Catenaries in Riemannian Surfaces

2023/06/06 by Luiz C. B. da Silva, da Silva, Luiz C. B., Rafael López +1
Mathematics · #49J05 #53A04 #53B20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2306.04013

openalex publication_date 2023/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The 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.

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