Abstract
It is first observed that on a 3-dimensional Sasakian manifold the torsion of a Legendre curve is identically equal to +1. It is then shown that, conversely, if a curve on a Sasakian 3-manifold has constant torsion +1 and satisfies the initial conditions at one point for a Legendre curve, it is a Legendre curve. Furthermore, among contact metric structures, this property is characteristic of Sasakian metrics. For the standard contact structure onR 3 with its standard Sasakian metric the curvature of a Legendre curve is shown to be twice the curvature of its projection to thexy-plane with respect to the Euclidean metric. Thus this metric onR 3 is more natural for the study of Legendre curves than the Euclidean metric.
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This work was done while the first author was a visiting scholar at Michigan State University.
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Baikoussis, C., Blair, D.E. On Legendre curves in contact 3-manifolds. Geom Dedicata 49, 135–142 (1994). https://doi.org/10.1007/BF01610616
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DOI: https://doi.org/10.1007/BF01610616