The catenary in space forms

Rafael López

Published 2022-08-29Version 1

In this paper, the notion of the catenary curve in the sphere and in the hyperbolic plane is introduced. In both spaces, a catenary is defined as the shape of a hanging chain when its weight is measured with respect to a given geodesic of the space. Several characterizations of the catenary are established in terms of the curvature of the curve and of the angle that its unit normal makes with a vector field of the ambient space. Furthermore, in the hyperbolic plane, we extend the concept of catenary substituting the geodesics by a horocycle or the hyperbolic distance by the horocycle distance.