Integrable geodesic flows on tubular sub-manifolds

TL;DR

This paper introduces a new class of 3D tubular surfaces with integrable geodesic flows by generalizing classical notions and deriving conditions for integrability using Jacobi fields, with diverse examples provided.

Contribution

It generalizes the concept of tubular surfaces to 3D manifolds and establishes conditions for integrability of geodesic flows using Jacobi fields.

Findings

Derived conditions for integrable geodesic flows on generalized tubular surfaces.

Constructed explicit examples of such surfaces with varied geometries.

Showed that these surfaces admit ignorable coordinates, simplifying geodesic analysis.

Abstract

In this paper we construct a new class of surfaces whose geodesic flow is integrable (in the sense of Liouville). We do so by generalizing the notion of tubes about curves to 3-dimensional manifolds, and using Jacobi fields we derive conditions under which the metric of the generalized tubular sub-manifold admits an ignorable coordinate. Some examples are given, demonstrating that these special surfaces can be quite elaborate and varied.