Abstract In this article Hasimoto surfaces in Galilean space $$G_{3}$$ G 3 will be considered, Gauss curvature ( K ) and Mean H of $$\chi =\chi (s,t)$$ χ = ( s , t</m...

The paper studies invariants for curves in three dimensional equiform Galilean geometry. We obtain Lie algebra structure space and concrete expressions of curve using Fels-Olver’s moving frame method. corresponding relationship between is received, the Frenet formula also showed.

‎In the present study‎, ‎we consider a regular curve in Galilean‎ ‎$4$-space $mathbb{G}_{4}$ whose position vector is written as a‎ ‎linear combination of its Frenet vectors‎. ‎We characterize such‎ ‎curves in terms of their curvature functions‎. ‎Further‎, ‎we obtain‎ ‎some results of rectifying‎, ‎constant ratio‎, ‎$T$-constant and‎ ‎$N$-constant curves in $mathbb{G}_{4}$‎.

We develop the local theory of surfaces immersed in the pseudo-Galilean space, a special type of Cayley-Klein spaces. We define principal, Gaussian, and mean curvatures. By this, the general setting for study of surfaces of constant curvature in the pseudo-Galilean space is provided. We describe surfaces of revolution of constant curvature. We introduce special local coordinates for surfaces of...