By means of a special choice of coordinate lines of the surface in four-dimensional Galilean space, the first and second quadratic shape of the surface is defined. It has been proved that the second-order surface equation in three-dimensional space can be converted to a canonical form by means of a special transformation, which is the rotation of the coordinate axes of three-dimensional Galilean space. Furthermore, the transformation matrix is an element of the Heisenberg group that is neither symmetric nor orthogonal. In four-dimensional space R41 - the concept of a surface indicator is introduced and the main curvature of the surface is defined.
Magnetic curves in the generalized Heisenberg group
