The lack of a serpentine belt will immediately cause the power steering pump to stop functioning, resulting in very stiff steering. However, this doesn't mean the pump itself is broken. It simply isn't being turned. The same applies to other components driven by the belt.
Possible reasons the serpentine belt came off:
* Broken belt: The belt itself may have cracked, frayed, or broken. This is the most likely cause.
* Loose or worn pulley: A pulley (on the power steering pump, alternator, etc.) could be loose or worn, causing the belt to slip off.
* Worn tensioner: The tensioner keeps the belt taut. A faulty tensioner can allow the belt to loosen and come off.
* Seized bearing in a driven component: A seized bearing in one of the components driven by the belt could put too much stress on the belt and cause it to come off. This would eventually damage the component but the immediate cause of the belt coming off would still be the seized bearing.
In short: Get the serpentine belt back on. If the steering remains stiff, then you *might* suspect a problem with the power steering pump, but other issues like low power steering fluid could also be the problem. The gearbox is completely unrelated to the serpentine belt; it's driven by a separate system.
