Hexagonal geodesic 3-webs

TL;DR

This paper characterizes surfaces with hexagonal geodesic 3-webs through cubic integrals of geodesic flow, demonstrating integrability of the governing PDEs and providing new examples and analogues of classical theorems.

Contribution

It establishes a precise criterion linking hexagonal geodesic webs to cubic first integrals and shows the integrability of the associated PDE system, including new metric examples.

Findings

Surfaces with hexagonal geodesic 3-webs admit cubic first integrals.

The PDE system for such metrics is integrable via the generalized hodograph transform.

New local examples of metrics with these properties are constructed.

Abstract

We prove that a surface carries a hexagonal 3-web of geodesics if and only if the geodesic flow on the surface admits a cubic first integral and show that the system of partial differential equations, governing metrics on such surfaces, is integrable by generalized hodograph transform method. We present some new local examples of such metrics, discuss known ones, and establish an analogue of the celebrated Graf and Sauer Theorem for Darboux superintegrable metrics.