HELICOIDS IN S-2 x R AND H-2 x R

Abstract

We provide two characterizations of helicoids in S-2 x R and in H-2 x R. First, we show that any nontrivial ruled minimal surface in S-2 x R and in H-2 x R 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-2 x R or H-2 x R.