We'll tell you that every position in every theater is different; what a PM does in one place can have an entirely different set of responsibilities at another.
The PM where I work has little to do directly with rehearsals and performances as they are going. He is responsible for the overall upkeep of the spaces, safety procedures, hiring production, making sure protocol is met, scheduling of production strikes/load-ins, the production budget...he's sort of the PSM of production dept.