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Surfaces with inflection points in Euclidean 4-space. (English) Zbl 1296.53023
From the authors’ introduction: The curvature ellipse is of interest in the study of a surface \(M\) in the Euclidean 4-space \(\mathbb R^4\). At a point \(p\in M\), the curvature ellipse \({\mathcal E}_p\) is defined by the image \[ \{\Pi(v,v)\in T^\perp_p M\mid v\in T_p M,|v|= 1\}, \] in the normal space \(T^\perp_p M\), of the unit circle in the tangent plane \(T_pM\) under the second fundamental form \(\Pi\). If it degenerates to a segment contained in a straight line passing through \(\mathbf{0}_p\) of \(T^\perp_pM\), we say that \(p\) is an inflection point.
In this paper, we present the following reduction theorem.
Theorem 1. Let \(X\) be a conformal immersion from a connected Riemann surface \(S\) into \(\mathbb R^4\). Assume that the Gauss curvature \(K\) does not vanish anywhere. If all points of \(S\) are inflection points, then the surface \(X(S)\) lies in an affine 3-space in \(\mathbb R^4\).
In order to prove this theorem, we introduce a new complex-valued local invariant \(\Lambda\) in Section 2. For the resultant \(\Delta_p\) of \(X\) at \(p\in S\) and the normal curvature \(K_N(p)\), \(\Lambda (p)\) satisfies \[ 4\Delta_p = (K_N(p))^2-4| \Lambda (p)|^2. \]

MSC:
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53A05 Surfaces in Euclidean and related spaces
53B25 Local submanifolds
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