Catenaries in Riemannian surfaces

• DOI: 10.1007/s40863-023-00399-z
• Publication Date: 2024-01-26T17:02:13Z
• Authors: Luiz C. B. da Silva; Rafael López
• Keywords: Mathematics - Differential Geometry; Surface of revolution; name=Statistics; Catenary; /dk/atira/pure/subjectarea/asjc/1800/1804; 53A04, 53B20, 49J05; 500; 600; α-catenary; 01 natural sciences; name=Computational Theory and Mathematics; name=General Mathematics; Differential Geometry (math.DG); Clairaut relation; FOS: Mathematics; Probability and Uncertainty; /dk/atira/pure/subjectarea/asjc/2600/2600; 0101 mathematics; Grušin plane; /dk/atira/pure/subjectarea/asjc/1700/1703

AbstractThe concept of catenary has been recently extended to the sphere and the hyperbolic plane by the second author (López, arXiv:2208.13694). In this work, we define catenaries on any Riemannian surface. A catenary on a surface is a critical point of the potential functional, where we calculate the potential with the intrinsic distance to a fixed reference geodesic. Adopting semi-geodesic coordinates around the reference geodesic, we characterize catenaries using their curvature. Finally, after revisiting the space-form catenaries, we consider surfaces of revolution (where a Clairaut relation is established), ruled surfaces, and the Grušin plane.