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Mean directionally curved lines on surfaces immersed in \(\mathbb{R}^4\). (English) Zbl 1069.53006

The second fundamental form of a surface immersed in 4-space is a quadratic form on the tangent space with values in the normal space. It maps the tangential unit circle onto an ellipse, covered twice, with the mean curvature vector \(H\) as its center. Generically, there is a pair of orthogonal tangential directions mapped into the direction of \(H\). Their integral curves are called mean directionally curved lines. They become singular, if \(H=0\), or the curvature ellipse degenerates. The paper studies questions of periodicity, singularity and structural stability of such lines.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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