There are a few things that make a track bike suitable for use on the velodrome. First is the geometry. Since that’s quite a complex topic, there’s a spectrum of suitable geometry and this is already a track bike, let’s set that aside. Second is that it is fixed gear and has no brakes. Riders on the track are very close to each other and you don’t want anyone decelerating quickly which would cause a crash. Third the wheels must bolt on to the frame. Quick release skewers aren’t allowed because the high torque sprinting can cause the wheel to pull out of the dropouts, which should be track style drops to adjust chain tension. These three points just take a quick exam of the bike to confirm.
Fourth is something many people miss: a bike ridden on the velodrome needs to have wheels and tires capable of higher Gs than typical and they’re directed sideways.
A typical road bike only sees 1G on a regular basis with some shock loading when you hit a pothole. But on the track, especially shorter tracks, the wheels will see increased Gs in every turn. Those high forces are in an angled sideways direction instead of vertical. The sideways force wants to rip your tire off the rim. The forces fighting that are from the tire pressure forcing the tire bead against the rim wall.
For example, take a cyclist at 30 mph (not hard to do on the track) and a 166 m track length (like the Lexus Velodrome, which is on the short side). The turn radius is roughly 13 m. That results in 1.4 Gs. If the rider is at 40 mph they would experience 2.5 Gs every 4.6 seconds. It’s intense!
To enable such high acceleration turns, the turns (and even the straights) of the track are banked (50 degrees in the turns on this 166 m track). Cyclists do need lean at an angle to stay upright. However, the lean angle and timing won’t match the track perfectly, so there are still increased lateral forces. And there will be increased radial forces on the wheel as well. The radial forces could cause a pinch flats with too low of a pressure.
Thus it’s recommended to use high tire pressures on the velodrome. Track tires are specifically designed for this and are often rated for 125-175 PSI (9-11 bar). However be very careful to consider the pressure limit of your rims too. The maximum pressure you can safely run is the lower value of the limits of your tires and rims.
The tires used on the track are quite narrow to minimize rolling resistance. A super smooth surface like a wood track has no benefit from wide tires or low pressures. Those would only increase rolling resistance. Typical tire width for velodrome racing is 19-23 mm. Both tubular and clincher type tires and wheels can be used.