Cessna 310 pilot reported the nose gear did not fully extend. During landing the aircraft nose contacted the ground resulting aircraft damage. No injuries were reported.

Narrative

The flight operated under Part 91 in day VFR conditions; departing from Runway XX. After a normal takeoff and positive rate of climb; the landing gear was selected up. Shortly after retraction; an unusual noise was heard; which appeared to originate from the floor of the aircraft.As a precaution; I slowed and re-extended the landing gear. The gear did not indicate a down-and-locked condition. When looking at the mirror on the left nacelle; I noticed the nose gear was not fully forward. I requested a visual confirmation from the tower and performed a low pass over Runway XY. The tower confirmed that the nose gear was not extended; however; the mains seemed to be extended.I prepared for a landing with no nose gear and advised ATC. On final approach to Runway XY; I configured the aircraft for a minimal-energy touchdown and kept the nose off the ground as long as possible. The main gear touchdown was uneventful. As the nose lowered; it made contact with the runway; and the aircraft came to a controlled stop on the centerline.There were no injuries. Post-flight inspection revealed that the damage was limited to the nose gear doors and propeller blades due to ground contact. No structural airframe damage was observed. The FAA and NTSB called me shortly after getting the plane put back in the hangar.