Spin connection and boundary states in a topological insulator

We study the surface resistivity of a three-dimensional topological insulator when the boundaries exhibit a nontrivial curvature. We obtain an analytical solution for a spherical topological insulator, and we show that a nontrivial quantum spin connection emerges from the three-dimensional band structure. We analyze the effect of the spin connection on the scattering by a bump on a flat surface. Quantum effects induced by the geometry lead to resonances when the electron wavelength is comparable to the size of the bump.