Your question is not entirely clear to me, but if it is necessary you can likely hire a local attorney to guide you through the process of an official name change in state court and then submit a certified copy of the judgment that corrects your name to USCIS. If you are trying to change the birth certificate itself, you may have to do that in Mumbai. Best of luck!
These are general thoughts - they not intended as legal advice - so you need and should contact an Attorney for actual advice on this situation. This post and all others I make on AVVO are for informational purposes only. Therefore, nothing I post as comments, answers, or other communications should be taken as legal advice for any individual case or situation. Postings on AVVO are not intended to create, and receipt or viewing of this information does not constitute, an attorney-client relationship.