Narrow Results

In this study, we examine the Umbrella matrix group in Galilean space, which possesses the stochastic matrix property (i.e. leaves the $1={\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill ...\hfill & \hfill 1\hfill \end{array}\right]}^T\in {ℝ}_1^n$ axis fixed) within matrix groups. In our examination, we first obtain the matrix Lie group and Lie algebra in $G^3$ space and then generalize this result. Furthermore, we present the Cayley formula, which gives the transition between the $SO(n)$ Lie group and Lie algebra, for the first time between the Galilean Umbrella matrix Lie group and Lie algebra. Then, we define a case of shear motion along the $1$ axis with the help of a special Galilean transformation and generate rotated surfaces in Galilean space using certain curves.

In this study, we handle the spherical product surface in Galilean $3$-space ${𝔾}_3$. We calculate the Laplacian operator of the Gauss map of this surface. Then we give the necessary and sufficient conditions for spherical product surface to have harmonic Gauss map and we give a visualization of this type surface in ${𝔾}_3$. Furthermore, we make a classification of spherical product surface having pointwise 1-type Gauss map in ${𝔾}_3$. Lastly, we visualized this type spherical product surface of the first kind in ${𝔾}_3$.

In this study, we provide a brief description of rotational surfaces in 4-dimensional (4D) Galilean space using a curve and matrices in $G_4$. That is, we provide different types of rotational matrices, which are the subgroups of $M$ by rotating a selected axis in $E^4$. Hence, we choose two parameter matrices groups of rotations and we give the matrices of rotation corresponding to the appropriate subgroup in Galilean 4-space and we generate rotated surfaces.

In this paper, we derive the necessary and sufficient conditions of surface pencil pair interpolating Bertrand pair as common asymptotic curves in Galilean space G₃. Moreover, the conclusion to ruled surface pencil pair is also obtained. And the examples are given to confirm that the proposed methods are effective in product manufacturing by adjusting the shapes of the surface pencil pair.

In this paper, we investigate the flow of curve and its equiform geometry in 4-dimensional Galilean space. We obtain that the Frenet equations and curvatures of inextensible flows of curves and its equiformly invariant vector fields and intrinsic quantities are independent of time. We find that the motions of curves and its equiform geometry can be defined by the inviscid and stochastic Burgers’ equations in 4-dimensional Galilean space.