Why didn't Norah Jones use her connection to THE Ravi Shankar?  

Maybe to prove that she's a great musician herself and not use someone else's name to talk people into liking her.

As far as I know, Norah was never close to her father. Her parents split when she was a child and she grew up with her mother in the US.

Like many others I think she wanted to be known for her body of work not the fact of to whom she is related. I am not sure that many people today have any idea who Ravi Shankar is.

I read somewhere that Nicolas Cage is actually related to Coppola. But he never uses this relation to get where he is right now. He made a short film when he was young already, with him and I think his brother in it. There could be several reasons why people do not use relationship to get to the top, one of them is that they believe in their talents even though no one else does. I admire people like these, thanks for telling us that Norah is one of them. I don't have her albums but I've heard her songs and I love them