The pilot of a retractable-landing gear-equipped airplane reported that he landed with the landing gear extended. He added that, during the landing, the airplane bounced, and the "landing gear retracted and the aircraft settled onto its belly." The airplane sustained substantial damage to the fuselage. The pilot reported that there were no preaccident mechanical failures or malfunctions with the airplane that would have precluded normal operation. The pilot reported that he "may have failed to deploy the landing gear" and did not recall hearing the gear warning horn. A flight instructor witness reported that, while his student was performing an engine run-up, he saw the airplane on final about 400 ft from the threshold with the landing gear retracted. He added that he thought the airplane was going to do a low approach, so he turned to check on his student, but when he looked back up, he saw "the [airplane] was on the runway" sliding. The Federal Aviation Administration inspector reported that, when he arrived on scene, he observed the gear handle in the "up" position.