It is well known that the helicoids are the only ruled minimal surfaces in ℝ3. 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 × ℝ for a 2-dimensional manifold M and prove that M ×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 ℝ.
Bibliographical noteFunding Information:
Heayong Shin was supported by NRF 2014R1A2A2A01007324.
© 2016 Korean Mathematical Society.
- Minimal surface
- Ruled surface
ASJC Scopus subject areas
- General Mathematics