I enter my comment as an answer. The paper On the normal bundle of submanifolds of $\mathbb{P}^n$ by Lucian Badescu contains some answers to the question. Here are the links:
Published version: http://www.ams.org/journals/proc/2008-136-05/S0002-9939-08-09255-1/
On arXiv version: http://arxiv.org/pdf/math/0701487v1.pdf