How should I act when I meet one of my female relatives in a public place (such as a family gathering, university, public transportation, or on the street), whether she is a mahram or not, in general, and especially with one whom I love?

It is permissible for a man to greet and shake hands with his mahram (definition) relatives and to serve them. As for non-mahram women, it is disliked to initiate the greeting, especially when there is a fear of fitna (temptation or discord). One must lower their gaze and not speak to them except out of necessity. It is forbidden to speak to them or look at them if there is a fear of fitna or infatuation.