No, you would not be obligated to pay it. Whether or not the manager had actual authority to offer this discount, she had "apparent authority": a reasonable outsider, like a tenant, would believe the apartment manager had the authority to do this, and it would be common or normal for a manager to have this authority. Therefore, you were entitled to rely on her offer of a rent discount, and her apparent authority binds her employer, the landlord. If the owner wants, it may be able to recover that money from the manager herself, if she clearly exceeded her authority, but that is not your concern.