They're not the carrier. From the bottom of the page:
Gotham Air is not a direct air carrier but rather provides technology and information services enabling consumers to obtain aircraft charter and shared aircraft charter services from FAA certified and DOT registered air carriers who exercise operational control over all flights .
Exactly I've seen a few services like this advertised exactly the same way (WSJ ads some run by former industry execs).
Interesting thought that the picture of the heli contradicts that (says "Gotham Air" on the side). That's no small point actually. Gives someone who doesn't read the fine print the idea that they do operate the aircraft. I understand why they do that but it is a bit deceptive.
Wouldn't the livery on that helicopter in the photo just be a form of corporate advertising. Couldn't you paint your aircraft however you want so long as the correct registration / licensing marks are visible?
I would imagine that the chance of getting sued is quite small. What individual would sue and what harm would they claim? Government FTC most likely would not care and if they did all they would have to do is change the picture.
Likewise if there was an accident someone would have to say they relied on that information and was mislead. Unlikely (especially in the case of death how do you prove that?). Plus it is common practice to operate like this by similar parties.
A significant chunk of US airline flights are operated by a company other than the one whose logo is on the airliner. It mostly applies to the smaller regional "feeder" names, like United Express (actually operated by Cape Air, CommutAir, ExpressJet, and a bunch of others), and American Eagle (operated by AVAir, Chaparral Airlines, Command Airways, and more).
I have no idea if Gotham plans to actually operate that way, but I'd be shocked if there were actually a legal problem if they did.
Gotham Air is not a direct air carrier but rather provides technology and information services enabling consumers to obtain aircraft charter and shared aircraft charter services from FAA certified and DOT registered air carriers who exercise operational control over all flights .