Problem: Consider a smooth, projective variety $X$ in $\mathbb{P}^n$, consider a vector bundle $E\xrightarrow{\pi}X$ os rank $r$. Suppose that $\mathbb{P}^k$ is a subvariety of $X$, and we know that for example $E|_{\mathbb{P}^k}=\mathcal{O}_{\mathbb{P}^k}(a)^{\oplus s}$ for some $a,s$ (I'm trying to be as vague as possible).
Question: Is there any chance to describe $E$ (or some hypothesis to put in order to have a result)? How can we parametrize the possible bundles on $X$ whose restrictions to a certain projective space contained in $X$ have a certain beahviour? (can we even do better? replacing $\mathbb{P}^k$ with a smooth projective subvariety $Y$?)
I apologize in advance forr the vagueness of the question: I'm often computing bundles but I always know some restrictions of them, thus I was wondering if there is some theory about it (maybe it's a well known area of research, or there's a preicse reference I don't know of: I tried to look online unsuccessfully). Thanks in advance!