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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
##### Keywords:
second fundamental form; curvature ellipses; Gauss maps
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