What is the ruling regarding a man’s oath to his wife, stating that "she is forbidden to him if she does such and such," and then the wife did this thing out of forgetfulness and inadvertence without his knowledge, and what are the implications of this oath?

The prohibition of the wife depends on the intention of the one who utters it. Scholars differ on whether the wife's forgetfulness of the action sworn against affects the breaking of the oath. The Shafi'is hold that if the wife was concerned about the matter and did it out of forgetfulness, it does not harm; however, if she was not concerned about it at all and did it, it is considered a breaking of the oath. If the wife performs the action after the reason you wanted to prevent her from it has ceased, then nothing is incumbent upon you.