**A. Pereira do Vale & M. R. Pinto (ed.),
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Proceedings of the 1st International Meeting on Geometry and Topology.

Proceedings of the conference held in Braga, September 11-13, 1997.

Universidade do Minho, Portugal

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Variation of the Total Normal Twist of Closed Curves in Euclidean Spaces

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Tarek F. Mersal and Bernd Wegner

Fachbereich Mathematik, TU Berlin, Strasse des 17. Juni 135, D - 10623 Berlin, Germany wegner@math.tu-berlin.de

**Abstract:** The existence of parallel mates to a given closed space curves in the sense of H.R. Farran and S.A. Robertson [J. London Math. Soc. (2) ** 35** (1987), 527-538] depends on its total normal twist, i.e. the angle of rotation of a normal vector subject to parallel transfer in the normal bundle after one period. Parallel mates exist iff this quantity is an integer multiple of $2\pi$. The aim of this note is to investigate the change of the total normal twist under isotopic deformations of closed curves. This enables us to characterize isotopies of space curves where the total normal twist remains invariant. Main result is a description of its change in terms of the oriented surface area covered by the unit tangents to all curves of the deformation family.

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