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On a Carathéodory’s conjecture on umbilics: representing ovaloids. (English) Zbl 0897.53003

Carathéodory’s conjecture states that every ovaloid in Euclidean 3-space has at least two umbilics. Apparently, this can now be considered to be proved in the analytic case, but the \(C^r\) case is still open. In this connection, the authors consider representations of ovaloids. For the so-called Bonnet chart they give the differential equation of the curvature lines. The support function, defined on \(S^2\), is arbitrarily extended to a neighborhood of \(S^2\). An explicit formula gives the points of the ovaloid in terms of such an extension.
Reviewer: E.Heil (Darmstadt)

MSC:

53A05 Surfaces in Euclidean and related spaces
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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References:

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