In this paper we study the Gauss map of surfaces in three-dimensional Heisenberg group using the Gans model of the hyperbolic plane. We establish a relationship between the tension fields of the Gauss maps and the mean curvatures of the surfaces in $\mathcal{H}_3$.

In the present article, an analysis was performed on the torque-free motion of a rigid body, developing Euler’s analytical solution and Poinsot’s geometric solution. The analytical solution for the angular velocity and Euler’s angles was described given some initial conditions. Besides, an animation of Poinsot’s geometric solution was elaborated and a study was carried out on the conditions in which the herpolhode forms a closed curve. Finally, an algorithm was developed that displays the obtained solutions, which also generates an animation of the geometric solution, and moreover provides an algorithm that produces closed herpolhodes.

This paper characterizes primitive substitution tiling systems for Lie groups for which every element of the associated tiling space generates coherent frames for corresponding representation spaces.

Here we demonstrate some of the benefits of a novel parameterization of the Lie groups $\mathrm{Sp}(2,\mathbb{R}) ≅ \mathrm{SL}(2,\mathbb{R})$. Relying on the properties of the exponential map $\mathfrak{sl}(2,\mathbb{R}) \rightarrow \mathrm{SL}(2,\mathbb{R})$, we have found a few explicit formulas for the logarithm of the matrices in these groups.

Additionally, the explicit analytic description of the ellipse representing their field of values is derived and this allows a direct graphical identification of various types.

The purpose of present paper is to study cosymplectic manifolds admitting certain special vector fields such as holomorphically planar conformal (in short HPC) vector field. First, we prove that an HPC vector field on a cosymplectic manifold is also a Jacobi-type vector field. Then, we obtain the necessary conditions for such a vector field to be Killing. Finally, we give an important characterization for a torse-forming vector field on such a manifold given as to be recurrent.