A surface ${\mathcal{M}}^2$ with position vector $r=r(s,t)$ is called a Hasimoto surface if the relation $r_t=r_s\wedge r_{ss}$ holds. In this paper, we first define the Beltrami-Laplace operator according to the three fundamental forms of the surface, then we classify the $J$-harmonic Hasimoto surfaces and their Gauss map in ${\mathbb{E}}^3$, for $J=II$ and $III$.

In this paper, we deal with the natural geometry of quadric surfaces in the isotropic space ${\mathbb{I}}^3$. We provided defining equations for the $3^{rd},4^{th}$ type ruled surfaces and $1^{st},2^{nd}$ type of quadric surfaces. In this paper, we investigated the finite Chen-type of ruled and quadric surfaces in the 3-dimensional simply isotropic space ${\mathbb{I}}^3$, corresponding to the third fundamental form of the surface.