Non-integrability of geodesic flow on certain algebraic surfaces

TL;DR

This paper rigorously proves that the geodesic flow on the cubic surface defined by xyz=1 is non-integrable, using advanced mathematical tools, and discusses implications and extensions of this result.

Contribution

It provides the first rigorous proof of non-integrability for the geodesic flow on this specific algebraic surface, addressing an open problem posed by V. Kozlov.

Findings

Geodesic flow on xyz=1 surface is non-integrable.

Application of Morales-Ramis theorem and Kovacic algorithm.

Implications for the dynamics on algebraic surfaces.

Abstract

This paper addresses an open problem recently posed by V. Kozlov: a rigorous proof of the non-integrability of the geodesic flow on the cubic surface $x y z=1$. We prove this is the case using the Morales-Ramis theorem and Kovacic algorithm. We also consider some consequences and extensions of this result.