Helicoids in 𝕊2 × ℝ and ℍ2 × ℝ

We provide two characterizations of helicoids in S{double-struck}² × R{double-struck} and in H² × R. First, we show that any nontrivial ruled minimal surface in S{double-struck}² × R{double-struck} and in H{double-struck}² × R{double-struck} is a part of a helicoid. Second, we also show that these surfaces can be characterized as the only surface with zero mean curvature with respect to both the Riemannian product metric and the Lorentzian product metric on S{double-struck}² × R{double-struck} or H{double-struck}² × R{double-struck}.