How to redirect to different url without changing url?

I have Domain A, and Domain B. Domains B is an alias/abbreviation for Domain A. I can use domain hosting provider to set this up. But is there a way to do it in nodejs.

So if user comes to Domain B then page that shows up is of Domain A. But url remains of Domain B.

And if possible there should be a way for all links in that site to point to Domain B (either using javascript or using server side templating if it's possible for server to know where did original request originated from).

I guess main question here is what 3xx header is to be used and how.

Edit: it doesn't have to be different domain, it can be different subdomain. or url like domain.com/thisisbigurlthatmaycontainmoreurls/ vs domain.com/tibutmcmu/