One of the five passengers on the doorless helicopter hovering high above Manhattan had just leaned back toward the pilot to snap a photograph of his feet dangling in midair — a coveted shot known on social media as a “shoe selfie” — when the trouble started.

The pilot heard and saw warnings that the single engine of the AS350 Eurocopter was losing power. He considered trying to land in Central Park but decided that there were “too many people,” he told federal investigators, according to a report issued on Monday.

So, after yelling at the passengers to return to their seats, he circled around and glided toward the East River, making a mayday call as he cleared the buildings at the island’s edge. The pilot, Richard Vance, said he deployed the floats on the helicopter’s skids in preparation for impact, then reached down for the emergency fuel shut-off lever.

But he found that it had already been tripped, and beneath it he spotted a possible culprit, the nylon tether that had allowed his front-seat passenger to move around the cabin and take pictures unimpeded by doors or windows — or a seatbelt. Mr. Vance said that he moved the lever back to the on position and tried to restart the engine.