Mannheim curves in 3-dimensional space forms. (English) Zbl 1284.53004

Choi, Jin Ho; Kang, Tae Ho; Kim, Young Ho

Bull. Korean Math. Soc. 50, No. 4, 1099-1108 (2013).

In Euclidean 3-space, Mannheim curves are characterised by the equation \[ \kappa=a(\kappa^2+\tau^2) \] for a constant \(a\neq 0\), where \(\kappa\) and \(\tau\) are the curvature and the torsion of the curve. Similarly, Mannheim partner curves are characterised by \[ \kappa'=\frac{\kappa}{a}(1+a^2\tau^2) \] where \(\kappa'\) represents the derivative of the curvature with respect to the arc length parameter.
In this paper the authors give a definition of Mannheim and Mannheim partner curves in Riemannian 3-manifolds and give characterisations for such curves in 3-dimensional space forms which generalise the characterisations for the Euclidean case.

Reviewer: Raúl Oset Sinha (São Carlos)

MSC:

53A04	Curves in Euclidean and related spaces
53A35	Non-Euclidean differential geometry

Keywords:

Mannheim curves; Mannheim partner curves; space form