Read the short help file section on Creating Projected Curves. I think this is a good way to create what you are looking for.
The basic idea is to make the base cylinder. Then create a sketch to represent the grooves. Then project the groove sketch onto the cylinder. Finish off by using the cut-sweep command to cut away the unwanted material.
It looks like the groove will need to be made with several cuts, don't try a single, complex path.
You might also be able to create the path by manually drawing a 3D sketch on the surface, but those are usually more difficult to do.