Principal normal indicatrices of closed space curves

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