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

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.

### MSC:

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

### Keywords:

Mannheim curves; Mannheim partner curves; space form
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