We study curves of AW(k)‐type in the Lie group G with a bi‐invariant metric. Also, we characterize general helices in terms of AW(k)‐type curve in the Lie group G.

In this work, we study Hasimoto surfaces for the second and third classes of curve evolution corresponding to a Frenet frame in Minkowski 3-space. Later, we derive two formulas for the differentials of the second and third Hasimoto-like transformations associated with the repulsive-type nonlinear Schrödinger equation.

In the present paper, we study helicoidal surfaces in the three-dimensional isotropic space $I^3$ and construct helicoidal surfaces satisfying a linear equation in terms of the Gaussian curvature and the mean curvature of the surface.

Our principal goal is to study the prescribed curvature problem in a manifold with density. In particular, we consider the Euclidean 3-space ${\mathbb{R}}^3$ with a positive density function $e^{\varphi }$ , where $\varphi =-x^2-y^2$ , $(x,y,z)\in {\mathbb{R}}^3$ and construct all the helicoidal surfaces in the space by solving the second-order non-linear ordinary differential equation with the weighted Gaussian curvature and the mean curvature functions. As a result, we give a classification of weighted minimal helicoidal surfaces as well as examples of helicoidal surfaces with some weighted Gaussian curvature and mean curvature functions in the space.