  We study the electronic transport through uniformly bent carbon nanotubes.
For this purpose, we describe the nanotube with the tight-binding model and
calculate the local current flow by employing non-equilibrium Green's functions
(NEGF) in the Keldysh formalism. In addition, we describe the low-energy
excitations using an effective Dirac equation in curved space with a
strain-induced pseudo-magnetic field which can be solved analytically for the
torus geometry in terms of the Mathieu functions. We obtain a perfect
quantitative agreement with the NEGF results. For nanotubes with an armchair
edge, already a weak bending of 1% substantially changes the electronic
properties. Depending on the valley, the current of the zero mode flows either
on the outer or the inner side of the torus and, therefore, can be used as a
valley splitter. In contrast, the zigzag nanotubes are largely unaffected by
the bending. Our findings are of importance for nanoelectronic applications of
carbon nanotubes and open new possibilities for valleytronics.
