## Project Details

### Description

By a more resent theorem due to J. Weiner (1991), which also is proven curve-theoreticly by B. Solomon (1996) it follows that the 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.

In this project it is proven that, when restricted to closed curves on the unit 2-sphere with integrated geodesic curvature zero that contains no sub arc with integrated geodesic curvature Pi, then the inverse problem formulated by W. Fenchel always has solutions, if one allows zero and negative curvature of space curves. Furthermore it is explained why this is not true if non-vanishing curvature is required.

By giving examples of closed curves on the unit 2-sphere with integrated geodesic curvature 2*Pi*z, for any integer z, which are not a principal indicatrix of a closed space curve (even when allowing zero and negative curvature) the general answer to W. Fenchels question is found to be to the negative.

Status | Finished |
---|---|

Effective start/end date | 01/01/1997 → 31/01/1999 |