David Remnick's bestseller The Bridge: The Life and Rise of Barack Obama doesn't mention the white New York girlfriend — that's too interesting for Remnick to pursue. Remnick's audience is primarily white women who have a bit of a thing for the President, so a discussion of Obama's disquiet about interracial relationships would be a downer for Remnick's readers.
But Remnick does say about his first Chicago stay (1985-1988):
"Toward the end of his time as an organizer, Obama had a steady girlfriend, a white University of Chicago student who was studying anthropology."
This is interesting for a couple of reasons. First, she sounds much like his mom, who got accepted by the U. of Chicago at age 15 back when they took younger students (but she didn't go) and got a Ph.D. in anthropology.
Second, according to his book, he had previously realized in New York that it wouldn't work with a white girl, so why another white girlfriend? (This is assuming that they aren't the same person, whom Obama has melded together for whatever reason). One might think that a 656 page biography might ask those kind of questions, but it doesn't.