Suppose that $V = \mathcal{O}(n_{1}) \oplus \dots \oplus \mathcal{O}(n_{k})$ is a vector bundle on $\mathbb{P}^{m}$ and Let $X = \mathbb{P}(V)$, with projection $\pi: X \rightarrow \mathbb{P}^{m}$, Let $D = \pi^{-1}(H)$ where $H$ is a hyper plane. Note that $D$ is isomorphic to a projectivised bundle on $\mathbb{P}^{m-1}$
Is there a canonical description of the normal bundle of $D$ in $X$?
($k = \mathbb{C}$ )
$$ N_{D/X} = N_{\pi^{-1}(H)/X} \cong \pi^*N_{H/\mathbb{P}^n} \cong \pi^*O(1). $$