Anosov Flow On (tangent Bundles Of) Riemann Surfaces
As an example, this section develops the case of the Anosov flow on the tangent bundle of a Riemann surface of negative curvature. This flow can be understood in terms of the flow on the tangent bundle of the Poincare half-plane model of hyperbolic geometry. Riemann surfaces of negative curvature may be defined as Fuchsian models, that is, as the quotients of the upper half-plane and a Fuchsian group. For the following, let H be the upper half-plane; let Γ be a Fuchsian group; let M=HΓ be a Riemann surface of negative curvature, and let T1M be the tangent bundle of unit-length vectors on the manifold M, and let T1H be the tangent bundle of unit-length vectors on H. Note that a bundle of unit-length vectors on a surface is the principal bundle of a complex line bundle.
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