to nanohelices. Motivated by the excitement in both topological states of matter and novel shape deformed nanostructures, we have now theoretically considered the possible interplay between curvature effects  on the electronic, topological and superconducting properties of the quantum states in low-dimensional nanomaterials. We will firstly discuss how geometric effects in low-dimensional nanomaterials can lead to metal-insulator transition and promote the generation of topological states of matter by considering the paradigmatic example of quantum wires with Rashba spin-orbit coupling, which are periodically corrugated at the nanometer scale . Then, we will present the intricate twist between spin texture and spin transport in shape deformed nanostructures. These topologically non trivial spin patterns affect the electron spin interference in the deformed ring, thereby resulting in novel spin-orbitronics device concepts . When considering superconducting (SC) nanostructures, we demonstrate that a local SC spin-singlet amplitude control can be achieved and that the geometric curvature generates non-trivial textures of the spin-triplet pairs through windings of the d-vector. Such findings points to topological emergent behaviors and unveil novel paths to manipulate the quantum structure of the SC state in RSOC nanostructures through their geometry .
 P. Gentile, M. Cuoco, C. Ortix, SPIN, Vol. 3, No. 2, 1340002 (2013).
 P. Gentile, M. Cuoco, C. Ortix, Phys. Rev. Lett. 115, 256801 (2015).
 Z.-J. Ying, P. Gentile, C. Ortix, and M. Cuoco, Phys. Rev. B 94, 081406(R) (2016).
 Z.-J. Ying, M. Cuoco, C. Ortix, and P. Gentile, arXiv:1704.00578 (2017).