Left-handed geodesic flow of spheres of revolution

TL;DR

This paper proves that geodesic flows on certain 2-spheres of revolution with specific curvature conditions are left-handed, and constructs examples showing the sharpness of the curvature bound related to this property.

Contribution

It establishes the left-handedness of geodesic flows on 2-spheres of revolution with 1/4-pinched curvature and provides counterexamples for lower pinching constants, confirming a conjecture.

Findings

Geodesic flow is left-handed on 2-spheres with 1/4-pinched curvature.

Counterexamples exist for spheres with curvature less than or equal to , not exhibiting left-handedness.

Supports the conjecture that 1/4 is the optimal pinching constant for left-handedness.

Abstract

A flow on a 3-manifold is left-handed if any two ergodic invariant measures have negative linking number. We prove that on a 2-sphere of revolution whose curvature is $1/4$-pinched the geodesic flow is left-handed. Conversely, for every $\delta\le 1/4$, we construct a 2-sphere whose curvature is $\delta$-pinched and whose geodesic flow is not left-handed. This gives credit to a conjecture of Florio and Hryniewicz that $1/4$ is the optimal pinching constant for left-handedness among arbitrary positively curved 2-spheres.