Phil, no there are no rims to keep it on. They are very tighly stretched. I think the crowing that keeps the blade in the middle of the wheel also keeps the tire on... The seller of the tires, Sulfer Grove, says that you can glue them down if you want, but probably isn't necessary. I pushed some thick old dry hard oak through to see if there would be any issues...and NONE. Worked like a charm.
This system is basically two tires and two "rings". You put the rings in the grooves, then the tires over the rings. The rings are slightly thicker (but the same width) as the groove. So, they protrude from the wheel surface a little...creating the crown when the tire is put on. The 1/2" strip of duct tape creates just a little more crown. You can add multiple layers of duct tape to create more crown if you want/need.