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Surfaces in $${\mathbb{S}^2\times\mathbb{R}}$$ with a canonical principal direction. (English) Zbl 1176.53031
First there is shown how one can define local coordinates on a surface in $$\mathbb{S}^2\times\mathbb{R}$$ that are adapted to the structure of $$\mathbb{S}^2 \times\mathbb{R}$$, and, in special, when the surface is minimal. It is proved that all flat and minimal surfaces are open parts of vertical cylinders on a geodesic in $$\mathbb{S}^2$$, which means surfaces for which the angle between the unit normal and the $$\mathbb{R}$$-direction is everywhere equal to $$\frac{\pi}{2}$$ and for which the intersection with $$\mathbb{S}^2$$ is a great circle. The condition is investigated that the projection of the canonical unit vector tangent to the $$\mathbb{R}$$-direction onto the tangent space of an immersed surface is a principal direction. It is shown that this is equivalent to the condition that the surface is normally flat if a surface in $$\mathbb{S}^2\times\mathbb{R}$$ is considered as a codimension 2 immersion of a surface in $$\mathbb{E}^4$$. Moreover, a characterization of these surfaces is given and a classification theorem under the additional assumption of minimality or flatness.

##### MSC:
 53B25 Local submanifolds
##### Keywords:
minimal surfaces; product manifolds; principal directions
Full Text:
##### References:
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