A theorem due to J. Weiner, which also is proven by B. Solomon,
implies that a principal normal indicatrix of a closed space curve
with non-vanishing curvature has integrated geodesic curvature
zero and contains no sub arc 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 non-vanishing 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.Keywords: An inverse to Jacobi's
theorem, differential geometry of closed space curves, Frenet
ApparatusAMS-classification (1991): 53A04

Number of pages | 12 |
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Publication status | Published - 1997 |
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