Thermostats without conjugate points

TL;DR

This paper extends Hopf's theorem to thermostats, establishing conditions under which thermostat curvature is non-positive and exploring the properties of Green bundles and projectively Anosov thermostats.

Contribution

It generalizes classical results to thermostats without conjugate points, introduces the first example of a non-Anosov projectively Anosov thermostat, and shows limitations of Hopf's rigidity.

Findings

Total thermostat curvature is non-positive and zero only if curvature is identically zero.

Green bundles are transverse everywhere if and only if the thermostat is projectively Anosov.

Hopf's rigidity theorem does not extend to thermostats on the 2-torus.

Abstract

We generalize Hopf's theorem to thermostats: the total thermostat curvature of a thermostat without conjugate points is non-positive and vanishes only if the thermostat curvature is identically zero. We further show that, if the thermostat curvature is zero, then the flow has no conjugate points and the Green bundles collapse almost everywhere. Given a thermostat without conjugate points, we prove that the Green bundles are transverse everywhere if and only if it is projectively Anosov. Finally, we provide an example showing that Hopf's rigidity theorem on the 2-torus cannot be extended to thermostats. It is also the first example of a projectively Anosov thermostat which is not Anosov.