Bulletin of the
Korean Mathematical Society BKMS

It is well known that the helicoids are the only ruled minimal surfaces in $\RRR ^3$. The similar characterization for ruled minimal surfaces can be given in many other 3-dimensional homogeneous spaces. In this note we consider the product space $M\times\RRR$ for a 2-dimensional manifold $M$ and prove that $M\times\RRR$ has a nontrivial minimal surface ruled by horizontal geodesics only when $M$ has a Clairaut parametrization. Moreover such minimal surface is the trace of the longitude rotating in $M$ while translating vertically in constant speed in the direction of $\RRR$.

Keywords: ruled surface, minimal surface, helicoid

MSC numbers: 53A35