Don't confuse your own religious views with those of everyone else. Probably your first mistake.

Have you considered the possibility that sometimes people give their spouse permission to have an affair?

Have you also considered the fact that you can get married without taking such an oath? A wedding ceremony is not how you get married. You get married by signing a document that says you are marrying the other person who signs the document, and then filing the document with the government. Marriage is a legal arrangement that primarily covers property rights and certain legal decisions (e.g. the presumption that your spouse can make decisions on your behalf if you are incapacitated). Even in the few remaining places where cheating has any meaningful legal significance divorce lawyers often advise clients not to bother -- it is rarely worth pursuing and most people just want to finish the divorce as quickly as possible and all filing for fault will do is raise your legal fees (for little tangible benefit).

I don't remember any oath about not sleeping with another woman, in my language and culture. Maybe there is one in his language in culture, but I am not sure about that. There is one about being together in good and bad times, but "cheating" does not imply separation or divorce, just having some time with someone else. It is even something agreed in a marriage in certain circumstances of health or age.

I have no idea what oath or commitments he made with his wife

Good point. Most people do make oaths to our partners. "Till death do we part."

But then we culturally decided divorce doesn't count. "Till death do we part" became "Till death or major or minor inconvenience or whim do we part." Then culturally we decided that marrying an oath-breaker is just Tuesday.