Involve: A Journal of Mathematics

  • Involve
  • Volume 12, Number 2 (2019), 235-255.

Curves of constant curvature and torsion in the 3-sphere

Debraj Chakrabarti, Rahul Sahay, and Jared Williams

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Abstract

We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded in the 3-sphere. This behavior is very different from that of helices in three-dimensional Euclidean space, which also have constant curvature and torsion.

Article information

Source
Involve, Volume 12, Number 2 (2019), 235-255.

Dates
Received: 23 June 2017
Revised: 13 October 2017
Accepted: 22 April 2018
First available in Project Euclid: 25 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.involve/1540432914

Digital Object Identifier
doi:10.2140/involve.2019.12.235

Mathematical Reviews number (MathSciNet)
MR3864216

Zentralblatt MATH identifier
06980500

Subjects
Primary: 53A35: Non-Euclidean differential geometry

Keywords
Frenet–Serret equations constant curvature and torsion geodesic curvature helix 3-sphere curves in the 3-sphere

Citation

Chakrabarti, Debraj; Sahay, Rahul; Williams, Jared. Curves of constant curvature and torsion in the 3-sphere. Involve 12 (2019), no. 2, 235--255. doi:10.2140/involve.2019.12.235. https://projecteuclid.org/euclid.involve/1540432914


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