On March 19, 2002, at approximately 1355 mountain standard time, a Cessna 310B, N5381A, was substantially damaged when its nose landing gear collapsed during landing roll at Provo Municipal Airport (PVU), Provo, Utah. The airline transport instructor pilot and commercial pilot receiving instruction were not injured. Visual meteorological conditions prevailed for this local training flight being conducted under Title 14 CFR Part 91. The flight originated at approximately 1130. According to the instructor, the landing gear failed to fully retract after takeoff. He stated that after lift-off, and with "no more usable runway," we "retracted the landing gear, looking for a green light gear indicator." The light did not come on. He looked outside at the mirror on the left main tank and identified the "gear malfunction." They continued the local flight to burn off excess fuel and then made a precautionary landing at Provo. During the landing roll, the nose landing gear collapsed. The airplane sustained substantial damage to the left and right forward nose landing gear well former (keel beam) assemblies. According to the manufacturer, the Cessna 310B, is equipped with a landing gear position light system. Two landing gear position lights are provided, one above and one below the landing gear switch. The "upper light is red, and is on at all times when the gear is retracted." The "lower light is green, and illuminates only when the landing gear is fully extended and locked." If the landing gear is not fully extended and locked, the green light will not illuminate, and if neither light is on, the landing gear is in an intermediate position.