Abstract
In this article we investigate the total torsion of closed lines of curvature on a surface in 3 and obtain the following results.(1) The total torsion of a closed line of curvature on a surface is kπ where k is an integer. Conversely, if the total torsion of a closed curve is kπ for an integer k, then the curve can appear as a line of curvature on a surface. In particular, if the total torsion of a closed curve is 2kπ, then it can appear as a line of curvature on a closed, oriented surface of genus 1.(2) The total torsion of a closed line of curvature on an ovaloid is zero.
Publisher
Cambridge University Press (CUP)
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