Timelike meridian surfaces of hyperbolic type in the Minkowski 4-space

We consider a special class of timelike surfaces in the four-dimensional Minkowski space which are one-parameter systems of meridians of rotational hypersurfaces with spacelike axis and call them meridian surfaces of hyperbolic type. We show that all timelike meridian surfaces of hyperbolic type are surfaces with flat normal connection and give the complete classification of those surfaces with constant Gauss curvature. We also classify all minimal timelike meridian surfaces of hyperbolic type and all timelike meridian surfaces of hyperbolic type with non-zero constant mean curvature (CMC-surfaces). We show that there are no timelike meridian surfaces of hyperbolic type with parallel mean curvature vector field other than CMC surfaces lying in a hyperplane. Finally, we describe the class of timelike meridian surfaces of hyperbolic type with parallel normalized mean curvature vector field, but non-parallel mean curvature vector.