A Topological and Geometric Framework for the Transformation from Graphene to Toroidal Carbon Nanotubes

We develop a unified geometric and topological framework to describe the transformation of planar graphene into cylindrical and toroidal carbon nanotubes while maintaining strict consistency with graphene's lattice structure and low-energy Dirac electronic theory. Starting from the continuum approximation of graphene, we construct cylindrical and toroidal compactifications through lattice-compatible boundary identifications and analyze the corresponding evolution of the fundamental group. For the toroidal case, we explicitly classify flat U(1) gauge connections via holonomy representations of the fundamental group, demonstrating the existence of two independent flux parameters defined modulo the flux quantum. We derive the momentum quantization conditions directly from Dirac spinor boundary compatibility with the underlying lattice and valley structure, avoiding heuristic plane-wave assumptions. The mechanical stability of the toroidal geometry is shown to follow from the Helfrich bending energy, whose small-deformation limit yields the curvature-dependent functional used in our analysis, and we further verify this stability through the eigenvalue spectrum of the bending operator. By formulating the Dirac Hamiltonian on a curved surface, we demonstrate that curvature induces an effective geometric gauge field through the spin connection, leading to a curvature-dependent modification of the quantization condition. The resulting unified quantization law incorporates lattice periodicity, electromagnetic flux, and geometric curvature on equal footing. We illustrate the framework with a graphene-scale numerical example, a doubly-quantized Dirac spectrum for representative metallic and semiconducting toroidal configurations, and a comparison with experimentally observed toroidal carbon nanotubes, and we discuss the embedding and mechanical-stability constraints that separate topologically valid but physically unrealistic configurations from experimentally realizable nanotori. This framework provides a mathematically consistent description of toroidal carbon nanotubes as compact, doubly-periodic quantum systems and clarifies the interplay between topology, curvature, and electronic structure in curved graphene-based nanostructures.