Tough question. You need to know exactly how you feel about various sexual orientations including your own. Your personal and family views as well as religious beliefs if any, have to be taken into account. You certainly have to have a heart to heart talk with your girlfriend once you are sure you know yourself and your family's positions. Sexual orientation is sensitive. You have a very difficult task ahead.
What difference. What is the difference if she were heterosexual? Monogamy isn't special to heterosexuals and trust is trust. How can she be sure you won't cheat on her with another woman? Maybe if you can answer that question, you can apply it to her. Keep the relationship happy, caring and loving to reduce any reason for cheating.