Оne-sided Surfaces in E4
Abstract
Let a normal vector be defined along a closed curve on the surfaces. If we return to the starting point and the normal direction coincides with the original, regardless of the choice of the curve, then the surface is called bilateral. Otherwise, we have a one-sided surface. The Mobius band is a one-sided surface in E3. A cross-cap and a Klein bottle are also one-sided surfaces in E3. In the paper, we present equations of the Mobius band, the Klein bottle and the cross-cap in E4. We define a closed curve s = s(v) and the normal vector n = n(v) that goes along the curve s and returns to the starting point with an opposite direction. The examined surface is the one-sided surface. With the help of a computer mathematics system we calculate normal curvature indicatrixes for examined surfaces along the defined closed curve. We demonstrate that in the case of the Mobius band and the Klein bottle it is a line segment or an ellipse passing through points of the curve. For the case of the cross-cap it is an ellipse that does not pass through points of the curve. Additionally, we plot a graph of scalar curvature for the Mobius band, the Klein bottle, and the cross-cap.
DOI 10.14258/izvasu(2015)1.1-21
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