Name on birth certificate different than green card and passport
I have always emphasized the importance of keeping your name consistent on all documents. If you have identical names on birth certificate (a very importance piece of document if you want to immigrate to America but not important at all for non immigrant visas), green card, and native passport. I am hearing from many readers who started their American journey with a student or visitor visa, or maybe found a job on H1, and then graduated to a permanent resident card. It is when they apply for naturalization that they realize that if there is even a minor discrepancy on your current documentation with a document that was issued at birth (which may have been decades ago for some), your application can be jeopardized or even delayed.
So what can you do if you realize that you have an inconsistency? Well, if you are starting out even with a tourist visa to the US, even though it may not be demanded, make sure that your birth certificate and passport have identical names. Once your name gets into an American database and assigned an alien number, the history will follow you for life. If you already made that error, start off by trying to get a new birth certificate that matches your documents in the American system. After all if your country has issued a passport with a certain name, they should be able to issue a birth certificate in that name as well, and if they cannot, use a court order to do so. In any case, do not apply for citizenship till you fix this mess.