Let's consider the projective spaces $CP^{1}$ and $CP^{2}$.Suppose $CP^{1}$ is the subspace of $CP^{2}$,then we have the normal bundle of $CP^{1}$ in $CP^{2}$. My question is that how can we get the transition functions of this normal bundle under the standard cover of projective space.
I know that the normal bundle should be the tangent vectors which are perpendicular to the tangent spaces of submanifold intuitively.Locally $CP^{1}$ in $CP^{2}$ is like $C^{1}$ in $C^{2}$ .At this point ,I don't know how to continue in accurate math language? Is my idea right or am I misunderstanding something?