Abstract
We show a way to choose nice coordinates on a surface in \({\mathbb{S}^2 \times \mathbb{R}}\) and use this to study minimal surfaces. We show that only open parts of cylinders over a geodesic in \({\mathbb{S}^2}\) are both minimal and flat. We also show that the condition that the projection of the direction tangent to \({\mathbb{R}}\) onto the tangent space of the surface is a principal direction, is equivalent to the condition that the surface is normally flat in \({\mathbb{E}^4}\) . We present classification theorems under the extra assumption of minimality or flatness.
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J. Fastenakels is a research assistant of the Research Foundation—Flanders (FWO).
J. Van der Veken is a postdoctoral researcher supported by the Research Foundation—Flanders (FWO).
This work was partially supported by project G.0432.07 of the Research Foundation—Flanders (FWO).
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Dillen, F., Fastenakels, J. & Van der Veken, J. Surfaces in \({\mathbb{S}^2\times\mathbb{R}}\) with a canonical principal direction. Ann Glob Anal Geom 35, 381–396 (2009). https://doi.org/10.1007/s10455-008-9140-x
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DOI: https://doi.org/10.1007/s10455-008-9140-x