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Let $S \subset \mathbb{P}^g$ be a smooth $K3$ surface of degree $2g-2$, $g\geq 3$. We classify all the cases for which$h^0(\mathcal{N}_{S/\mathbb{P}^g}(-2)) \neq 0$ and the cases for which$h^0(\mathcal{N}_{S/\mathbb{P}^g}(-2)) <h^0(\mathcal{N}_{C/\mathbb{P}^{g-1}}(-2))$ for $C \subset \mathbb{P}^{g-1}$ ageneral canonical curve section of $S$.

Global sections of twisted normal bundles of K3 surfaces and their hyperplane sections