RULED MINIMAL SURFACES IN PRODUCT SPACES

It is well known that the helicoids are the only ruled minimal surfaces in ${\mathbb{R}}^3$. 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 $M{\times}{\mathbb{R}}$ for a 2-dimensional manifold M and prove that $M{\times}{\mathbb{R}}$ 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 ${\mathbb{R}}$.