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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
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