Thermostats without conjugate points

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.

💡 Research Summary

The paper studies smooth thermostats on closed oriented Riemannian surfaces, i.e. flows generated by the vector field F = X + λV where X is the geodesic vector field, V the vertical field on the unit tangent bundle SM, and λ∈C^∞(SM) a prescribed “thermostat function”. A thermostat geodesic γ satisfies ∇_{\dotγ}\dotγ = λ(γ,\dotγ)J\dotγ, preserving speed and defining a flow φ_t on SM.

The authors introduce a family of curvature-like quantities κ_p depending on a gauge function p∈C^1 along the flow: κ_p = π^*K_g − Hλ + λ² + F p + p(p − Vλ), where K_g is the Gaussian curvature, H=