Yes, you should definitely report your injury to the Property Manager, if you’re injured in a slip and fall. The reason being, that by doing that, you give notice to the Property Manager, and through them, the owner, that an incident occurred, that you were injured. They would, hopefully, and they should, document it. They may even give you some report or something that states that they acknowledge that you did report it right away. The difference being that if you didn’t report it right away, they could later state that we know nothing about what happened, no one told us, and they could more plausibly deny that anything happened at all, in fighting against your case.
So definitely approach them right away, if possible, if you’re not, for example, taken to the emergency room in an ambulance, and you couldn’t do it. That would be different. You should go back as soon as possible to report it, or maybe call them, or have some family member or friend call them, but you want to make a record as quickly as you can, of the accident with the other party.