Weingarten equations for surfaces on Helmholtz-type groups

In this paper, we study surfaces on three-dimensional Helmholtz-type Lie groups that define the actions of groups of motions of Helmholtz geometries, which are geometries of local maximal mobility. In this paper, we present left-invariant metrics and Levi-Civita connections for these Lie groups, which were found earlier. For surfaces of Helmholtz-type Lie groups, we calculate the spinors that generate them, which satisfy the Dirac and Weingarten equations. We also derive compatibility conditions for the Weingarten equations.