Is this Jewish tradition? What's the traditional greeting in Breslev or Chasidism with hand or no?

Because of the sanctity of marriage in Judaism, men and women who are not married do not have friendly physical contact, not even shaking hands, but rather it's accepted to smile and bow slightly in mutual respect. In certain cases, if the other person doesn't know Jewish religious law, and puts out their hand to shake, and it would cause public embarrassment (which is also prohibited) not to shake their hand, some halachic authorities allow to shake their extended hand, so as not to humiliate them in public (unless the situation leaves room for explanation of the Jewish law, which will be understood by all those around, and will alleviate the embarrassment). Men can shake hands, hug or kiss men (obviously not on the lips!), and women to women, according to local custom and manners. It is not considered respectful, even for a man to hug or kiss a respected rabbi, unless they are personally already close, but the Sephardic Jews from Oriental communities will often respectfully kiss the hand of great rabbis. Also in ancient times, the Jewish greeting is to say: Shalom, but I'm not familiar regarding any source indicating whether our forefathers shook hands or not. Today, it's becoming more common to see friendly men hug eachother as a greeting, and so it is among chassidim, especially Breslav, as well.