With the aid of particle image velocimetry (PIV) momentary two-dimensional velocity fields of dilute cationic surfactant solutions are measured in an annular gap between coaxial cylinders. In the shear-induced state the velocity profiles of the surfactant solutions differ from a Couette profile drastically: Time dependent flows with high shear rates near the walls are observed. The flow is almost one-dimensional, velocity components in radial or axial direction cannot be detected. The velocity profile can exhibit a local maximum. A macroscopic heterogeneous structure is observed, domains with high viscosity alternate with low viscosity regions. Elastic phenomena are observed in a relaxation experiment. Existing rheometric data of cationic surfactant solutions in the shear-induced state give only apparent viscosities.