Geometry of umbrella matrices in Galilean space

In this study, we examine the Umbrella matrix group in Galilean space, which possesses the stochastic matrix property (i.e. leaves the [Formula: see text] axis fixed) within matrix groups. In our examination, we first obtain the matrix Lie group and Lie algebra in [Formula: see text] space and then generalize this result. Furthermore, we present the Cayley formula, which gives the transition between the [Formula: see text] 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 [Formula: see text] axis with the help of a special Galilean transformation and generate rotated surfaces in Galilean space using certain curves.