We analyze spin scattering in ballistic transport of electrons through a ripple at a normal incidence of an electron flow. The model of a ripple consists of a curved graphene surface in the form of an arc of a circle connected from the left-hand and right-hand sides to two flat graphene sheets. At certain conditions the curvature induced spin-orbit coupling creates a transparent window for incoming electrons with one spin polarization simultaneously with a backscattering of those with opposite polarization. This window is equally likely transparent for electrons with spin up and spin down that move in opposite directions. The spin filtering effect being small in one ripple becomes prominent with the increase of N consequently connected ripples that create a graphene sheet of the sinusoidal type. We present the analytical expressions for spin up (down) transmission probabilities as a function of N connected ripples.

The Hamiltonian for nanocones with curvature-induced spin-orbit coupling have been derived. The effect of curvature-induced spin-orbit coupling on the electronic properties of graphitic nanocones is considered. Energy spectra for different numbers of the pentagonal defects in the tip of the nanocones are calculated. It was shown that the spin-orbit interaction considerably affects the local density of states of the graphitic nanocone. This influence depends on the number of defects present at the tip of the nanocone. This property could be applied in atomic force microscopy for the construction of the probing tip.

The field-theory model is proposed to study the electronic states near the Fermi energy in spheroidal fullerenes. The low energy electronic wavefunctions obey a two-dimensional Dirac equation on a spheroid with two kinds of gauge fluxes taken into account. The first one is so-called K spin flux which describes the exchange of two different Dirac spinors in the presence of a conical singularity. The second flux (included in a form of the Dirac monopole field) is a variant of the effective field approximation for elastic flow due to twelve disclination defects through the surface of a spheroid. We consider the case of a slightly elliptically deformed sphere which allows us to apply the perturbation scheme. It is shown exactly how a small deformation of spherical fullerenes provokes an appearance of fine structure in the electronic energy spectrum as compared to the spherical case. In particular, two quasi-zero modes in addition to the true zero mode are predicted to emerge in spheroidal fullerenes. An additional 'hyperfine' splitting of the levels (except the quasi-zero-mode states) is found.