Consider an abstract surface $(M,g)$ embedded into $\mathbb{R}^3$ via $f:M \to \mathbb{R}^3$. Denote by $N:M \to \mathbb{R}^3$ the Gauss map (normal field) of the surface. Write the Laplace Beltrami operator as $\Delta^{(1)}$.
Define the *vector* Laplace Beltrami operator acting on the normal field as

$$\Delta^{(3)} N := (\Delta^{(1)}N_1,\Delta^{(1)}N_2,\Delta^{(1)}N_3).$$

For Euclidean space, this concept is described e.g. in [1].

Is the extension to surfaces meaningful? What can we say about this operation on the Gauss map? Can we for example characterize normal and tangent part of $\Delta^{(3)} N$?