Invariant Surfaces under Hyperbolic Translations in Hyperbolic Space

Abstract

We consider hyperbolic rotation ($G_0$), hyperbolic translation ($G_1$), and horocyclic rotation ($G_2$) groups in ${\mathbb{H}}^3$, which is called Minkowski model of hyperbolic space. Then, we investigate extrinsic differential geometry of invariant surfaces under subgroups of $G_0$ in ${\mathbb{H}}^3$. Also, we give explicit parametrization of these invariant surfaces with respect to constant hyperbolic curvature of profile curves. Finally, we obtain some corollaries for flat and minimal invariant surfaces which are associated with de Sitter and hyperbolic shape operator in ${\mathbb{H}}^3$.