Principal normal indicatrices of closed space curves

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    Abstract

    A theorem due to J. Weiner, which is also proven by B. Solomon, implies that a principal normal indicatrix of a closed space curve with nonvanishing curvature has integrated geodesic curvature zero and contains no subarc with integrated geodesic curvature pi. We prove that the inverse problem always has solutions if one allows zero and negative curvature of space curves and explain why this not is true if nonvanishing curvature is required. This answers affirmatively an open question asked by W. Fenchel in 1950 under the above assumptions but in general this question is found to be answered to the negative.
    Original languageEnglish
    JournalGeometriae Dedicata
    Volume74
    Issue number1
    Pages (from-to)49-60
    ISSN0046-5755
    DOIs
    Publication statusPublished - 1999

    Keywords

    • An inverse to Jacobi's theorem
    • Differential geometry of closed space curves
    • Frenet apparatus

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