Name on GC is different from Passport

[randy_gc]

In India, the passport office made a mistake in writing my name years ago, and I continued with that name for long, and received GC with that name.

After applying for Citizenship, in the hope that I may get my citizenship soon, I got my name corrected from a local court.

While waiting for Citizenship, my Indian passport came up for renewal. This time I submitted the court documentation for name correction and received the Indian passport with the correct name.

Now the name on GC does not match with the passport.

Does it create any problems during travel? while leaving USA or entering USA?
Do I need to apply for GC renewal/re-issue with the correct name?

Thanks in advance.