## 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

#### 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