  We consider magnetic properties of a long, thin-walled ferromagnetic
nanotube. We assume that the tube consists of isotropic homogeneous magnet
whose spins interact via the exchange energy, the dipole-dipole interaction
energy, and also interact with an external field via Zeeman energy. Possible
stable states are the parallel state with the magnetization along the axis of
the tube, and the vortex state with the magnetization along azimuthal
direction. For a given material, which of them has lower energy depends on the
value \gamma=R^2d/(L \lambda_x^2), where R is the radius of the tube, d is its
thickness, L is its length and \lambda_x is an intrinsic scale of length
characterizing the ration of exchange and dipolar interaction. At \gamma<1 the
parallel state wins, otherwise the vortex state is stable. A domain wall in the
middle of the tube is always energy unfavorable, but it can exist as a
metastable structure. Near the ends of a tube magnetized parallel to the axis a
half-domain structure transforming gradually the parallel magnetization to a
vortex just at the edge of the tube is energy favorable. We also consider the
equilibrium magnetization textures in an external magnetic field either
parallel or perpendicular to the tube. Finally, magnetic fields produced by a
nanotube and an array of tubes is analyzed.
