After a cross-country flight, the flight crew configured the airplane for landing, but the nose landing gear did not extend. The flight crew attempted to trouble shoot the problem by cycling the landing gear several times, with no results. The flight crew also tried to lower the nose landing gear with the hydraulic hand pump, but that was not successful. The flight crew went through the pre-landing checklist one more time, but the nose landing gear still did not come down. After trying unsuccessfully to extend the nose landing gear, the flight crew decided to perform a landing with the nose landing gear still retracted and the main landing gear extended. The airplane touched down on the runway and a section of the fuselage just above the retracted nose landing gear sustained substantial damage. The pilot mentioned in his statement that the nose landing gear doors might have been frozen shut because the flight originated from an area of high moisture, and the landing gear was lowered in an area of very cold temperatures. This could not be proven and the manufacturer stated that they did not have any previous reports of the nose landing gear doors freezing shut with this model of airplane. After the accident, the landing gear was tested using both the handpump and low and high hydraulic pressure in the airplane systems. All landing gear extended and retracted normally and provided the appropriate landing gear configuration indications on the instrument panel. There were no mechanical anomalies with the landing gear system that would have precluded normal operations.