Vol. 12, No. 2, 2019

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Curves of constant curvature and torsion in the 3-sphere

Debraj Chakrabarti, Rahul Sahay and Jared Williams

Vol. 12 (2019), No. 2, 235–255
DOI: 10.2140/involve.2019.12.235

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.

Frenet–Serret equations, constant curvature and torsion, geodesic curvature, helix, 3-sphere, curves in the 3-sphere
Mathematical Subject Classification 2010
Primary: 53A35
Received: 23 June 2017
Revised: 13 October 2017
Accepted: 22 April 2018
Published: 8 October 2018

Communicated by Colin Adams
Debraj Chakrabarti
Department of Mathematics
Central Michigan University
Mt. Pleasant, MI
United States
Rahul Sahay
University of California
Berkeley, CA
United States
Jared Williams
Department of Physics and Astronomy
University of Missouri
Columbia, MO
United States